supplemental material for identification-robust subvector

Supplemental Material for IDENTIFICATION-ROBUST SUBVECTOR INFERENCE

By

Donald W. K. Andrews

September 2017Updated September 2017

COWLES FOUNDATION DISCUSSION PAPER NO. 2105S

COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY

Box 208281 New Haven, Connecticut 06520-8281

http://cowles.yale.edu/

http://cowles.yale.edu/

Contents

10 Outline 2

11 Time Series Observations 3

12 Veri�cation of Assumptions on the Estimator Set b�1n 6

13 Veri�cation of Assumptions for the First-Step AR CS 15

14 Veri�cation of Assumptions on the Second-Step

Data-Dependent Signi�cance Level 18

15 Veri�cation of Assumptions for the Second-Step C(�)-AR Test 24

16 Veri�cation of Assumptions for the Second-Step C(�)-LM Test 30

17 Veri�cation of Assumptions for the Second-Step C(�)-QLR1 Test 41

18 Amalgamation of High-Level Conditions 49

19 Proof of Theorem 8.1 56



22 Additional Simulation Results 83

23 Additional Second-Step C(�) Tests 88

1

10 Outline

References to sections with section numbers less than 10 refer to sections of the main pa-

per �Identi�cation-Robust Subvector Inference.�Similarly, all theorems and lemmas with section

numbers less than 10 refer to results in the main paper.

Section 11 generalizes the asymptotic results in Section ?? for the moment condition model

from i.i.d. observations to strictly stationary strong mixing time series observations.

Sections 12-14 of this Supplemental Material (SM) provide high-level su¢ cient conditions for

the parts of Assumptions B, C, and OE (stated in Section ??) that concern (i) the estimator set,

(ii) the �rst-step AR CS, and (iii) the data-dependent second-step signi�cance level, respectively,

in the moment condition model. Sections 15-17 provide high-level su¢ cient conditions for the parts

of Assumptions B, C, and OE that concern the second-step C(�)-AR, C(�)-LM, and C(�)-QLR1

tests, respectively, in the moment condition model. Section 18 amalgamates the conditions in

Sections 12-17 for the two-step AR/AR, AR/LM, and AR/QLR1 subvector tests. Sections 19 and

20 prove Theorems ?? and ?? by using primitive conditions to verify the su¢ cient conditions in

Section 18.

Section 21 proves the times series results in Theorem 11.1.

Section 22 provides some additional simulation results to those presented in the main paper.

For illustrative purposes, Section 23 de�nes C(�) versions of the CQLR tests in Andrews and

Guggenberger (2015) and I. Andrews and Mikusheva (2016), but does not verify the high-level

conditions in Section ?? for these tests.

In this SM, a null sequence S is de�ned as in (??), i.e., S := f(��n; Fn) : (�1�n; Fn) 2 FSV ;�2�n = �20; n � 1g; except in some sections where S is de�ned with a speci�c parameter space, suchas FAR=AR; in place of the generic parameter space FSV : For an assumption that is stated for asequence S; we say that it holds for a subsequence Sm if the subsequence version of the assumption

holds.

Throughout the SM, we use the following notational convention when considering tests of H0 :

�2 = �20: For any function A(�) of � = (�01; �02)0; we de�ne

A(�1) := A(�1; �20) and A := A(�1�n; �20); (10.1)

where �20 is the null value of �2 and �1�n is the true value of �1:

We let B(�1; ") denote a closed ball in Rp1 centered at �1 with radius " > 0:

We let � jsn for s = 1; :::; pj denote the singular values of �1=2n Gjn written in nonincreasing

order, for j = 1; 2; where Gjn 2 Rk�pj and n 2 Rk�k are (nonrandom) population matrices

2

that correspond to the sample Jacobian (wrt �j) bGjn 2 Rk�pj and sample variance matrix bn;respectively. Let Gn := [G1n : G2n] 2 Rk�p: Because they arise frequently below, for notationalsimplicity, we let

� jn := � jpjn = smallest singular value of �1=2n Gjn for j = 1; 2 and

�n := smallest singular value of �1=2n Gn: (10.2)

The quantity � jn is a measure of the (local) strength of identi�cation of �j at � = (�01�n; �020)

0 and

�n is a measure of the (local) strength of identi�cation of � at � = (�01�n; �020)

0:

Similarly, we let ��jsn for s = 1; :::; pj denote the singular values of �1=2n Gjn�jn written in

nonincreasing order, for j = 1; 2; where �jn 2 Rpj�pj is the (nonrandom) population matrix thatcorresponds to b�jn 2 Rpj�pj de�ned in (??). For notational simplicity, we let

��jn := ��jpjn = smallest singular value of �1=2n Gjn�jn for j = 1; 2 and

��n := smallest singular value of �1=2n GnDiagf�1;�2g: (10.3)

The quantity ��jn is another (equivalent) measure of the (local) strength of identi�cation of �j at

� = (�01�n; �020)

0:

In Sections 12 and 13 in this SM, the results are designed to hold not just for the moment

condition model, but also for minimum distance models and moment condition models where the

moments may depend on n1=2-consistent and asymptotically normal preliminary estimators. But,

the de�nitions of Gjn(�) for j = 1; 2 and n(�) di¤er across these models. In consequence, for

generality, Gjn(�) and n(�) are de�ned in these sections by the conditions they must satisfy in the

various results given, rather than by explicit expressions. For sample moments bgn(�) (without anypreliminary estimators), this leads to Gjn(�) = EFn bGjn(�) and n(�) = V arFn(n

1=2bgn(�)): Thelatter de�nitions are employed in Section 15 and the sections that follow it, which consider only

the moment condition model.

Many results in this SM are stated to hold for both a sequence S and a subsequence Sm: For

brevity, we only prove these results for a sequence S: For a subsequence Sm; the proofs only require

the minor notational adjustment of changing n to mn:

11 Time Series Observations

In this section, we generalize the results of Theorems ?? and ?? from i.i.d. observations to

strictly stationary strong mixing observations. In the time series case, F denotes the distribution

3

of the stationary in�nite sequence fWi : i = :::; 0; 1; :::g: Asymptotics under drifting sequences oftrue distributions fFn : n � 1g are used to establish the correct asymptotic size of the two-stepAR/AR, AR/LM, and AR/QLR1 tests. Under such sequences, the observations form a triangular

array of row-wise strictly stationary observations. In the time series case, we de�ne F (�) di¤erently

from its de�nitions in (??) for the i.i.d. case:

F (�) :=

1Xm=�1

�EF gi(�)g

0i�m(�)� EF gi(�)EF gi(�)0

�: (11.1)

Note that F (�) = limV arF (n�1=2Pni=1 gi(�)): We let f�F (m) : m � 1g denote the strong mixing

numbers of the observations under the distribution F:

The time series analogue FTS;AR=AR of the space of distributions FAR=AR; de�ned in (??), is

FTS;AR=AR := f(�1; F ) : EF gi(�1) = 0k; �1 2 �1�; fWi : i = :::; 0; 1; :::g are stationary

and strong mixing under F with �F (m) � Cm�d for some d > (2 + )= ;

EF jjgi(�1)jj2+ �M; EF jjvec(G1i(�1))jj2+ �M; EF �2+ 1i �M;

�min(F (�1)) � �; V arF (jjG1si(�1)jj) � � 8s = 1; :::; p1g (11.2)

for some ; � > 0 and M < 1; where F (�) is de�ned in (11.1). For the two-step AR/LM and

AR/QLR1 tests with time series observations, we use the parameter space FTS;AR=LM;QLR1; whichis de�ned as in (??), but with FTS;AR=AR in place of FAR=AR:

For CS�s, we use the parameter spaces F�;TS;AR=AR and F�;TS;AR=LM;QLR1; which are de-�ned as in (??) and (??), respectively, but with FTS;AR=AR(�2) in place of FAR=AR(�2); whereFTS;AR=AR(�20) denotes FTS;AR=AR with its dependence on �20 made explicit.

Next, we de�ne the second-step C(�)-AR, C(�)-LM, and C(�)-QLR1 tests in the time series

context. To do so, we let

VF (�) := limV arF

0@n�1=2 nXi=1

0@ gi(�)

vec(Gi(�))

1A1A=

1Xm=�1

EF

0@ gi(�)� EF gi(�)vec(Gi(�)� EFGi(�))

1A0@ gi�m(�)� EF gi�m(�)vec(Gi�m(�)� EFGi�m(�))

1A0 : (11.3)The second equality holds for all (�; F ) 2 F�;TS;AR=AR:

The test statistics depend on an estimator bVn(�) of VF (�): This estimator (usually) is a het-eroskedasticity and autocorrelation consistent (HAC) variance estimator based on the observations

4

ffi(�) � bfn(�) : i � ng; where fi(�) := (gi(�)0; vec(Gi(�))0)0 and bfn(�) := (bgn(�)0; vec( bGn(�))0)0:

There are a number of HAC estimators available in the literature, e.g., see Newey and West (1987)

and Andrews (1991b). The asymptotic properties of the tests are the same for any consistent HAC

estimator. Hence, for generality, we do not specify a particular estimator bVn(�): Rather, we stateresults that hold for any estimator bVn(�) that satis�es the following consistency conditions. TheAssumptions V and V-CS that follow are applied with two-step tests and CS�s, respectively.

Assumption V. 8K <1; sup�12B(�1�n;K=n1=2) jjbVn(�1)� VFn(�1)jj !p 0 under any null sequence

f(�1�n; Fn) 2 FTS;AR=AR : n � 1g for which VFn(�1�n)! V1 for some pd matrix V1:

Assumption V-CS. 8K <1; sup�12B(�1�n;K=n1=2) jjbVn(�1; ��2n)� VFn(�1; ��2n)jj !p 0 under any

sequence f(��n; Fn) 2 F�;TS;AR=AR : n � 1g for which VFn(��n)! V1 for some pd matrix V1:

We write the (p+ 1)k � (p+ 1)k matrix bVn(�) in terms of its k � k submatrices:

bVn(�) =26666664

bn(�) b�11n(�)0 � � � b�2p2n(�)0b�11n(�) bVG11n(�) � � � bV 0Gp1n(�)...

.... . .

...b�2p2n(�) bVGp1n(�) � � � bVGppn(�)

37777775 ; (11.4)

where the subscripts on b�jsn(�) run from (j; s) = (1; 1); :::; (1; p1); (2; 1); :::; (2; p2):

In the time series case, for the two-step C(�)-AR, C(�)-LM, and C(�)-QLR1 tests, we use the

same de�nitions as in Section ?? for the moment condition model and Section ??, but with bn(�)and b�jsn(�) for j = 1; :::; pj ; j = 1; 2 de�ned as in Assumption V and (11.4), rather than as in

(??) and (??). The two-step C(�)-AR, C(�)-LM, and C(�)-QLR1 CS�s in the time series case are

de�ned using (??), the de�nitions just given for the corresponding tests, and Assumption V-CS in

place of Assumption V.

In the time series case, we employ the following assumption in addition to Assumption SI.

Assumption SI-TS. (i) For the null sequence S; the strong mixing numbers satisfyP1m=1 �

1=q�1=rFn

(m) < 1 for some q = maxfp1 + �1; 2g and r = q + �1 for some �1 > 0; where

r is as in Assumption SI, and

(ii) sup�12�1 jjbn(�1) � Fn(�1)jj = op(1) for bn(�) and F (�) de�ned in (11.4) and (11.1),respectively.

For the time series case, the asymptotic results are as follows.

Theorem 11.1 Suppose the two-step AR/AR, AR/LM, and AR/QLR1 tests and CS�s are de�ned

as in this section and Assumption V or V-CS holds. Then, the results of Theorems ?? and ?? hold

5

with the parameter spaces FTS;AR=AR; FTS;AR=LM;QLR1; F�;TS;AR=AR; and F�;TS;AR=LM;QLR1 inplace of FAR=AR; FAR=LM;QLR1; F�;AR=AR; and F�;AR=LM;QLR1; respectively, and with AssumptionSI augmented by Assumption SI-TS (everywhere Assumption SI appears in Theorems ?? and ??).

Comment: Theorem 11.1 shows that the results of Theorems ?? and ?? for i.i.d. observations

generalize to strictly stationary strong mixing observations, provided the spaces of distributions are

adjusted suitably and the variance estimator bVn(�) of VF (�) is de�ned appropriately.12 Veri�cation of Assumptions on the Estimator Set b�1n

12.1 Estimator Set Results

The estimator set (ES), b�1n; is de�ned in (??). The following lemma veri�es Assumption

C(i) under high-level assumptions on b�1n including the assumption that there exist n1=2-consistentsolutions f�1n : n � 1g to the FOC�s given in (??) (which implies that �1 is locally stronglyidenti�ed given �20): Lemma 12.2 below provides su¢ cient conditions for the existence of such

solutions f�1ng:

Assumption ES1. For the null sequence S; there exist solutions f�1n 2 �1 : n � 1g to the FOC�sgiven in (??) that satisfy n1=2(�1n � �1�n) = Op(1):

Assumption ES2. For the null sequence S; (i) bgn(�1) is di¤erentiable on B(�1�n; ") for some " > 0(for all sample realizations) 8n � 1; (ii) bgn = Op(n�1=2); (iii) sup�12B(�1�n;") jj bG1n(�1)jj = Op(1) forsome " > 0; (iv) ncn ! 1 for fcn : n � 1g in (??), and (v) cW1n is a symmetric psd k � k matrixthat satis�es cW1n = Op(1):

Lemma 12.1 Suppose b�1n is de�ned in (??) and bQn(�) is the criterion function de�ned in (??).Let S be a null sequence (or Sm a null subsequence) that satis�es Assumptions ES1 and ES2. Then,b�1n is non-empty wp!1 and Assumption C(i) holds for the sequence S (or subsequence Sm).

The following lemma provides su¢ cient conditions for Assumption ES1 for sequences S that

satisfy lim �1n > 0 (i.e., for �1-locally-strongly-identi�ed sequences). Let �1 = (�11; :::; �1p1)0:

Assumption FOC. For the null sequence S and some " > 0; (i) lim infn!1 �1n > 0; (ii)

sup�12B(�1�n;") jjbgn(�1)� gn(�1)jj = op(1) for some nonrandom Rk-valued functions fgn(�) : n � 1g;(iii) gn = 0k 8n � 1; (iv) �1�n ! �1�1 for some �1�1 2 �1; (v) bgn(�1) is twice continuously di¤er-entiable on B(�1�n; ") (for all sample realizations) 8n � 1; (vi) bgn = Op(n�1=2); (vii) gn(�1) is twicecontinuously di¤erentiable on B(�1�n; ") 8n � 1; (viii) sup�12B(�1�n;") jj bG1n(�1)�G1n(�1)jj = op(1)for some nonrandom Rk�p1-valued functions fG1n(�) : n � 1g; (ix) sup�12B(�1�n;") jjG1n(�1)jj =

6

O(1); (x) sup�12B(�1�n;"n) jjG1n(�1)�G1njj = o(1) for all sequences of positive constants "n ! 0; (xi)

G1n ! G11 for some matrix G11 2 Rk�p1 ; (xii) G1n(�1) = (@=@�01)gn(�1) 8�1 2 B(�1�n; ");8n �1; (xiii) lim infn!1 �min(n) > 0; (xiv) b�1n = Op(1); where b�1n := maxs;u�p1 sup�12B(�1�1;")

jj(@2=@�1s@�1u)bgn(�1)jj; (xv) �1n = O(1); where �1n := maxs;u�p1 sup�12B(�1�1;")

jj(@2=@�1s@�1u)gn(�1)jj; and (xvi) cW1n is symmetric and psd and cW1n !p W11 for some non-

random nonsingular matrix W11 2 Rk�k:

In the moment condition model (where bgn(�1) = n�1Pni=1 g(Wi; �1)); we have gn(�1) =

EFnbgn(�1) in Assumption FOC(ii), G1n(�1) = EFn bG1n(�1) in Assumption FOC(viii), and As-sumptions FOC(vii), (xii), and (xv) (with " replaced by "=2 in Assumptions FOC(xii) and (xv),

which does not matter in Lemma 12.2 below) are implied by Assumptions FOC(v) and (xiv) and

EFnb�1n = O(1) (for b�1n de�ned in Assumption FOC(xiv)).1 In this case, by Assumption FOC(xii),

G1n(�1) = EFn bG1n(�1) = (@=@�01)EFnbgn(�1):Lemma 12.2 Let S be a null sequence (or Sm a null subsequence) that satis�es Assumption FOC.

Then, Assumption ES1 holds for the sequence S (or subsequence Sm).

Comments: (i). The result of Lemma 12.2 is similar to classical results in the statistical literature,

see Cramér (1946), Aitchison and Silvey (1959), and Crowder (1976). The proof of Lemma 12.2

follows that of van der Vaart (1998, Thm. 5.42, p. 69).

(ii) The estimator bn plays no role in Lemma 12.2. Nevertheless, Assumption FOC(xiii)

involves n because Assumptions FOC(i) and (xiii) imply that the limit of the smallest singular

value of G1n is positive (since �1n is the smallest singular value of �1=2n G1n):

The next lemma provides su¢ cient conditions for Assumption OE(i).

Lemma 12.3 Let S be a null sequence (or Sm a null subsequence) for which (i) dH(�1�n; b�1n) =Op(n

�1=2) and (ii) dH(�1�n; CS1n [ f�1�ng) = Op(n�1=2): Then, dH(�1�n; CS+1n) = Op(n�1=2) holdsfor the sequence S (or subsequence Sm).

Comments: 1. Lemmas 12.4 and 13.2 below provide primitive su¢ cient conditions for conditions

(i) and (ii), respectively, of Lemma 12.3.

2. Condition (i) of Lemma 12.3 requires that b�1n 6= ? wp!1 by the de�nition of dH :1To see this, suppose Assumptions FOC(v) and (xiv) hold and EFnb�1n = O(1): Taking expections under Fn in

the �rst line of (12.6) below gives Assumption FOC(vii). By a similar expansion to that in (12.6), but about a pointin B(�1�n; "=2); rather than �1�n; gives (@=@�01)gn(�1) = EFn bG1n(�1) = G1n(�1) 8�1 2 B(�1�n; "=2); which impliesAssumption FOC(xii), and (@2=@�1s@�1`)gn(�1) = EFn(@

2=@�1s@�1`)bgn(�1) 8�1 2 B(�1�n; "=2); which implies that�1n � EFnb�1n = O(1) and Assumption FOC(xv) holds.

7

Now we verify dH(�1�n; b�1n) = Op(n�1=2) for b�1n de�ned in (??) for sequences S with lim

infn!1 �1n > 0; i.e., for �1-locally strongly-identi�ed sequences.

Assumption ES3. For the null sequence S; (i) lim infn!1 �1n > 0; (ii) b�1n is non-empty wp!1,(iii) bgn(�1) is di¤erentiable on B(�1�n; ") for some " > 0 (for all sample realizations) 8n � 1;

(iv) bgn = Op(n�1=2); (v) sup�12B(�1�n;") jj bG1n(�1) � G1n(�1)jj = op(1) for some " > 0 for some

nonrandom Rk�p1-valued functions fG1n(�) : n � 1g; (vi) sup�12B(�1�n;"n) jjG1n(�1) � G1njj = o(1)for all sequences of positive constants "n ! 0; (vii) G1n ! G11 for some matrix G11 2 Rk�p1 ; (viii)bn � n !p 0 for some nonrandom matrices fn 2 Rk�k : n � 1g; (ix) lim infn!1 �min(n) > 0;(x) cn ! 0 for fcn : n � 1g in (??), and (xi) cW1n is symmetric and psd and cW1n !p W11 for some

nonrandom nonsingular matrix W11 2 Rk�k:

Note that Assumption ES3(ii) is implied by Assumptions ES1 and ES2 by Lemma 12.1 and,

hence, by Assumptions FOC and ES2 by Lemmas 12.1 and 12.2.

Assumption ES4. For the null sequence S; (i) sup�12�1 n1=2jjbgn(�1)� gn(�1)jj = Op(1) for some

nonrandom Rk-valued functions fgn(�) : n � 1g and (ii) lim infn!1 inf�1 =2B(�1�n;") jjgn(�1)jj > 0

8" > 0:

Lemma 12.4 Suppose b�1n is of the form in (??) with bQn(�) as in (??). Let S be a null sequence (orSm a null subsequence) that satis�es Assumptions ES3 and ES4. Then, dH(�1�n; b�1n) = Op(n�1=2)for the sequence S (or subsequence Sm).

12.2 Proofs of Lemmas 12.1-12.4

Proof of Lemma 12.1. Let �1n be as in Assumption ES1. If �1n 2 b�1n wp!1, thend(�1�n; CS

+1n) � d(�1�n; b�1n) � d(�1�n; f�1ng) = jj�1�n � �1njj = Op(n�1=2); (12.1)

where the �rst two inequalities hold because f�1ng � b�1n � CS+1n wp!1 using the de�nition ofCS+1n in (??) and the last inequality holds by Assumption ES1. Hence, Assumption C(i) holds andb�1n is non-empty wp!1.

It remains to show �1n 2 b�1n wp!1. By Assumption ES1, �1n satis�es the �rst condition inthe de�nition of b�1n in (??). Hence, it remains to show that the second condition in the de�nitionof b�1n in (??) holds for �1n: That is, we need to show

bQn(�1n) � inf�12�1

bQn(�1) + cn wp! 1. (12.2)

8

Element-by-element mean-value expansions of bgn(�1n) about �1�n givebgn(�1n) = bgn + bG1n(e�1n)(�1n � �1�n) = Op(n�1=2) +Op(1)Op(n�1=2) = Op(n�1=2); (12.3)

where, as de�ned above, bgn := bgn(�1�n); e�1n lies between �1n and �1�n and may di¤er across therows of bG1n(e�1n); the �rst equality uses Assumption ES2(i), and the second equality holds byAssumptions ES1, ES2(ii), and ES2(iii). Equations (??) and (12.3) and Assumption ES2(v) yieldbQn(�1n) = Op(n�1): Hence, we have

bQn(�1n)� cn = Op(n�1)� cn < 0 � inf�12�1

bQn(�1); (12.4)

where the strict inequality holds wp!1 because ncn ! 1 by Assumption ES2(iv). Hence, (12.2)

holds. �

Proof of Lemma 12.2. First, we establish the existence of consistent (as opposed to n1=2-

consistent) solutions to the FOC�s. Let

bn(�1) := bG1n(�1)0cW1nbgn(�1) and n(�1) := G1n(�1)0W11gn(�1): (12.5)

We use essentially the same argument as in van der Vaart (1998, Thm. 5.42, p. 69), but withbn(�1) and n(�1) in place of van der Vaart�s n(�) and (�); respectively. The main di¤erencesare that the population quantity n(�1) here depends on n; whereas van der Vaart�s population

quantity (�) does not; bn(�1) is a product of three random matrices none of which needs to be a

sample average, whereas van der Vaart�s n(�) is a sample average; and n(�1) here is the product

of three population matrices, whereas van der Vaart�s (�) is a single population matrix.

For �1 2 B(�1�n; ") (with " > 0 as in Assumption FOC), element-by-element two-term Taylor

expansions of bgn(�1) about �1�n givebgn(�1) = bgn + bG1n � (�1 � �1�n) + 1

2

p1Xs=1

(�1s � �1�ns)@

@�1sbG1n(e�1n)(�1 � �1�n)

= op(1) +G1n � (�1 � �1�n) +Op(1)jj�1 � �1�njj2; (12.6)

where e�1n lies between �1n and �1�n and may di¤er across rows of (@=@�1s) bG1n(e�1n); �1 = (�11; :::;�1p1)

0; the op(1) and Op(1) terms (in (12.6) and below) hold uniformly over �1 2 B(�1�n; ") as

n ! 1; �1�n = (�1�n1; :::; �1�np1)0; the �rst equality uses Assumption FOC(v), and the second

equality uses Assumptions FOC(vi), (viii), and (xiv).

Similarly, for �1 2 B(�1�n; "); element-by-element two-term Taylor expansions of gn(�1) about

9

�1�n give

gn(�1) =@

@�01gn � (�1 � �1�n) +

1

2

p1Xs=1

(�1s � �1�ns)@2

@�1s@�01

gn(e�1n)(�1 � �1�n)= G1n � (�1 � �1�n) +O(1)jj�1 � �1�njj2; (12.7)

where the O(1) term (in (12.7) and below) holds uniformly over �1 2 B(�1�n; ") as n ! 1; the�rst equality uses Assumption FOC(vii) and gn = 0k (by Assumption FOC(iii)), and the second

equality holds by Assumptions FOC(xii) and (xv).

For �1 2 B(�1�n; "); element-by-element mean-value expansions of bG1n(�1) about �1�n givebG1n(�1) = bG1n + p1X

s=1

@

@�1sbG1n(�y1n)(�1s � �1�ns) = G1n + op(1) +Op(1)jj�1 � �1�njj; (12.8)

where �y1n lies between �1 and �1�n and may di¤er across rows of (@=@�1s) bG1n(�y1n); the mean-value expansions use Assumption FOC(v), and the second equality uses Assumptions FOC(viii)

and (xiv).

For �1 2 B(�1�n; "); element-by-element mean-value expansions of G1n(�1) about �1�n give

G1n(�1) = G1n +

p1Xs=1

@

@�1sG1n(�

41n)(�1s � �1�ns) = G1n +O(1)jj�1 � �1�njj; (12.9)

where �41n lies between �1n and �1�n and may di¤er across rows of (@=@�1s)G1n(�41n); the �rst

equality uses Assumptions FOC(vii) and (xii) and the second equality holds using Assumption

FOC(xv) (since (@=@�1s)G1n(�1) = (@2=@�1s@�01)gn(�1) by Assumption FOC(xii)).

Combining (12.5), (12.6), and (12.8) gives: For �1 2 B(�1�n; ");

bn(�1) = (G1n + op(1) +Op(1)jj�1 � �1�njj)0(W11 + op(1))

�(op(1) +G1n � (�1 � �1�n) +Op(1)jj�1 � �1�njj2) (12.10)

= G011W11G11 � (�1 � �1�n) + op(1) + op(1)jj�1 � �1�njj+Op(1)jj�1 � �1�njj2;

where the �rst equality uses Assumption FOC(xvi) and the second equality uses Assumption

FOC(xi).

10

Combining (12.5), (12.7), and (12.9) gives: for �1 2 B(�1�n; ");

n(�1) = (G1n +O(1)jj�1 � �1�njj)0W11(G1n � (�1 � �1�n) +O(1)jj�1 � �1�njj2)

= G01nW11G1n � (�1 � �1�n) +O(1)jj�1 � �1�njj2

= G011W11G11 � (�1 � �1�n) + o(1)jj�1 � �1�njj+O(1)jj�1 � �1�njj2; (12.11)

where the third equality uses Assumption FOC(xi).

Di¤erentiability of gn(�1) and Gn(�1) on B(�1�n; ") holds by Assumptions FOC(vii) and (xii).

This implies di¤erentiability of n(�1) on B(�1�n; "): The derivative matrix of n(�1) is

@

@�01n(�1) = G1n(�1)

0W11G1n(�1) (12.12)

+

��@

@�11G1n(�1)

�0W11g1n(�1); :::;

�@

@�1p1G1n(�1)

�0W11g1n(�1)

�:

This matrix is uniformly continuous on B(�1�n; ") by Assumption FOC(vii).

Now we show that for some "1 2 (0; "];

lim infn!1

inf�12B(�1�n;"1)

�min

�@

@�01n(�1)

�> 0: (12.13)

We have lim infn!1 �min(G01nW11G1n) > 0 (by Assumptions FOC(i), (xiii), and (xvi) because �1n

is the smallest singular value of �1=2n G1n): Using this and (12.9), we obtain: for some "2 2 (0; "];

lim infn!1

inf�12B(�1�n;"2)

�min(G1n(�1)0W11G1n(�1)) > 0: (12.14)

Next, by (12.7), gn(�1) = O(1)jj�1 � �1�njj 8�1 2 B(�1�n; "): Hence,

maxs�p1

� @

@�1sG1n(�1)

�0W11g1n(�1)

= O(1)jj�1 � �1�njj 8�1 2 B(�1�n; ") (12.15)

using Assumptions FOC(xii) and (xv). This and (12.14) imply (12.13) for some su¢ ciently small

"1 > 0:

Now, by the inverse function theorem applied to n(�1); for every su¢ ciently small � > 0; there

exists an open neighborhoodMn� of �1�n such thatMn� � B(�1�n; "1) and the map n : cl(Mn�)!B(0p1 ; �) is a homeomorphism, where cl(Mn�) denotes the closure ofMn� and B(0p1 ; �) is the closed

ball at 0p1 with radius �: The diameter of cl(Mn�) is bounded by a multiple of � that does not

11

depend on n by the following argument:

sup�12Mn�

jj�1 � �1�njj = sup�2B(0p1 ;�)

jj�1n (�)��1n (0p1)jj = sup�2B(0p1 ;�)

jj(@=@�0)�1n (e�n)�jj= sup

�2B(0p1 ;�)jj[(@=@�01)n(�1)j�1=�1n (e�n)]�1�jj � �= inf

�12B(�1�n;"1)�min((@=@�

01)n(�1)) = O(1)�

(12.16)

for some e�n 2 B(0p1 ; �) that may di¤er across the rows of (@=@�0)�1n (e�n); where the �rst equalityholds because n : cl(Mn�)! B(0p1 ; �) is a homeomorphism and �1n (0

p1) = �1�n by Assumption

FOC(iii), the second equality holds by element-by-element mean-value expansions of �1n (�) about

0p1 ; the third equality holds by the standard formula for the derivative matrix of an inverse function,

and the last equality holds by (12.13).

Combining (12.10), (12.11), and (12.16) gives

sup�12cl(Mn�)

jjbn(�1)�n(�1)jj = op(1) + op(1)� +Op(1)�2; (12.17)

where the op(1) and Op(1) terms are uniform in � for � small. (This equation is analogous to the

second displayed equation on p. 69 of van der Vaart (1998). The remainder of the proof of the

existence of consistent solutions to the FOC�s is the same as in van der Vaart (1998), although for

completeness we provide more details here.)

Because PFn(op(1) + op(1)� > �=2) ! 0 8� > 0; there exists a sequence �n # 0 such thatPFn(op(1) + op(1)�n > �n=2) ! 0: Let Kn� := fsup�12cl(Mn�)

jjbn(�1) � n(�1)jj < �g: Then, wehave

PFn(Kn�n) := PFn( sup�12cl(Mn�n )

jjbn(�1)�n(�1)jj < �n)= PFn(op(1) + op(1)�n +Op(1)�

2n < �n)

= PFn(op(1) + op(1)�n +Op(1)�2n < �n; B

cn) + o(1)

� PFn(�n=2 +Op(1)�2n < �n; B

cn) + o(1)

! 1; (12.18)

where the second equality uses (12.17), the third equality holds by writing PFn(An) = PFn(An \Bcn)+PFn(An\Bn) for Bn = fop(1)+op(1)�n > �n=2g and PFn(Bn)! 0; the inequality holds using

the condition in Bcn; and the convergence holds because PFn(Bcn) ! 1 and PFn(�n=2 + Op(1)�

2n <

�n)! 1 using �n ! 0:

12

On the event Kn�; the map � ! � � bn(�1n (�)) maps B(0p1 ; �) into itself (because 8� 2B(0p1 ; �); jj�� bn(�1n (�))jj � jj��n(�1n (�))jj+sup�2B(0p1 ;�) jjbn(�1n (�))�n(�1n (�))jj � �;where the second inequality uses the de�nition of Kn� and the fact that � 2 B(0p1 ; �) implies that�1n (�) 2 cl(Mn�)): This map is continuous. Hence, by Brouwer�s �xed point theorem, it possesses

a �xed point in B(0p1 ; �): That is, there exists �n 2 B(0p1 ; �) such that �n � bn(�1n (�n)) = �n:For �1n := �1n (�n) 2 cl(Mn�n); this gives bn(�1n) = 0p1 : Because the set Mn�n contains �1�n; the

diameter of Mn�n is bounded by a multiple (that does not depend on n) of �n; and �n # 0; we have�1n � �1�n !p 0

p1 : Hence, �1n is a consistent solution to the FOC�s bn(�1) = 0p1 :Given the consistency of the solutions f�1n : n � 1g to the FOC�s in (??), we now establish

the n1=2-consistency of f�1n : n � 1g: The FOC�s, mean-value expansions around �1�n; and �1n ��1�n !p 0 give

0p1 = bG1n(�1n)0cW1nbgn(�1n) = bG1n(�1n)0cW1n

�bgn + bG1n(e�1n)(�1n � �1�n)� and�1n � �1�n = �

� bG1n(�1n)0cW1nbG1n(e�1n)��1 bG1n(�1n)0cW1nbgn = Op(n�1=2); (12.19)

where e�1n lies between �1n and �1�n and may di¤er across the rows of bG1n(e�1n) and the last equalityholds because bG1n(�1n)0cW1n

bG1n(e�1n)!p G011W11G11 (by Assumptions FOC(viii), (x), (xi), and

(xvi) and �1n� �1�n !p 0); G011W11G11 is nonsingular (since W11 is nonsingular by Assumption

FOC(xvi) and G11 has full column rank p1 by Assumptions FOC(i), (xi), and (xiii) because �1n is

the smallest singular value of �1=2n G1n); bG1n(�1n)0cW1n = Op(1) (by Assumptions FOC(viii), (x),

(xi), and (xvi) and �1n � �1�n !p 0); and bgn = Op(n�1=2) (by Assumption FOC(vi)). Equation

(12.19) completes the proof of the lemma. �

Proof of Lemma 12.3. We have

dH(�1�n; CS+1n) = dH(�1�n; CS1n [ b�1n)1(CS1n = ?) + dH(�1�n; CS1n [ b�1n)1(CS1n 6= ?)

� dH(�1�n; b�1n)[1(CS1n = ?) + 1(CS1n 6= ?)] + dH(�1�n; CS1n)1(CS1n 6= ?)� dH(�1�n; b�1n) + dH(�1�n; CS1n [ f�1�ng)= Op(n

�1=2); (12.20)

where the �rst inequality holds using straightforward manipulations, the second inequality holds

because CS1n 6= ? implies dH(�1�n; CS1n) = dH(�1�n; CS1n [ f�1�ng); and the last equality holdsby conditions (i) and (ii) of the lemma. �

Proof of Lemma 12.4. Let fb�1n : n � 1g be a sequence in b�1n (for all n � 1 and all sample

13

realizations) for which jj�1�n �b�1njj = dH(�1�n; b�1n) + op(n�1=2): Such a sequence exists wp!1 byAssumption ES3(ii).

Let qn := inf�1 =2B(�1�n;") jjgn(�1)jj: By Assumption ES4(ii), lim infn!1 qn > 0: By the de�nitionof qn; b�1n =2 B(�1�n; ") implies jjgn(b�1n)jj � qn 8n � 1: Hence, we have

PFn(b�1n =2 B(�1�n; "))

� PFn(jjgn(b�1n)jj � qn)� PFn(jjbgn(b�1n)jj+Op(n�1=2) � qn)� PFn(bgn(b�1n)0cW1nbgn(b�1n)=�min(cW1n) � (qn �Op(n�1=2))2)

� PFn( inf�12�1

bgn(�1)0cW1nbgn(�1) + cn � �(qn �Op(n�1=2))2)� PFn(bg0ncW1nbgn + cn � �(qn �Op(n�1=2))2)= o(1); (12.21)

where the second inequality holds by Assumption ES4(i) and the triangle inequality, the third

inequality uses Assumption ES3(xi), the fourth inequality holds by the de�nition of b�1n in (??)and because Assumption ES3(xi) implies that �min(cW1n) � � wp!1 for some constant � > 0; thelast inequality holds because �1�n 2 �1; and the equality holds because bg0ncW1nbgn = op(1) using

Assumptions ES3(iv) and (xi), � > 0; cn ! 0 by Assumption ES3(x), and lim infn!1 qn > 0 by

Assumption ES4(ii).

Equation (12.21) implies that b�1n � �1�n !p 0:

Next, the FOC�s in (??), mean-value expansions around �1�n; and b�1n � �1�n !p 0 give

0p1 = bG1n(b�1n)0cW1nbgn(b�1n) = bG1n(b�1n)0cW1n

�bgn + bG1n(e�1n)(b�1n � �1�n)� and sob�1n � �1�n = �

� bG1n(b�1n)0cW1nbG1n(e�1n)��1 bG1n(b�1n)0cW1nbgn = Op(n�1=2); (12.22)

where e�1n lies between b�1n and �1�n and may di¤er across the rows of bG1n(e�1n); the mean-value expansions use Assumption ES3(iii), and the second equality on the second line holds be-

cause bG1n(b�1n)0cW1nbG1n(e�1n)!p G

011W11G11; G011W11G11 is nonsingular, and bG1n(b�1n)0cW1n =

Op(1) by Assumptions ES3(i), (v)�(ix) and (xi), b�1n� �1�n !p 0 (which implies that there exists a

sequence of positive constants "n ! 0 for which PFn(jjb�1n � �1�njj > "n)! 0; so that Assumption

ES3(vi) can be applied), and bgn = Op(n�1=2) by Assumption ES3(iv). (Note that G11 has full

column rank p1 by Assumptions ES3(i) and (vii)�(ix) because �1n is the smallest singular value of

�1=2n G1n and lim infn!1 �1n > 0:)

14

Given the de�nition of b�1n; (12.22) implies that d(�1�n; b�1n) = Op(n�1=2): �13 Veri�cation of Assumptions for the First-Step AR CS

13.1 First-Step AR CS Results

First, we provide a result that veri�es Assumption B(i) for the �rst-step (FS) AR CS, de�ned

in (??).

Assumption FS1AR. For the sequence S; (i) bn � n !p 0 for some variance matrices fn 2Rk�k : n � 1g; (ii) n ! 1 for some variance matrix 1 2 Rk�k; (iii) n1=2bgn !d Z1 �N(0k;1); and (iv) lim infn!1 �min(n) > 0:

Lemma 13.1 Let S be a null sequence (or Sm a null subsequence) that satis�es Assumption

FS1AR: Then, under the sequence S (or subsequence Sm), the nominal level 1 � �1 �rst-step ARCS, CSAR1n ; has asymptotic coverage probability 1� �1 and, hence, satis�es Assumption B(i).

Next, we provide a result, Lemma 13.2, that is useful, in conjunction with Lemmas 12.3 and

12.4, for verifying Assumption OE(i) for the �rst-step AR CS, which equals

CS1n = f�1 2 �1 : nbgn(�1)0b�1n (�1)bgn(�1) � �2k(1� �1)g: (13.1)

Lemma 13.2 provides conditions under which dH(�1�n; CS1n [ f�1�ng) = Op(n�1=2) for a sequenceS and the AR CS CS1n in (13.1).

When verifying Assumption OE(i) for a sequence S with CS1n in place of CS+1n; where CS1n is as

in (13.1), we use the following global strong-identi�cation condition: For all sequences fKn : n � 1gfor which Kn !1;

limn!1

inf�1 =2B(�1�n;Kn=n1=2)

n1=2jjgn(�1)jj =1: (13.2)

Assumption FS2AR. For the sequence S; (i) sup�12�1 n1=2jjbgn(�1) � gn(�1)jj = Op(1) for some

nonrandom Rk-valued functions fgn(�) : n � 1g; (ii) (13.2) holds, (iii) sup�12�1 jjbn(�1)�n(�1)jj =op(1) for some nonrandom Rk�k-valued functions fn(�) : n � 1g; (iv) sup�12�1 jjn(�1)jj = O(1);and (v) lim infn!1 inf�12�1 �min(n(�1)) > 0:

Lemma 13.2 Suppose CS1n is of the form in (13.1). Let S be a null sequence (or Sm a null

subsequence) that satis�es Assumption FS2AR: Then, dH(�1�n; CS1n [ f�1�ng) = Op(n�1=2) for thesequence S (or subsequence Sm).

15



nbg0nb�1n bgn !d Z01

�11 Z1 � �2k; (13.3)

where the convergence in distribution holds because n1=2bgn !d Z1 by Assumption FS1AR(iii)

and b�1n !p �11 by Assumptions FS1AR(i), (ii), and (iv), and the �2k distribution arises because

Z1 � N(0k;1) by Assumption FS1AR(iii). Hence, we have

PFn(�1�n 2 CSAR1n ) = PFn(nbg0nb�1n bgn � �2k(1� �1))! 1� �1; (13.4)

which establishes the result of the lemma. �

Proof of Lemma 13.2. Let fb�1n : n � 1g be a sequence in CS1n [ f�1�ng (for all n � 1 and allsample realizations) for which jj�1�n�b�1njj = dH(�1�n; CS1n[f�1�ng)+op(n�1=2): Such a sequencealways exists because CS1n [ f�1�ng is non-empty for all n � 1:

We establish the result of the lemma by contradiction. Suppose dH(�1�n; CS1n [ f�1�ng) 6=Op(n

�1=2): Then, by de�nition of b�1n;jjb�1n � �1�njj 6= Op(n�1=2): (13.5)

If a sequence of random variables f�n : n � 1g satis�es �n = Op(1); then 8" > 0; 9K" < 1 such

that lim supn!1 PFn(j�nj > K") < ": Hence, (13.5) implies: 9" > 0 such that 8K <1;

lim supn!1

PFn(jjb�1n � �1�njj > K=n1=2) � ": (13.6)

For n � 1 and 0 < K <1; de�ne

Pn(K) := PFn(jjb�1n � �1�njj > K=n1=2) and Ln(K) := inf�1 =2B(�1�n;K=n1=2)

n1=2jjgn(�1)jj: (13.7)

Let fKn : n � 1g be a sequence such that Kn ! 1 as n ! 1; e.g., Kn = ln(n): Let m0 = 0:

For a given positive integer n; let mn (< 1) be a positive integer for which Pmn(Kn) > "=2 and

mn > mn�1: Such an mn always exists because (13.6) can be rewritten as lim supm!1 Pm(K) � ":The subsequence fmng satis�es

Pmn(Kn) > "=2 8n � 1: (13.8)

16

By the de�nition of Ln(K) in (13.7),

�1 =2 B(�1�n;K=n1=2) implies jjn1=2gn(�1)jj � Ln(K): (13.9)

Equation (13.9) implies that, for all n � 1;

(i) b�1mn =2 B(�1�mn ;Kn=m1=2n ) implies jjm1=2

n gmn(b�1mn)jj � Lmn(Kn) and b�1mn 2 CS1mn ;

(ii) PFmn (b�1mn =2 B(�1�mn ;Kn=m

1=2n ))

� PFmn (jjm1=2n gmn(

b�1mn)jj � Lmn(Kn) & b�1mn 2 CS1mn); and (13.10)

(iii) "=2 < Pmn (Kn) � PFmn (fjjm1=2n gmn(

b�1mn)jj � Lmn(Kn)g & b�1mn 2 CS1mn);

where we choose to take the subscript on Kn to be n throughout (rather than mn) because we

use (13.8) in the last line, the �rst line uses fb�1mn : n � 1g is a sequence in CS1mn [ f�1�mngby de�nition and dH(�1�mn ; fb�1mng) = 0 if b�1mn = �1�mn (so b�1mn =2 B(�1�mn ;Kn=m

1=2n ) impliesb�1mn 2 CS1mn); the �rst inequality on the last line holds by (13.8), and the second inequality on

the last line holds by the inequality in (ii) and the de�nition of Pn(K) in (13.7).

Given the de�nition of the AR statistic, ARn(�1); in (??), we have

AR1=2n (b�1n) � inf�12�1

�1=2min

�b�1n (�1)� jjn1=2bgn(b�1n)jj� �jjn1=2gn(b�1n) +Op(1)jj� �(jjn1=2gn(b�1n)jj �Op(1)); (13.11)

where bn(�1) is nonsingular 8�1 2 �1 wp!1 by Assumptions FS2AR(iii) and (v), which guar-antees that the AR statistic on the left-hand side (lhs) of the �rst line is well de�ned wp!1,the second inequality holds for some � > 0 wp! 1 by Assumptions FS2AR(i), (iii), and (iv) be-

cause inf�12�1 �min(b�1n (�1)) = 1=(sup�12�1 �max(bn(�1))) and sup�12�1 �max(bn(�1)) = sup�12�1�max(n(�1)) + op(1) = O(1) + op(1) by Assumptions FS2AR(iii) and (iv), and the third inequality

holds by the triangle inequality. Hence, for all n � 1;

PFmn (ARmn(b�1mn) > �

2k(1� �1) & b�1mn 2 CS1mn)

� PFmn (�(jjm1=2n gmn(

b�1mn)jj �Op(1)) > (�2k(1� �1))1=2

& jjm1=2n gmn(

b�1mn)jj � Lmn(Kn) & b�1mn 2 CS1mn)

� PFmn (�(Lmn(Kn)�Op(1)) > (�2k(1� �1))1=2 & jjm1=2n gmn(

b�1mn)jj � Lmn(Kn)

& b�1mn 2 CS1mn): (13.12)

17

We show below that limLmn(Kn) =1: Note that limLmn(Kn) =1 implies that �(Lmn(Kn)�Op(1)) > (�

2k(1� �1))1=2 wp!1. This and (13.12) yield

lim infn!1

PFmn (ARmn(b�1mn) > �

2k(1� �1) & b�1mn 2 CS1mn)

� lim infn!1

PFmn (jjm1=2n gmn(


> "=2; (13.13)

where the second inequality holds by the last line of (13.10). Equation (13.13) is a contradiction

because ARmn(b�1mn) > �2k(1 � �1) implies that b�1mn =2 CS1mn : That is, (13.13) asserts that

0 = lim infn!1 PFmn (b�1mn =2 CS1mn & b�1mn 2 CS1mn) > "=2 > 0:

It remains to show that limLmn(Kn) =1:Given the de�nition of Ln(K) in (13.7), the conditionin (13.2) (i.e., Assumption FS2AR(ii)) states that for all sequences fKng for which Kn ! 1;limLn(Kn) = 1: Hence, for all sequences fKmng for which Kmn ! 1; limn!1 Lmn(Kmn) = 1:Given fKng; we show that there exists a sequence fK�

mng such that K�

mn= Kn 8n � 1: Becausemn

is strictly increasing in n; n! mn is a one-to-one map. Let �(m) be the corresponding inverse map

for m 2 M := fmn : n � 1g: For any m 2 M; de�ne K�m = K�(m): Then, Lmn(Kn) = Lmn(K

�mn)

8n � 1 because �(mn) = n: In consequence, limn!1 Lmn(Kn) = limn!1 Lmn(K�mn) = 1; where

the second equality holds by (13.2). �


Data-Dependent Signi�cance Level

14.1 Data-Dependent Signi�cance Level Results

Here we verify Assumptions B(iii) and OE(ii) for the second-step signi�cance level (SL) b�2n(�1)de�ned as in (??)�(??).

The results in this section and the sections that follow apply only to moment condition models.

In consequence, in these sections,

gn(�) := EFnbgn(�) 2 Rk;Gjn(�) := EFn bGjn(�) 2 Rk�pj for j = 1; 2; andn(�) := V arFn(n

1=2bgn(�)) 2 Rk�k: (14.1)

18

We de�ne

b�jn(�) := n�1nXi=1

vec(Gji(�)� bGjn(�))gi(�)0 2 R(pjk)�k and�jn(�) := n�1

nXi=1

EFnvec(Gji(�)� EFnGji(�))gi(�)0 2 R(pjk)�k for j = 1; 2: (14.2)

Note that b�jn(�) = [b�j1n(�)0 : � � � : b�jpjn(�)0]0 for b�jsn(�) de�ned in (??) for s = 1; :::; p1 and

j = 1; 2:

We de�ne

Gjsi(�) :=@

@�jsgi(�) 2 Rk;

�2jsn(�) := V arFn(jjGjsi(�)jj) 8s = 1; :::; pj ; and

�jn(�) := Diagf��1j1n(�); :::; ��1jpjn

(�)g for j = 1; 2: (14.3)

First, we provide a lemma that veri�es Assumption B(iii) under high-level conditions. The

following assumption is employed when the second-step test is the C(�)-AR test de�ned in (??).

Assumption SL1AR. For the null sequence S; (i) lim ��1n < KL (where KL < 1 appears in

the de�nition of b�2n(�1) in (??)), (ii) bG1n � G1n !p 0 for fG1n := G1n(�1�n) : n � 1g de�nedin (14.1), (iii) lim supn!1 jjG1njj < 1; (iv) bn � n !p 0 for fn := n(�1�n) : n � 1g de�nedin (14.1), (v) lim infn!1 �min(n) > 0; (vi) lim supn!1 jjnjj < 1; (vii) b�21sn � �21sn !p 0 for

f�21sn := �21sn(�1�n) : n � 1g de�ned in (14.3) 8s = 1; :::; p1; and (viii) lim infn!1 �21sn > 0

8s = 1; :::; p1:

The following assumption is employed when the second-step test is the C(�)-LM or C(�)-QLR1

test de�ned in (??) and (??), respectively.

Assumption SL1LM;QLR1. For the null sequence S; (i) lim ��n < KL; (ii) Assumptions SL1AR(ii)�

(viii) hold, (iii) b�22sn � �22sn !p 0 for f�22sn := �22sn(�1�n) : n � 1g de�ned in (14.3) 8s = 1; :::; p2;and (iv) lim infn!1 �22sn > 0 8s = 1; :::; p2:

Lemma 14.1 Suppose b�2n(�1) is de�ned in (??)�(??) for the second-step C(�)-AR, C(�)-LM, orC(�)-QLR1 test. Let S be a null sequence (or Sm a null subsequence) that satis�es Assumption

SL1AR for the second-step C(�)-AR test and Assumption SL1LM;QLR1 for the second-step C(�)-LM

or C(�)-QLR1 test. Then, Assumption B(iii) holds for the sequence S (or subsequence Sm).

Next, we provide high-level conditions under which a sequence S satis�es Assumption OE(ii).

The following assumption is employed when the second-step test is the C(�)-AR test.

19

Assumption SL2AR. For the null sequence S; 8K < 1; (i) lim infn!1 ��1n > KU (where

KU > 0 appears in the de�nition of b�2n(�1) in (??)), (ii) sup�12B(�1�n;K=n1=2) jj bG1n(�1)�G1n(�1)jj!p 0 for fG1n(�) : n � 1g de�ned in (14.1), (iii) sup�12B(�1�n;K=n1=2) jjG1n(�1) � G1njj ! 0;

(iv) G1n = O(1); (v) sup�12B(�1�n;K=n1=2) jjbn(�1) � n(�1)jj !p 0 for fn(�) : n � 1g de-�ned in (14.1), (vi) sup�12B(�1�n;K=n1=2) jjn(�1) � njj ! 0; (vii) lim infn!1 �min(n) > 0; (viii)

sup�12B(�1�n;K=n1=2) jb�21sn(�1)� �21sn(�1)j !p 0 for f�21sn(�) : n � 1g de�ned in (14.3) 8s = 1; :::; p1;(ix) sup�12B(�1�n;K=n1=2) j�

21sn(�1) � �21snj ! 0 8s = 1; :::; p1; and (x) lim infn!1 �21sn > 0 8s =

1; :::; p1:

The following assumption is employed with the second-step C(�)-LM and C(�)-QLR1 tests.

Assumption SL2LM;QLR1. For the null sequence S; 8K < 1; (i) lim infn!1 ��n > KU ; (ii) As-sumptions SL2AR(v)�(x) hold, (iii) sup�12B(�1�n;K=n1=2) jj bGn(�1)�Gn(�1)jj !p 0 for some nonran-

dom Rk�p-valued functions fGn(�) : n � 1g; (iv) sup�12B(�1�n;K=n1=2) jjGn(�1)�Gnjj ! 0; (v) Gn =

O(1); (vi) sup�12B(�1�n;K=n1=2) jb�22sn(�1)��22sn(�1)j !p 0 for some nonrandom real-valued functions

f�22sn(�) : n � 1g de�ned in (14.3) 8s = 1; :::; p2; (vii) sup�12B(�1�n;K=n1=2) j�22sn(�1) � �22snj ! 0

8s = 1; :::; p2; and (viii) lim infn!1 �22sn > 0 8s = 1; :::; p2:

Lemma 14.2 Suppose b�2n(�1) is de�ned in (??)�(??) for the second-step C(�)-AR, C(�)-LM, orC(�)-QLR1 test. Let S be a null sequence (or Sm a null subsequence) that satis�es Assumption


or C(�)-QLR1 test. Then, Assumption OE(ii) holds for the sequence S (or subsequence Sm).

14.2 Proofs of Lemmas 14.1 and 14.2

Proof of Lemma 14.1. First, we prove the lemma for the second-step C(�)-AR test under

Assumption SL1AR. De�ne �1n := Diagf��111n; :::; ��11p1n

g: We write a SVD of �1=2n G1n�1n as

C�1n��1nB

�01n; where C

�1n and B

�1n are k � k and p1 � p1 orthogonal matrices, respectively, and ��1n

is a k � p1 matrix with the singular values of �1=2n G1n�1n on its main diagonal in nonincreasing

order and zeros elsewhere. The smallest singular value of �1=2n G1n�1n is ��1n; see (10.3), and it

appears as the (p1; p1) element of ��1n: Let �1n 2 Rp1 be such that jj�1njj = 1 and B�01n�1n = ep1 :=(0; :::; 0; 1)0 2 Rp1 : Then, ��1nB�01n�1n = ep1��1n:

20

We have

ICS21n

= �min

�b�1n( bG1n �G1n +G1n)0b�1n ( bG1n �G1n +G1n)b�1n�= �min

�(�1n + op(1))(

�1=2n G1n + op(1))

0h1=2n

b�1n 1=2n i(�1=2n G1n + op(1))(�1n + op(1))

�= inf

�:jj�jj=1

��0(�1nG

01n

�1=2n + op(1)) [Ik + op(1)] (

�1=2n G1n�1n + op(1))�

�� (�01n�1nG

01n

�1=2n + op(1)) [Ik + op(1)] (

�1=2n G1n�1n�1n + op(1))

= (��1ne0p1C

�01n + op(1))

0 [Ik + op(1)] (C�1nep1�

�1n + op(1))

= (��1n)2 + op(1); (14.4)

where the second equality uses Assumptions SL1AR(ii), (v), and (vii), the third equality uses

Assumptions SL1AR(iii)�(vi) and (viii) (where Assumptions SL1AR(v) and (viii) imply that �1=2n =

O(1) and �1n = O(1); respectively), the inequality holds with �1n de�ned as above, the second

last equality holds by the calculations above concerning �1n; and the last equality holds using

Assumption SL1AR(i).

Equation (14.4) implies that ICS1n � lim ��1n + " wp!1 under the sequence S; 8" > 0: UsingAssumption SL1AR(i) this implies that ICS1n � KL wp!1 under the sequence S: By the de�nitionof b�2n(�1) in (??) and (??), this implies that b�2n = �2 wp!1 under the sequence S: That is,Assumption B(iii) holds for the sequence S under Assumption SL1AR.

Next, we prove the lemma for the second-step C(�)-LM or C(�)-QLR1 test under Assumption

SL1LM;QLR1. The proof is the same as that given above for the C(�)-AR test but with all quantities

involving bGn; Gn; b�n; �n; and ��n ; rather than bG1n; G1n; b�1n; �1n; and ��1n; respectively. Thesechanges require the use of Assumption SL1LM;QLR1(i) (i.e., lim ��n < KL); rather than Assumption

SL1AR(i) (i.e., lim ��1n < KL); and of Assumptions SL1LM;QLR1(iii) and (iv) (to obtain the analogues

of the second and third equalities in (14.4) for the C(�)-LM and C(�)-QLR1 test cases). �

Proof of Lemma 14.2. First, we prove the lemma for the second-step C(�)-AR test under As-

sumption SL2AR: Let ��1n(�) denote the smallest singular value of �1=2n (�)G1n(�)�1n(�); where

�1n(�) 2 Rp1�p1 is de�ned in (14.3). For notational simplicity, let bG1n�1 ; G1n�1 bn�1 ; n�1 ; b�1n�1 ;�1n�1 ; and �

�1n�1

denote bG1n(�1); G1n(�1); bn(�1); n(�1); b�1n(�1); �1n(�1); and ��1n(�1); respec-tively. Let inf�1 abbreviate inf �12B(�1�n;K=n1=2) and likewise with sup�1 : Let op(�1; "n); Op(�1; "n);

and o(�1; "n) denote k � p1; k � k; or p1 � p1 matrices that depend on �1 and are op("n); Op("n);and o("n); respectively, uniformly over �1 2 B(�1�n;K=n1=2) for a sequence of positive constantsf"ng:

21

First, we show

lim infn!1

inf�1��1n�1=�

�1n = 1; (14.5)

Given Assumption SL2AR(i), (14.5) holds if

lim infn!1

inf�1

�(��1n�1)

2 � (��1n)2�= 0 8K <1 (14.6)

because inf�1�(��1n�1)

2 � (��1n)2�� 0 8n � 1:

Let �1n�1 2 Rp1 be such that jj�1n�1 jj = 1 and �min(�1n�1G01n�1�1n�1G1n�1�1n�1) = �

01n�1�1n�1

�G01n�1�1n�1G1n�1�1n�1�1n�1 : Let LHS denote the lhs of (14.6). We have

0 � LHS = lim infn!1

inf�1

��min(�1n�1G

01n�1

�1n�1G1n�1�1n�1)� �min(�1nG01n�1n G1n�1n)

�� lim inf

n!1inf�1

��01n�1�1n�1G

01n�1

�1n�1G1n�1�1n�1�1n�1 � �01n�1�1nG

01n

�1n G1n�1n�1n�1

�= lim inf

n!1inf�1�01n�1 [�1n�1G

01n�1

�1n�1G1n�1�1n�1 � �1nG01n�1n G1n�1n]�1n�1

= 0; (14.7)

where the �rst equality holds because the square of the smallest singular value of an k� p1 matrixA with p1 � k equals the smallest eigenvalue of A0A and the last equality holds by Assumption

SL2AR(iii), (a) sup�1 jj�1n�1

� �1n jj ! 0 8K <1; (b) sup�1 jj�1n�1 � �1njj ! 0 8K <1; and (c)all of the multiplicands �1n; G1n; and �1n are O(1): Condition (a) holds because

sup�1

jj�1n�1 � �1n jj = sup

�1

jj � �1n�1 [n�1 � n]�1n jj = o(1); (14.8)

where the last equality holds by Assumptions SL2AR(vi) and (vii) (since Assumptions

SL2AR(vi) and (vii) imply lim infn!1 inf�1 �min(n(�1)) > 0): Condition (b) holds by the same

argument as for condition (a) using Assumption SL2AR(ix) and (x). This completes the proof of

(14.6) and, in turn, (14.5).

22

Next, we have

inf�1ICS21n(�1)

= inf�1�min

�b�1n�1 � bG1n�1 �G1n�1 +G1n�1�0 b�1n�1 � bG1n�1 �G1n�1 +G1n�1� b�1n�1�= inf

�1�min

�(�1n�1 + op(�1; 1))(

�1=2n�1

G1n�1 + op(�1; 1))0h�1=2n�1

bn�1�1=2n�1

i�1�(�1=2n�1

G1n�1 + op(�1; 1))(�1n�1 + op(�1; 1))�

= inf�1�min

�(

�1=2n�1

G1n�1�1n�1 + op(�1; 1))0 [Ik + op(�1; 1)] (

�1=2n�1

G1n�1�1n�1 + op(�1; 1))�

= inf�1

inf�:jj�jj=1

(�0�1n�1G01n�1

�1n�1G1n�1�1n�1�+ �

0�1n�1G01n�1

�1=2n�1

op(�1; 1)�1=2n�1

G1n�1�1n�1�

+2�0op(�1; 1)0 [Ik + op(�1; 1)]

�1=2n�1

G1n�1�1n�1�+ �0op(�1; 1)

0 [Ik + op(�1; 1)] op(�1; 1)�)

� inf�1

inf�:jj�jj=1

�0�1n�1G01n�1

�1n�1G1n�1�1n�1�

� sup�1

sup�:jj�jj=1

j�0�1n�1G01n�1�1=2n�1

op(�1; 1)�1=2n�1

G1n�1�1n�1�)j

�2 sup�1

sup�:jj�jj=1

j�0op(�1; 1)0�1=2n�1G1n�1�1n�1�j � sup

�1

sup�:jj�jj=1

j�0op(�1; 1)�j

= inf�1(��1n�1)

2 + op(1)

= (��1n)2 + op(1); (14.9)

where the second equality holds using Assumptions SL2AR(ii) and (vi)�(x), the third equality holds

using Assumptions SL2AR(iii)�(vii), (ix), and (x), the second last equality holds using Assumptions

SL2AR(iii), (iv), (vi), (vii), (ix), and (x), the de�nition of ��1n�1 ; and the fact that the square of

the smallest singular value of a k � p1 matrix A with p1 � k equals the smallest eigenvalue of

A0A; and the last equality holds by Assumptions SL2AR(i), (iv), (vii), and (x) and (14.5) (where

Assumptions SL2AR(i), (iv), (vii), and (x) imply that f��1n : n � 1g is bounded away from 0 and

1):Equation (14.9) and Assumption SL2AR(i) imply that inf�1 ICS1n(�1) � KU wp!1. Hence,

given the de�nition of b�2n(�1) in (??) and (??) for the second-step C(�)-AR test, Assumption

OE(ii) holds for the sequence S:

Lastly, we prove the lemma for the second-step C(�)-LM and C(�)-QLR1 tests under As-

sumption SL2LM;QLR1. The proof is the same as that given above but with all quantities involvingbGn(�1); Gn(�1); b�n(�1); and �n(�1); rather than bG1n�1 ; G1n�1 ; b�1n�1 ; and �1n�1 ; respectively. Thesechanges require the use of Assumption SL2LM;QLR1(i) (i.e., lim infn!1 ��n > KU ); rather than As-

sumption SL2AR(i) (i.e., lim infn!1 ��1n > KU ) and the use of Assumptions SL2LM;QLR1(iii)�(viii)

23

(to obtain the analogues of the second, third, and last equalities of (14.9) for the C(�)-LM and

C(�)-QLR1 test cases). �

15 Veri�cation of Assumptions for the Second-Step C(�)-AR Test

15.1 Second-Step C(�)-AR Test Results

This section veri�es Assumptions B(ii) and C(ii)-C(v) for the second-step C(�)-AR test de�ned

in Section ??.

The following lemma provides conditions under which Assumptions B(ii), C(ii), and C(iii) hold

for the second-step AR test for a sequence S (whether lim �1n > 0 or lim �1n = 0): Assumption

C(iv) automatically holds for the second-step AR test provided p1 < k because its nominal level �

critical value is the 1�� quantile of the �2k�p1 distribution which is nondecreasing in � for � 2 (0; 1)when p1 < k:

For a full column rank matrix A 2 Rk�p1 ; let MA = Ik �A(A0A)�1A0:We write a singular value decomposition (SVD) of �1=2n G1n as

�1=2n G1n = C1n�1nB01n; (15.1)

where C1n 2 Rk�k and B1n 2 Rp1�p1 are orthogonal matrices and �1n 2 Rk�p1 has the singularvalues �11n; :::; �1p1n of

�1=2n G1n in nonincreasing order on its diagonal and zeros elsewhere. We

specify the compact SVD of �1=2n G1n given in (??) with � = (�01�n; �020)

0 to be the compact SVD

that is obtained from the SVD in (15.1) by deleting the non-essential rows and columns of C1n;

�1n; and B1n: Suppose limn1=2�1sn 2 [0;1] exists for s = 1; :::; p1: Let q1 (2 f0; :::; p1g) be suchthat

limn1=2�1sn =1 for 1 � s � q1 and limn1=2�1sn <1 for q1 + 1 � s � p1: (15.2)

De�ne

S1n := Diagf(n1=2�11n)�1; :::; (n1=2�1q1n)�1; 1; :::; 1g 2 Rp1�p1 and

S11 := Diagf0; :::; 0; 1; :::; 1g 2 Rp1�p1 ; (15.3)

where q1 zeros appear in S11:We have S1n ! S11: In the case of strong or semi-strong identi�cation

of �1 given �20; q1 = p1 and S11 = 0p1�p1 : In the case of weak identi�cation of �1 given �20;

S11 6= 0p1�p1 :

24

For the second-step (SS) C(�)-AR test, we use the following assumption.

Assumption SS1AR. For the null sequence S; (i) limn1=2�1sn 2 [0;1] exists 8s � p1; (ii)

n1=2(bg0n; vec( bG1n�EFn bG1n)0)0 !d (Z01; Z

0G11)

0 � N(0(p1+1)k; V11) for some variance matrix V11 2R(p1+1)k�(p1+1)k whose �rst k rows are denoted by [1 : �011] for 1 2 Rk�k and �11 2 R(p1k)�k;(iii) 1 is nonsingular, (iv) b�1n !p �11 for �11 as in condition (ii), (v) bn � n !p 0

k�k for

fn := n(�1�n; �20) : n � 1g de�ned in (14.1), (vi) n ! 1 for 1 as in condition (ii), (vii)

C1n ! C11 for some matrix C11 2 Rk�k; and (viii) B1n ! B11 for some matrix B11 2 Rp1�p1 :

Lemma 15.1 Suppose bgn(�1) are moment conditions, bD1n(�) is de�ned in (??), cM1n(�1) is de�ned

in (??) with a > 0; and p1 < k: Let S be a null sequence (or Sm a null subsequence) that satis�es

Assumption SS1AR: Then, for the sequence S (or subsequence Sm),

(a) AR2n !d AR21 := Z 01�1=21 M�

a11�1=21 Z1 � �2k�p1

for some (possibly) random k � p1 matrix �a11 that is independent of Z1; where �

a11 has full

column rank p1 a.s. and

(b) for all � 2 (0; 1); PFn(�AR2n (�1�n; �) > 0)! �:

Comments: (i). Lemma 15.1 establishes Assumptions B(ii), C(ii), and C(iii) for the second-step

AR test. It veri�es Assumption C(iii) because the �2k�p1 distribution is absolutely continuous on

R when p1 < k:

(ii). The de�nition of the limit random matrix �a11 is complicated and its form, beyond having

full column rank a.s., is not important. In consequence, for brevity, �a11 is de�ned in the proof of

Lemma 15.1 below, see (15.6), rather than in Lemma 15.1 itself.

(iii). A key result of Lemma 15.1(a) is that �a11 has full column rank. This uses the full rank

perturbation an�1=2�1 introduced in the de�nition of cM1n in (??).

(iv). Under strong and semi-strong identi�cation, the term an�1=2�1 in the de�nition of cM1n

has no e¤ect on the asymptotic distribution in Lemma 15.1(a).

(v). The proof of Lemma 15.1 uses Lemmas 10.2 and 10.3 and Corollary 16.2 in the SM to

Andrews and Guggenberger (2017) (AG1) to obtain the asymptotic distribution of cM1n:

The following lemma provides conditions under which Assumption C(v) holds for the second-

step C(�)-AR test for a sequence S with lim �1n > 0; where �1n := �1p1n is the smallest singular

value of �1=2n G1n: Let �1 = (�11; :::; �1p1)0:

Assumption SS2AR. For the null sequence S; 8K < 1; (i) lim infn!1 �1n > 0; (ii) bgn(�1)is twice continuously di¤erentiable on B(�1�n; ") (for all sample realizations) 8n � 1 for some

" > 0; (iii) bgn = Op(n�1=2); (iv) sup�12B(�1�n;K=n1=2) jj bG1n(�1) � G1n(�1)jj !p 0 for fG1n(�) : n �

25

1g de�ned in (14.1), (v) lim supn!1 sup�12B(�1�n;K=n1=2) jjG1n(�1)jj < 1; (vi) sup�12B(�1�n;K=n1=2)jj(@2=@�1s@�01)bgn(�1)jj = Op(1) for s = 1; :::; p1; (vii) sup�12B(�1�n;K=n1=2) jjb�1n(�1) � �1n(�1)jj =op(1) for f�1n(�) : n � 1g de�ned in (14.2), (viii) sup�12B(�1�n;K=n1=2) jj�1n(�1) � �1njj ! 0; (ix)

jj�1njj = O(1); (x) sup�12B(�1�n;K=n1=2) jjbn(�1)�n(�1)jj !p 0 for fn(�) : n � 1g de�ned in (14.1),(xi) sup�12B(�1�n;K=n1=2) jjn(�1)� njj ! 0; (xii) lim infn!1 �min(n) > 0; and (xiii) n = O(1):

Lemma 15.2 Suppose bgn(�1) are moment conditions, bD1n(�) is de�ned in (??), and cM1n(�1) is

de�ned in (??) with a � 0: Let S be a null sequence (or Sm a null subsequence) that satis�es

Assumption SS2AR: Then, under the sequence S (or subsequence Sm), for all constants K <1;(a) sup�12B(�1�n;K=n1=2) jjcM1n(�1)� cM1njj = op(1);(b) sup�12B(�1�n;K=n1=2)

n1=2cM1n(�1)b�1=2n (�1)bgn(�1)� n1=2cM1n(�1)b�1=2n (�1)bgn = op(1);(c) sup�12B(�1�n;K=n1=2)

n1=2cM1n(�1)b�1=2n (�1)bgn(�1)� n1=2cM1nb�1=2n bgn = op(1); and

(d) sup�12B(�1�n;K=n1=2) jAR2n(�1)�AR2nj = op(1):

Comments: (i). Lemma 15.2(d) establishes Assumption C(v) for the second-step C(�)-AR test

for a sequence S with lim infn!1 �1n > 0:

(ii). Lemma 15.2 does not require a > 0; but Lemma 15.1 above does.


Proof of Lemma 15.1. We have b�1=2n !p �1=21 and �1=21 is nonsingular by Assumptions

SS1AR(iii), (v), and (vi).

We write

V11 =

24 1 �011

�11 G11

35 ; where 1 2 Rk�k; �11 2 R(p1k)�k; and G11 2 R(p1k)�(p1k): (15.4)

By the argument in the proof of Lemma 10.2 in Section 15 of the SM to AG1, we have

n1=2

0@ bgnvec( bD1n � EFn bG1n)

1A !d

0@ Z1

ZG11 � �11�11 Z1

1A� N

0@0(p1+1)k;0@ 1 0k�(p1k)

0(p1k)�k D11

1A1A ; whereD11 : = G11 � �11�11 �011; (15.5)

using Assumptions SS1AR(ii)�(vi).

26

We partition B11 and C11 (de�ned in Assumption SS1AR) and de�ne �11 and �a11 as follows:

B11 = [B11;q1 : B11;p1�q1 ]; C11 = [C11;q1 : C11;k�q1 ];

L�p1�q1 :=

26640q1�(p1�q1)

Diagflimn1=2�1(q1+1)n; :::; limn1=2�1p1ng0(k�p1)�(p1�q1)

37752 Rk�(p1�q1);vec(D11) := ZG11 � �11�11 Z1 for D11 2 Rk�p1 ;

�11 = [�11;q1 : �11;p1�q1 ] 2 Rk�p1 ; �11;q1 := C11;q1 ;

�11;p1�q1 := C11L�p1�q1 +

�1=21 D11B11;p1�q1 ; and

�a11 := �11 + a�1B11S11; (15.6)

where B11;q1 2 Rp1�q1 ; B11;p1�q1 2 Rp1�(p1�q1); C11;q1 2 Rk�q1 ; C11;k�q1 2 Rk�(k�q1); �11;q1 2Rk�q1 ; �11;p1�q1 2 Rk�(p1�q1); and S11 is de�ned in (15.3).2 The limits in L�p1�q1 exist by

Assumption SS1AR(i). Note that �11;q1 (:= C11;q1) has full column rank q1 because C11 is an

orthogonal matrix (since C1n ! C11 by Assumption SS1AR(vii) and C1n is orthogonal for all n by

de�nition).

Using (15.5), by the proof of Lemma 10.3 in Section 16 of the SM to AG1 with p; bDn; cWn; WFn ;bUn; UFn ; Dh; �h; h2; h3; and h�1;p�q in AG1 set equal to p1; bD1n; b�1=2n ; n; Ip1 ; Ip1 ; D11; �11;

B11; C11; and L�p1�q1 ; respectively, we have

n1=2�1=2nbD1nT1n !d �11; where T1n := B1nS1n: (15.7)

This result uses Assumptions SS1AR(i)�(viii).

We have

T1n := B1nS1n ! B11S11 (15.8)

using S1n ! S11 and Assumption SS1AR(viii).

We have b�1=2n !p �1=211 by Assumptions SS1AR(iii), (v), and (vi). This, (15.7), and (15.8)

combine to yield

n1=2(b�1=2nbD1n + an�1=2�1)T1n = n1=2b�1=2n

bD1nT1n + a�1T1n !d �11 + a�1B11S11 =: �a11:

(15.9)

2For simplicity, there is some abuse of notation here, e.g., B11;q1 and B11;p1�q1 denote di¤erent matrices even ifp1 � q1 happens to equal q1:

27

Using the notation introduced in (15.6), we can write the limit random matrix in (15.9) as

�a11 := �11 + a�1B11S11 = [�11;q1 : �11;p1�q1 + a�1B11;p1�q1 ] (15.10)

because B11S11 = [0p1�q1 : B11;p1�q1 ] by the de�nition of S11 in (15.3). As noted above, �11;q1

has full column rank q1: In addition, �1B11;p1�q1 2 Rk�(p1�q1) is a matrix of independent standardnormal random variables (because B11 is an orthogonal matrix) and �1B11;p1�q1 is independent

of �11;p1�q1 : By Corollary 16.2 of AG1, these results and a > 0 imply that �a11 has full column

rank p1 a.s.

The matrix �a11 is independent of Z1 because �

a11 is a nonrandom function of (D11; �1); �1

is independent of (Z1; D11) by de�nition, and D11 is independent of Z1 since they are jointly

normal with zero covariance (because Evec(D11)Z 01 = E(ZG11 � �11�11 Z1)Z 01 = 0(p1k)�k)

using (15.6) and Assumption SS1AR(ii)).

Given a matrix A; the projection matrix PA is invariant to the multiplication of A by any

nonzero constant and the post-multiplication of A by any nonsingular matrix. In consequence, by

the continuous mapping theorem,

cM1n := Ik � Pb�1=2nbD1n+an�1=2�1 = Ik � Pn1=2[b�1=2n

bD1n+an�1=2�1]T1n !d M�11+a�1B11S11=:M�

a11;

(15.11)

where the second equality holds for n large because T1n is nonsingular for n large (because B1n

is orthogonal and S1n is nonsingular for n large by its de�nition in (15.3) and the de�nition of

q1 in (15.2)) and the convergence uses (15.9) and the fact, established above, that �a11 has full

column rank p1 � k (which implies that the function J(�a11) = (�

a011�

a11)

�1 is well-de�ned and

continuous a.s. so the continuous mapping theorem is applicable). The convergence in (15.11)

holds jointly with n1=2bgn !d Z1 (using Assumption SS1AR(ii)).

The result of part (a) follows from (15.11), b�1=2n !p �1=21 ; and Assumption SS1AR(ii) using

the continuous mapping theorem. We have Z 01�1=21 M�

a11�1=21 Z1 � �2k�p1 conditional on �

a11

(because, as shown above, �a11 and Z1 are independent and �

a11 has full column rank p1 a.s.

and, by Assumption SS1AR(ii), �1=21 Z1 � N(0k; Ik)) and, hence, unconditionally as well.

Part (b) follows immediately from part (a) because �AR2n (�1�n; �) = AR2n � �2k�p1(1 � �) and�2k�p1(1� �) is the 1� � quantile of the �

2k�p1 distribution. �

Proof of Lemma 15.2. For any �1 2 B(�1�n;K=n1=2); element-by-element mean-value expansions

28

give

bgn(�1) = bgn + @

@�1bgn(�1)(�1 � �1�n) + � @

@�01bgn(e�1n)� @

@�01bgn(�1)� (�1 � �1�n)

= bgn + @

@�01bgn(�1)(�1 � �1�n) +Op(n�1); (15.12)

where e�1n lies between �1 and �1�n and may di¤er across the rows of (@=@�01)bgn(e�1n) and, hence,satis�es e�1n� �1�n = Op(n�1=2) (because �1 2 B(�1�n;K=n1=2)), the �rst equality uses AssumptionSS2AR(ii), and the second equality uses mean-value expansions of (@=@�01)bgn(e�1n) and (@=@�01)bgn(�1)about �1�n and Assumption SS2AR(vi).

For part (a), given the de�nition of cM1n(�1) in (??), it su¢ ces to show that

(I) sup�12B(�1�n;K=n1=2)

jj bD1n(�1)� bD1njj = op(1);(II) sup

�12B(�1�n;K=n1=2)jjbn(�1)� bnjj = op(1); (15.13)

(III) bD1n has singular values that are bounded away from 0 and1 wp!1, (IV) bn has eigenvaluesthat are bounded away from 0 and 1 wp!1, and (V) an�1=2�1 = op(1):

Condition (II) holds by Assumptions SS2AR(x) and (xi). Condition (IV) holds by Assumptions

SS2AR(x)�(xiii). Condition (V) holds because a and �1 do not depend on n: Because bD1n(�) is asimple function of bG1n(�); b�1n(�); b�1n (�); and bgn(�); see (??), condition (I) holds if

sup�12B(�1�n;K=n1=2)

jj bG1n(�1)� bG1njj = op(1); sup�12B(�1�n;K=n1=2)

jjb�1n(�1)� b�1njj = op(1);sup

�12B(�1�n;K=n1=2)jjbgn(�1)� bgnjj = op(1); (15.14)

and conditions (II) and (IV) hold (because bG1n; b�1n; and bgn are Op(1)): The �rst condition in (15.14)holds by mean-value expansions of the elements of bG1n(�1) about �1�n using Assumptions SS2AR(ii)and (vi). The second condition in (15.14) holds by Assumptions SS2AR(vii) and (viii). The third

condition in (15.14) holds by (15.12) and Assumptions SS2AR(iv) and (v). Hence, condition (I)

holds.

To establish condition (III), we have

bD1n = G1n + op(1); (15.15)

by the de�nition of bD1n in (??) and Assumptions SS2AR(iii), (iv), (vii), (ix), (x), and (xii). The29

singular values of G1n are bounded away from 0 and 1 by Assumptions SS2AR(i) and (v) because

�1n is the smallest singular value of �1=2n G1n and the eigenvalues of

�1=2n are bounded away from

0 and 1 by Assumptions SS2AR(xii) and (xiii). This and (15.15) establish condition (III), which

completes the proof of part (a).

Part (b) is established as follows: For all �1 2 B(�1�n;K=n1=2);

n1=2cM1n(�1)b�1=2n (�1)bgn(�1)� n1=2cM1n(�1)b�1=2n (�1)bgn= n1=2cM1n(�1)b�1=2n (�1)

@

@�01bgn(�1)(�1 � �1�n) + cM1n(�1)b�1=2n (�1)Op(n

�1=2)

= n1=2cM1n(�1)(b�1=2n (�1) bD1n(�1) + an�1=2�1)(�1 � �1�n)� n1=2cM1n(�1)an�1=2�1(�1 � �1�n)

+n1=2cM1n(�1)b�1=2n (�1)[b�11n(�1) : ::: : b�1p1n(�1)]�Ip1 b�1n (�1)bgn(�1)� (�1 � �1�n)+Op(n

�1=2)

= op(1); (15.16)

where the Op(n�1=2) terms holds uniformly over �1 2 B(�1�n;K=n1=2); the �rst equality uses

(15.12), the second equality uses the de�nition of bD1n(�1) in (??) and the fact that

sup�12B(�1�n;K=n1=2) jjcM1n(�1)jj = Op(1) because the eigenvalues of cM1n(�1) equal zero or one (sincecM1n(�1) is a projection matrix) and sup�12B(�1�n;K=n1=2) jjb�1=2n (�1)jj = Op(1) by Assumptions

SS2AR(x)�(xii), and the third equality uses (1) cM1n(�1)[b�1=2n (�1) bD1n(�1)+an�1=2�1] = 0k�p1 (be-cause cM1n(�1) projects onto the orthogonal complement of the space spanned by b�1=2n (�1) bD1n(�1)+an�1=2�1; see (??)), (2) sup�12B(�1�n;K=n1=2) jjcM1n(�1)jj = Op(1) as above, (3) sup�12B(�1�n;K=n1=2)

jjb�jn (�1)jj = Op(1) for j = 1=2; 1 as above, (4) sup�12B(�1�n;K=n1=2) jjbgn(�1)jj = op(1) (by (15.12) andAssumptions SS2AR(iii)�(v)), (5) sup�12B(�1�n;K=n1=2) jj�1 � �1�njj = O(n�1=2); and

(6) sup�12B(�1�n;K=n1=2)b�1n(�1) = sup�12B(�1�n;K=n1=2)[

b�11n(�1)0 : � � � : b�1p1n(�1)0]0 = Op(1) by As-

sumptions SS2AR(vii)�(ix).

Part (c) holds by parts (a) and (b), bgn = Op(n�1=2) (which holds by Assumption SS2AR(iii)),and sup�12B(�1�n;K=n1=2) jjb�1=2n (�1)� b�1=2n jj = op(1) (which is implied by Assumptions SS2AR(x)�(xii)).

Part (d) follows from part (c) and n1=2cM1nb�1=2n bgn = Op(1) (which holds by (2) and (3) above

and bgn = Op(n�1=2)) given the de�nition of AR2n(�) in (??). �

30

16 Veri�cation of Assumptions for the Second-Step C(�)-LM Test

16.1 Second-Step C(�)-LM Test Results

This section veri�es Assumptions B(ii) and C(ii)-C(v) for the second-step C(�)-LM test de�ned

in Section ??. The results in this section apply only to moment condition models.

We employ the same de�nitions as in Sections ??, 14.1, and 15.1. In addition, we de�ne

�21n; :::; �2p2n; q2; C2n; �2n; B2n; S2n; S21; and �a21 as �11n; :::; �1p1n; q1; C1n; �1n; B1n; S1n; S11;

and �a11 are de�ned in Section 15.1, respectively, but with subscripts 2 in place of 1 throughout.

Given the de�nitions above, C2n�2nB02n is a SVD of �1=2n G2n and its singular values are

�21n; :::; �2p2n: We choose the compact SVD of �1=2n G2n speci�ed in (??) with � = (�01�n; �020)

0 to

be the compact SVD that is obtained from the SVD C2n�2nB02n by deleting the non-essential rows

and columns of C2n; �2n; and B2n: Given the de�nition of q2; we have S2n ! S21: In the case of

(local) strong or semi-strong identi�cation, q2 = p2 and S21 = 0k�p2 : In the case of (local) weak

identi�cation, S21 6= 0k�p2 :As de�ned in (10.2), �n is the smallest singular value of

�1=2n Gn 2 Rk�p; where p = p1 + p2:

We let rjn := rjFn for rjF de�ned in (??) for j = 1; 2 and C�n := C�Fn for C�F de�ned in (??).

For the second-step (SS) C(�)-LM test, we use the following assumptions.

Assumption SS1LM. For the null sequence S; (i) limn1=2�2sn 2 [0;1] exists 8s � p2; (ii)

n1=2(bg0n; vec( bG1n � EFn bG1n)0; vec( bG2n � EFn bG2n)0)0 !d (Z01; Z

0G11; Z

0G21)

0 � N(0(p+1)k; V1) for

some variance matrix V1 2 R(p+1)k�(p+1)k whose �rst k rows are denoted by [1 : �011 : �021]

for 1 2 Rk�k and �j1 2 R(pjk)�k for j = 1; 2; (iii) b�2n !p �21 for �21 as in condition (ii),

(iv) C2n ! C21 for some matrix C21 2 Rk�k; (v) B2n ! B21 for some matrix B21 2 Rp2�p2 ;(vi) bGn � Gn !p 0 and Gn ! G1 for some matrix G1 2 Rk�p; where Gn := EFn bGn; and (vii)b�2jsn � �2jsn !p 0 for f�2jsn : n � 1g de�ned in (14.3) and �2jsn ! �2js1 for some scalars �2js1 > 0

8s = 1; :::; pj ; 8j = 1; 2:

Assumption SS2LM. For the null sequence S; 8K < 1; (i) lim infn!1 ��n > K�U for K

�U > 0

de�ned in (??), (ii) bgn(�1; �2) is di¤erentiable in �2 at �20 and (@=@�02)bgn(�1; �20) is di¤erentiablein �1 with both holding 8�1 2 B(�1�n; ") (for all sample realizations), 8n � 1; for some " > 0;

(iii) sup�12B(�1�n;K=n1=2) jj bG2n(�1) � G2n(�1)jj !p 0 for fG2n(�) : n � 1g de�ned in (14.1), (iv)lim supn!1 sup�12B(�1�n;K=n1=2) jjG2n(�1)jj < 1; (v) sup�12B(�1�n;K=n1=2) jj(@

2=@�1s@�02)bgn(�1; �20)jj

= Op(1) for s = 1; :::; p1; (vi) sup�12B(�1�n;K=n1=2) jjb�2n(�1) � �2n(�1)jj = op(1) for f�2n(�) : n � 1gde�ned in (14.2), and (vii) sup�12B(�1�n;K=n1=2) jj�2n(�1)� �2njj ! 0:

31

Given the quantities 1; G1; and �2js1 in Assumption SS1LM ; we de�ne

ICS�1 := �1=2min(�

01G

01

�11 G1�1); �1 := Diagf��1111; :::; �

�11p11; �

�1211; :::; �

�12p21g 2 R

p�p;

WI1 := 1� s�ICS�1 �K�

L

K�U �K�

L

�; and Dy1 := (M�

a11+WI1P�a11)�

a21: (16.1)

As de�ned, WI1 = 0 if ICS�1 � K�U ; WI1 = 1 if ICS�1 � K�

L; and WI1 2 [0; 1] otherwise.The following lemma veri�es Assumptions B(ii), C(ii), and C(iii) for the second-step C(�)-LM

test.

Lemma 16.1 Suppose bgn(�1) are moment conditions, bDjn(�) is de�ned in (??) for j = 1; 2;cM1n(�1) is de�ned in (??) with a > 0 and p2 � 1: Let S be a null sequence (or Sm a null subse-

quence) that satis�es Assumptions SS1AR and SS1LM : Then, for the sequence S (or subsequence

Sm),

(a) Dy1 has full column rank p2 a.s.,

(b) LM2n !d LM21 := Z 01�1=21 P

Dy1�1=21 Z1 � �2p2 ; where (�

a11;�

a21; D

y1) is independent

of Z1 and �aj1 has full column rank pj a.s. for j = 1; 2; and

(c) for all � 2 (0; 1); lim supn!1 PFn(�LM2n (�1�n; �) > 0) = �:

Comments: (i). For the second-step LM test, Lemma 16.1(c) establishes Assumptions B(ii)

and C(ii). Lemma 16.1(b) establishes Assumption C(iii) because the �2p2 distribution is absolutely

continuous on R when p2 � 1: Assumption C(iv) automatically holds for the second-step LM test

provided p2 � 1 because its nominal level � critical value is the 1�� quantile of the �2p2 distributionwhich is nondecreasing in � for � 2 (0; 1):

(ii). The result of Lemma 16.1(a) is key because it allows one to use the continuous mapping

theorem to obtain the asymptotic distribution of the LM2n statistic.

(iii). When lim infn!1 �n > 0 (i.e., under strong local identi�cation of �); �aj1 reduces to

�j1 for j = 1; 2; where �11 is de�ned in (15.6) and �21 is de�ned analogously, and the terms

an�1=2�1 and an�1=2�2 do not a¤ect the asymptotic distribution of LM2n:

(iv). The proof of Lemma 16.1 uses Lemmas 10.2 and 10.3 in the SM to AG1 to obtain the

asymptotic distribution of bD2n after suitable rescaling and right-hand side (rhs) rotation, but notrecentering.

(v). To prove that the result in Comment (iii) to Theorem ?? holds (which considers the

pure C(�)-LM test (in which case WIn(�) := 0), we establish below that Lemma 16.1 holds with

WI1 = 0 provided Assumptions SS1LM (vi) and (vii) are replaced by (vi) rjn (:= rjFn) = rj1 for

all n su¢ ciently large for some constant rj1 2 f0; :::; pjg for j = 1; 2; and (vii) �min(C 0�nC�n) � �

32

8n � 1 for some � > 0:

The next lemma provides conditions under which Assumption C(v) holds for the second-step

C(�)-LM test for sequences S with lim infn!1 ��n > K�U :

Lemma 16.2 Suppose bgn(�1) are moment conditions, bDjn(�) is de�ned in (??) for j = 1; 2 andcM1n(�1) is de�ned in (??) with a � 0: Let S be a null sequence (or Sm a null subsequence)

that satis�es Assumptions SS1AR; SS2AR; SS1LM ; SS2LM ; and SL2LM;QLR1 with SL2LM;QLR1(i)

deleted. Then, under the sequence S (or subsequence Sm), for all constants K <1;(a) sup�12B(�1�n;K=n1=2) jjb�1=2n (�1) bD2n(�1)� b�1=2n

bD2njj = op(1) and(b) sup�12B(�1�n;K=n1=2) jLM2n(�1)� LM2nj = op(1):

Comments: (i). Lemma 16.2(b) establishes Assumption C(v) for the second-step C(�)-LM test

for a sequence S with lim infn!1 ��n > K�U (> 0):


(iii). Lemma 16.2 holds for the pure C(�)-LM test (in which caseWIn(�) := 0) with the condi-

tion lim infn!1 ��n > K�U (> 0) in Assumption SS2LM (i) replaced by the condition lim infn!1 �n >

0:


Proof of Lemma 16.1. We write

V1 =

26641 �011 �021

�11 G11 0G2G11

�21 G2G11 G21

3775 ; where 1 2 Rk�k; �j1 2 R(pjk)�k; Gj1 2 R(pjk)�(pjk);

(16.2)

and G2G11 2 R(p2k)�(p1k) for j = 1; 2: By the argument in the proof of Lemma 10.2 in Section 15of the SM to AG1, we have

n1=2

0BB@bgn

vec( bD1n � EFn bG1n)vec( bD2n � EFn bG2n)

1CCA (16.3)

!d

0BB@Z1

ZG11 � �11�11 Z1ZG21 � �21�11 Z1

1CCA � N

0BB@0(p+1)k;0BB@

1 0k�(p1k) 0k�(p2k)

0(p1k)�k D11 0D2D11

0(p2k)�k D2D11 D21

1CCA1CCA ; where

Dj1 := Gj1 � �j1�11 �0j1 for j = 1; 2 and D2D11 := G2G11 � �21�11 �011

33

using (16.2) and Assumptions SS1AR(iii)�(vi) and SS1LM (ii).

We de�ne B21;q2 ; B21;p2�q2 ; C21;q2 ; C21;k�q2 ; L�p2�q2 ; D21; �21; �21;q2 ; �21;p2�q2 ; and

�a21 using the de�nitions in (15.6) with subscripts 2 in place of 1: The limits in L�p2�q2 exist

by Assumption SS1LM (i). We de�ne T2n as in (15.8) with the subscript 2 in place of 1: As in

(15.7)�(15.9) with subscripts 2 in place of 1; we have

n1=2(b�1=2nbD2n + an�1=2�2)T2n = n1=2b�1=2n

bD2nT2n + a�2T2n !d �21 + a�2B21S21 =: �a21

(16.4)

using Assumptions SS1AR(iii), (v), and (vi) and SS1LM (i)�(v). By the same argument as given just

below (15.10), the limit random matrix �a21 has full column rank p2 a.s., as stated in part (b).

The convergence results in (15.11) and (16.4) hold jointly (using Assumption SS1LM (ii)). In

addition, by the same argument as in the paragraph following (15.10), �a11 and �

a21 are jointly

independent of Z1; as stated in part (b).

Now, we prove part (a). We have

ICS�n := �1=2min

�b�n bG0nb�1n bGnb�n�!p �1=2min

��1G

01

�11 G1�1

�=: ICS�1 and

WIn !p WI1; (16.5)

where the �rst and last de�nitions in the �rst line are given in (??) and (16.1), respectively, the

convergence in probability in the �rst line holds using Assumptions SS1AR(iii), (v), and (vi) (which

yield b�1n !p �11 ), Assumptions SS1LM (vi) and (vii) (which yield b�n !p �1 and bGn !p G1),

and Slutsky�s Theorem (because the smallest eigenvalue of a matrix is a continuous function of the

matrix), and the second line holds by the �rst line, the de�nition of WIn(�) in (??), and Slutsky�s

Theorem (because the function s(�) in (??) is assumed to be continuous).When ICS�1 � K�

L; we have WI1 = 1;

Dy1 := (M�a11+WI1P�a11)�

a21 = �

a21; (16.6)

and �a21 has full rank p2 by the same argument as used to prove Lemma 15.1(a) in (15.10), which

uses Corollary 16.2 of AG1, with �a21 in place of �

a11:

When ICS�1 > 0; we have �1=2min(�j1G0j1

�11 Gj1�j1) � �

1=2min(�1G

01

�11 G1�1) := ICS

�1 >

0 for j = 1; 2 using the de�nition of a minimum eigenvalue. Since �2js1 > 0 8s = 1; :::; pj ;

8j = 1; 2 by Assumption SS1LM (vii), this implies that �1=2min(G

0j1

�11 Gj1) > 0 for j = 1; 2:

Using Assumptions SS1AR(vi) and SS1LM (vi), this implies that � jn := �1=2min(G

0jn

�1n Gjn) !

�1=2min(G

0j1

�11 Gj1) > 0; where G1 = [G11 : G21] for Gj1 2 Rk�pj ; � jn is de�ned in (10.2),

34

and Gjn := EFn bGjn; see (14.1), for j = 1; 2: In turn, this gives qj = pj for j = 1; 2 because, by

de�nition, qj satis�es limn1=2� jsn =1 for 1 � s � qj (see (15.2) and the second paragraph of thissection). Finally, qj = pj for j = 1; 2 implies that

�aj1 = �j1 for j = 1; 2 (16.7)

using the de�nition of �a11 in (15.6) and the analogous de�nition of �

a21 speci�ed in the second

paragraph of this section.

By an analogous argument, when ICS�1 > 0; we have �n = �1=2min(G

0n

�1n Gn)! �

1=2min(G

01

�11 G1)

> 0; where �n is de�ned in (10.2).

When ICS�1 > K�L � 0; we have

Dy1 := (M�a11+WI1P�a11)�

a21 =

�Ik � s1P�a11

��a21 =

�Ik � s1P�11

��21; where

s1 := s

�ICS�1 �K�

L

K�U �K�

L

�> 0; (16.8)

the last equality on the �rst line holds by (16.7), and s1 > 0 because s(�) is a strictly increasingcontinuous function on [0; 1] with s(0) = 0; see (??).

Suppose (Ik � s1P�11)�21 has rank less than p2: Then, 9� 2 Rp2 with jj�jj = 1 such that

�21� = s1P�11�21�: Because s1 > 0; this occurs only if �21� 2 col(�11); where col(�)denotes the column space of a matrix, because the right-hand side of the equation is in col(�11):

But, �21� 2 col(�11) is a contradiction because [�11 : �21] has full column rank a.s., which we

now show.

Thus, to prove part (a), it remains to show that [�11 : �21] has full column rank a.s. when

ICS�1 > K�L � 0: It su¢ ces to show [�11 : �21] has full column rank a.s. when lim �n > 0 and

qj = pj for j = 1; 2 (because it is shown above that ICS�1 > 0 implies both conditions). We have

�j1 = Cj1;pj (which is nonrandom) when qj = pj for j = 1; 2 by (15.6) and the second paragraph

of this section, where Cj1 = [Cj1;pj : Cj1;k�pj ] 2 Rk�k:We have Cjn ! Cj1 and Bjn ! Bj1 by Assumptions SS1AR(vii) and (viii) and SS1LM (iv)

and (v), where Cjn = [Cjn;pj : Cjn;k�pj ] and Bjn are the k � k and pj � pj orthogonal matricesin the singular value decompositions �1=2n Gjn = Cjn�jnB

0jn for j = 1; 2; see (15.1). The k � pj

diagonal matrix �jn of singular values of �1=2n Gjn can be written as [�jn;pj : 0

pj�(k�pj)]0; where

�jn;pj 2 Rpj�pj is a diagonal matrix with positive diagonal elements for n su¢ ciently large (since itssmallest diagonal element is � jn and � jn ! �

1=2min(G

0j1

�11 Gj1) > 0 for j = 1; 2 as shown above). In

consequence, the singular value decomposition Cjn�jnB0jn equals Cjn;pj�jn;pjB0jn; where Cjn;pj !

35

Cj1;pj and Bjn ! Bj1 for j = 1; 2: Furthermore, �1=2n Gjn ! �1=21 Gj1: Hence, �jn;pj =

C 0jn;pj (Cjn;pj�jn;pjB0jn)Bjn ! C 0j1;pj (

�1=21 Gj1)Bj1 := �j1;pj ; where �j1;pj is a pj�pj diagonal

matrix with nonnegative elements because �jn;pj has these properties n � 1: In consequence,

�1=21 Gj1 = Cj1;pj�j1;pjB0j1: (16.9)

Suppose [�11 : �21] = [C11;p1 : C21;p2 ] has column rank less than p: Then, there is a vector

� 2 Rk with jj�jj = 1 such that [C11;p1 : C21;p2 ]� = 0: Let � = (�01; �02)0 for �j 2 Rpj : Let

�j = Bj1��1j1;pj�j for j = 1; 2 and � = (�

01; �

02)0: We have

0k = [C11;p1 : C21;p2 ]� = C11;p1�1 + C21;p2�2

= C11;p1�11;p1B011B11�

�111;p1�1 + C21;p2�21;p2B

021B21�

�121;p2�2

= C11;p1�11;p1B011�1 + C21;p2�21;p2B

021�2

= �1=21 G11�1 +�1=21 G21�2

= lim�1=2n G1n�1 + lim�1=2n G2n�2

= lim�1=2n Gn�

6= 0k; (16.10)

where the fourth equality holds by the de�nition of �j ; the �fth equality holds by (16.9), the sixth

equality holds because lim�1=2n Gn = �1=21 G1 by Assumptions SS1AR and SS1LM ; Gn = [G1n :

G2n]; and G1 = [G11 : G21]; the last equality holds because Gn = [G1n : G2n]; and the inequality

holds because lim �n > 0 (shown above), �n is the smallest singular value of �1=2n Gn; and � 6= 0

(because Bj1 is an orthogonal matrix, ��1j1;pj is nonsingular, and � 6= 0): Equation (16.10) is a

contradiction, which completes the proof that [�11 : �21] has full column rank p when lim �n > 0:

This completes the proof of part (a).

Next, we prove part (a) for the case of a pure C(�)-LM test, in which case WIn(�) := 0;

WI1 := 0; and Dy1 =M�a11�a21; when Assumptions SS1LM (vi) and (vii) are replaced by the two

conditions (vi) and (vii) in Comment (v) to Lemma 16.1. If [�a11 : �

a21] has full column rank p;

then the matrixM�a11�a21 has rank p2: This can be proved by showing that ifM�

a11�a21 has rank

less than p2; then [�a11 : �

a21] has column rank less than p: Let (�)+ denote the Moore-Penrose

generalized inverse. Suppose M�a11�a21 has rank less than p2: Then, there exists a nonzero vector

36

' 2 Rp2 such that �a21' = P�a11�a21': That is,

�a21' = �

a11�; where � := (�

a011�

a11)

+�a011�

a21'; and

[�a11 : �

a21]� = 0

k; where � := (�0;�'0)0 6= 0k: (16.11)

In this case, [�a11 : �

a21] has column rank less than p; which establishes the claim in the �rst

sentence of the paragraph.

To prove part (a) in the pure C(�)-LM case, it remains to show that [�a11 : �

a21] has full

column rank p: By its de�nition and (15.10),

�aj1 = [Cj1;qj : �j1;pj�qj + a�jBj1;pj�qj ] for j = 1; 2: (16.12)

Suppose

[C11;q1 : C21;q2 ] has full column rank q1 + q2: (16.13)

Then, by Corollary 16.2 of AG1, [�a11 : �

a21] has full column rank p1 + p2 = p a.s. conditional

on �j1;pj�qj for j = 1; 2 and, hence, unconditionally as well. This holds because, conditional on

�j1;pj�qj for j = 1; 2;

[�11;p1�q1 + a�1B11;p1�q1 : �21;p2�q2 + a�2B21;p2�q2 ] 2 Rk�(p1�q1+p2�q2) (16.14)

has a multivariate normal distribution with identity variance matrix multiplied by a constant (since

� := [�1 : �2] has a multivariate normal distribution with identity variance matrix by assumption

and Bj1;pj�qj has orthonormal columns for j = 1; 2): This is su¢ cient to verify the condition on the

variance matrix of M2�p�q��2 in Corollary 16.2 of AG1 (where M2�p�q��2 is de�ned in Corollary

16.2 of AG1).

Thus, to prove part (a) in the pure C(�)-LM case, it remains to show (16.13). First, we show

qj � rj1 for j = 1; 2: By the de�nition of qj (see (15.2)), we have n1=2� jsn ! 1 8s � qj and

lim supn!1 n1=2� jsn <1 8s > qj : Because rjn := rjFn is the rank of

�1=2Fn

EFn bGjn (see (??)) and� jsn is the sth largest singular value of the same matrix, � jsn = 0 8s > rjn; 8n � 1: By condition(vi) in Comment (v) to Lemma 16.1, rjn = rj1 for some rj1 2 f0; :::; pjg for all n su¢ ciently large.The latter two results imply that lim supn!1 n

1=2� jsn <1 8s > rj1: In consequence, qj � rj1:Let Cjn;qj denote the �rst qj columns of Cjn for j = 1; 2: Let Cjn;rj1 denote the �rst rj1

37

columns of Cjn for j = 1; 2: Now, we have

�min([C11;q1 : C21;q2 ]0[C11;q1 : C21;q2 ])

= lim�min([C1n;q1 : C2n;q2 ]0[C1n;q1 : C2n;q2 ])

� lim�min([C1n;r11 : C2n;r11 ]0[C1n;r11 : C2n;r11 ])

= lim�min([C�1Fn : C�2Fn ]0[C�1Fn : C�2Fn ])

= lim�min(C0�FnC�Fn)

� �

> 0; (16.15)

where the �rst equality holds by Assumptions SS1AR(vii) and SS1LM (iv) (because the smallest

eigenvalue of a matrix is a continuous function of the matrix), the �rst inequality holds because

qj � rj1; the second equality holds by the argument given in the following paragraph, the third

equality holds by de�nition, see (??), and the last two inequalities hold by condition (vii) in

Comment (v) to Lemma 16.1 and C�n := C�Fn :

The second equality of (16.15) holds by the following argument. As stated above (see the

third paragraph of this section), we choose the compact SVD of �1=2n Gjn speci�ed in (??) with

� = (�01�n; �020)

0 to be the compact SVD that is obtained from the SVD Cjn�jnB0jn by deleting the

non-essential rows and columns of Cjn; �jn; and Bjn for j = 1; 2: This implies that the matrix

containing the �rst rjn columns of Cjn; which is denoted by Cjn;rjn ; equals C�jFn (de�ned in (??)).

Since rjn = rj1 for all n su¢ ciently large by condition (vi) in Comment (v) to Lemma 16.1,

we obtain Cjn;rj1 = C�jFn for all n su¢ ciently large for j = 1; 2; which establishes the second

equality in (16.15). This completes the proof of part (a) in the pure C(�)-LM case because (16.15)

establishes (16.13).

Next, we complete the proof of part (b) using the result of part (a). By the convergence results

in (15.11), (16.4), and (16.5) which hold jointly (using Assumption SS1LM (ii)), the continuous

mapping theorem gives

PDyn= P

(cM1n+WIn bP1n)n1=2(b�1=2nbD2n+an�1=2�2)T2n !d P(M�

a11

+WI1P�a11)�

a21= P

Dy1; (16.16)

where the equality holds by the de�nition of Dyn(�) in (??) and because a projection matrix PA

is invariant to the multiplication of A by any nonzero constant and the post-multiplication of A

by any nonsingular matrix and the continuous mapping theorem applies because of the a.s. full

column rank property of Dy1 established in part (a) of the lemma. The convergence in (16.16)

38

holds jointly with n1=2bgn !d Z1 by (16.3), which uses Assumption SS1LM (ii).

The convergence result of part (b) follows from (16.16), n1=2bgn !d Z1; and b�1=2n !p �1=21

using the continuous mapping theorem. The limit of the test statistic LM2n; de�ned in (??), is

Z 01�1=21 P

Dy1�1=21 Z1: This limit has a �2p2 distribution conditional on D

y1 and, hence, is uncon-

ditionally �2p2 as well, because (i) �1=21 Z1 � N(0k; Ik); (ii)

�1=21 Z1 and Dy1 are independent

(because Dy1 is a deterministic function of �aj1 for j = 1; 2); and (iii) Dy1 has full rank p2 a.s.

This completes the proof of part (b).

Part (c) holds because

limP (�LM2n (�1�n; �) > 0)

= limP (LM2n > �2p2(1� �))

= P (Z 01�1=21 P

Dy1�1=21 Z1 > �2p2(1� �))

= �; (16.17)

where the �rst equality holds by the de�nition of �LM2n (�1�n; �) in (??) with � = (�01�n; �020)

0; and

the second and third equalities hold by part (b) of the lemma. �

Proof of Lemma 16.2. Part (a) of the Lemma holds by condition (I) in (15.13) in the proof

of Lemma 15.2 with the subscripts 1 replaced by subscripts 2 (which holds using Assumptions

SS2LM (ii) and (iii)�(vii) in place of Assumptions SS2AR(ii) and (iv)�(viii)), combined with condi-

tions (II) and (IV) in (15.13) (which hold because Assumption SS2AR is imposed in the present

lemma).

Now, we prove part (b). Assumption SS2LM (i) implies that lim infn!1 �2n = lim infn!1 �2p2n >

0; where �2n is de�ned in (10.2). In consequence, q2 = p2; where q2 is de�ned as q1 is de�ned in

(15.2) with subscripts 1 replaced by subscripts 2: By de�nition T2n := B2nS2n; where B2n is or-

thogonal and S2n = Diagf(n1=2�21n)�1; :::; (n1=2�2p2n)�1g using q2 = p2; see (15.3) with the leadingsubscripts 1 replaced by 2: These results give n1=2T2n = O(1): This and part (a) of the lemma give

sup�12B(�1�n;K=n1=2)

jjn1=2b�1=2n (�1) bD2n(�1)T2n � n1=2b�1=2nbD2nT2njj = op(1): (16.18)

Next, we have

ICS�n !p ICS�1 > K�

U (16.19)

using the result in (16.5), the fact that ICS�1 := �1=2min(�

01G

01

�11 G1�1) = lim �

�n (using Assump-

tions SS1AR(vi) and SS1LM (vi) and (vii)), and the condition lim infn!1 ��n > K�U (i.e., Assumption

39

SS2LM (i)). Hence, we obtain

ICS�n > K�U wp! 1 and WIn := 1� s

�ICS�n �K�

L

K�U �K�

L

�= 0 wp! 1; (16.20)

where the second result uses the �rst result and the conditions on s(�) in (??) (which imply thats(x) = 1 for all x � 1):

We now show that

sup�12B(�1�n;K=n1=2)

jWIn(�1)j = 0 wp! 1 (16.21)

given that lim infn!1 ��n > K�U :

By the same argument as used to show (14.5) and (14.9) in the proof of Lemma 14.2 in Section

14, but with ��n�1 (:= ��n (�1)) and �

�n in place of �

�1n�1

and ��1n; respectively, we have

lim infn!1

inf�1��n�1=�

�n = 1 and inf

�1ICS�2n (�1) � (��n )2 + op(1); (16.22)

where inf�1 abbreviates inf�12B(�1�n;K=n1=2) and likewise with sup�1 :

By the same argument as used to show (14.5), but with �n; Gn; �n�1 ; Gn�1 ; lim supn!1 sup�1 ;

and � in place of �1n; G1n; �1n�1 ; G1n�1 ; lim infn!1 inf�1 ; and �; respectively, in (14.6) and (14.7),and with �n in place of �1n�1 ; where �n 2 Rp is such that jj�njj = 1 and �min(�nG0n�1n Gn�n) =�0n�nG

0n

�1n Gn�n�n; we obtain lim supn!1 sup�1 �

�n�1=��n = 1: Combining this with the �rst result

in (16.22), we get

limn!1

sup�1

j��n�1 � ��n j = 0: (16.23)

Using (16.23), by the same argument as used to show (14.9), but with sup�1 ; + inf�1 ; and � inplace of inf�1 ; � sup�1 ; and �; respectively, we get sup�1 ICS�2n (�1) � (��n )2 + op(1): This and thesecond result in (16.22) give

sup�1

jICS�n(�1)� ICS�nj = op(1) (16.24)

using ICS�n = ��n + op(1) by (16.19). In turn, this establishes (16.21) using the same argument as

above to show the second result in (16.20).

Now, (16.21) implies that, for �1 2 B(�1�n;K=n1=2); the C(�)-LM statistic LM2n(�1) can be

written as in (??) and (??) but with WIn(�1) = 0 wp! 1: That is, wp! 1; the C(�)-LM statistic

40

can be written as

LM2n(�1) = (n1=2egn(�1)0cM1n(�1))

�n1=2[b�1=2n (�1) bD2n(�1) + an�1=2�2]T2n�

��n1=2[b�1=2n (�1) bD2n(�1) + an�1=2�2]T2n�0 cM1n(�1)n

1=2[b�1=2n (�1) bD2n(�1) + an�1=2�2]T2n��1��n1=2[b�1=2n (�1) bD2n(�1) + an�1=2�2]T2n�0 cM1n(�1)n

1=2egn(�1): (16.25)

Each of the multiplicands in (16.25) di¤ers from its counterpart evaluated at �1�n by op(1) uni-

formly over �1 2 B(�1�n;K=n1=2) 8K < 1 by Lemma 15.2(a) and (c) and (16.18). We have

jjn1=2egn(�1�n)jj2 = nbg0nb�1n bgn = Op(1) by (13.3). In addition, using (15.11) and (16.4),cM1nn

1=2[b�1=2nbD2n + an�1=2�2]Tn !d M�

a11�a21 (16.26)

and the latter has full column rank p2 by Lemma 16.1(a) under Assumptions SS1AR; SS2AR; SS1LM ;

and SS2LM ; sinceWI1 = 0 and Dy1 =M�a11�a21 in the present case by (16.1), (16.5), and (16.20).

In consequence, when �1 = �1�n; the term in the second line of (16.25) that is inverted converges in

distribution to a matrix that is nonsingular a.s. (using cM21n =

cM1n): This, (16.25), and the results

immediately following (16.25) establish the result of part (b). �

17 Veri�cation of Assumptions for the Second-Step C(�)-QLR1

Test

17.1 Second-Step C(�)-QLR1 Test Results

This section veri�es Assumptions B(ii) and C(ii)-C(v) for the second-step C(�)-QLR1 test

de�ned in Section ??. The results in this section apply only to moment condition models.

We employ the same de�nitions as in Sections ??, 14.1, 15.1, and 16.1. In particular, the

following quantities, which appear in the asymptotic distribution in Lemma 17.1 below, are de�ned

as follows: Z1 and 1 are de�ned in Assumption SS1LM (ii), the (possibly random) k� p1 matrix�a11 is de�ned in (15.6), and the (possibly random) k � p2 matrix �

a21 is de�ned in (15.6) with

subscripts 2 in place of 1 throughout. As de�ned, �1=21 Z1 � N(0k; Ik):

41

By de�nition,

n(�1) := Fn(�1) := V arFn(n1=2bgn(�1)) 2 Rk�k;

G2n(�1) := EFn bG2n(�1) 2 Rk�p2 ;G2si(�1) :=

@

@�02sgi(�1) 2 Rk 8s = 1; :::; p2;

�22sn(�1) := V arFn(jjG2si(�1)jj) 2 R 8s = 1; :::; p2; and

�2n(�1) := Diagf��121n(�1); :::; ��12p2n

(�1)g 2 Rp2�p2 : (17.1)

We write a SVD of �1=2n G2n�2n (:= �1=2n (�1�n)G2n(�1�n)�2n(�1�n)) as

�1=2n G2n�2n = C�2n�

�2nB

�02n; (17.2)

where C�2n 2 Rk�k and B�2n 2 Rp2�p2 are orthogonal matrices and ��2n 2 Rk�p2 has the singularvalues ��21n; :::; �

�2p2n

of �1=2n G2n�2n in nonincreasing order on its diagonal and zeros elsewhere.

Suppose limn1=2��2sn 2 [0;1] exists for s = 1; :::; p2: (This is Assumption SS1QLR1(i) below.) Letq�2 (2 f0; :::; p2g) be such that

limn1=2��2sn =1 for 1 � s � q�2 and limn1=2��2sn <1 for q�2 + 1 � s � p2: (17.3)

Note that q�2 = q2 under Assumption SS1QLR1 below, where q2 is de�ned in (15.2) with subscripts

2 in place of 1:3 For notational simplicity, let ��2n := ��2p2n: That is, ��2n is the smallest singular

value of �1=2n G2n�2n:

De�ne

h�21;s := limn1=2��2sn <1 8s = q2 + 1; :::; p2;

S�2n := Diagf(n1=2��21n)�1; :::; (n1=2��2q1n)�1; 1; :::; 1g 2 Rp2�p2 and

S�21 := Diagf0; :::; 0; 1; :::; 1g 2 Rp2 ; (17.4)

where q�2 zeros appear in S�21: We have S

�2n ! S�21: In the case of local strong or semi-strong

identi�cation of �2 given �1�n; q�2 = p�2 and S

�21 = 0k�p2 : In the case of local weak identi�cation of

�2 given �1�n; S�21 6= 0k�p2 :For the second-step (SS) C(�)-QLR1 test, we use the following assumptions.

3This holds because f(��2sn)2 : s = 1; :::; p2g and f(�2sn)2 : s = 1; :::; p2g are the eigenvalues of �2nG02n

�1n G2n�2n

and G02n�1n G2n; respectively, and �1 � lim infn!1 �min(�2n) � lim supn!1 �sup(�2n) � �2 for some constants

�1 > 0 and �2 <1 by Assumption SS1QLR1:

42

Assumption SS1QLR1. For the null sequence S; (i) limn1=2��2sn 2 [0;1] exists 8s � p2; (ii)

C�2n ! C�21 for some matrix C�21 2 Rk�k; and (iii) B�2n ! B�21 for some matrix B�21 2 Rp2�p2 :

Assumption SS2QLR1. For the null sequence S; 8K < 1; (i) sup�12B(�1�n;K=n1=2) jb�22sn(�1) ��22sn(�1)j !p 0 for f�22sn(�) : n � 1g de�ned in (17.1) 8s = 1; :::; p2; (ii) sup�12B(�1�n;K=n1=2) j�

22sn(�1)�

�22snj ! 0 8s = 1; :::; p2; and (iii) �22sn ! �22s1 for some constant �22s1 2 (0;1) 8s = 1; :::; p2:

The following quantities appear in the expression for the asymptotic distribution of rk2n that

is speci�ed below. De�ne ZD21 2 Rk�p2 by

vec(ZD21) := ZG21 � �21�11 Z1 2 Rp2k; (17.5)

where Z1; ZG21; and �21 are de�ned in Assumption SS1LM : The matrix ZD21 has a nor-

mal distribution and is independent of Z1 because of the joint normality of ZG21 and Z1 and

Cov(vec(ZD21); Z1) = E(ZG21 � �21�11 Z1)Z 01 = EZG21Z01 � �21 = 0(p2k)�k: Partition the

(nonrandom) matrices B�21 and C�21 as

B�21 = (B�21;q2 ; B�21;p2�q2) and C

�21 = (C�21;q2 ; C

�21;k�q2); (17.6)

where q2 = q�2 ; B�21;q2 2 Rp2�q2 ; B�21;p2�q2 2 Rp2�(p2�q2); C�21;q2 2 Rk�q2 ; and C�21;k�q2 2

Rk�(k�q2): For simplicity, there is some abuse of notation here, e.g., B�21;q2 and B�21;p2�q2 denote

di¤erent matrices even if p2 � q2 happens to equal q2:Next, de�ne the (possibly random) matrix �

�21 as follows:

��21 = (�

�21;q2 ;�

�21;p2�q2) 2 R

k�p2 ; ��21;q2 := C

�21;q2 2 R

k�q2 ;

��21;p2�q2 := C�21h

��21;p2�q2 +

�1=21 ZD21�21B

�21;p2�q2 2 R

k�(p2�q2);

h��21;p2�q2 :=

26640q2�(p2�q2)

Diagfh�21;q2+1; :::; h�21;p2g

0(k�p2)�(p2�q2)

37752 Rk�(p2�q2); and�21 := Diagf��1211; :::; �

�12p21g 2 R

p2�p2 : (17.7)

When limn1=2��2n < 1; the lemma below shows that the asymptotic distribution of rk2n is

given by

rk21 := �min(K��021;p2�q2C

�21;k�q2C

�021;k�q2�

�21;p2�q2): (17.8)

43

De�ne

ARy21 := Z 01�1=21 (M�

a11+WI1P�a11)

�1=21 Z1;

LM21 := Z 01�1=21 P

Dy1�1=21 Z1; and

QLR121 :=1

2

�ARy21 � rk21 +

q(ARy21 � rk21)2 + 4LM21 � rk21

�: (17.9)

The random variables ARy21 and LM21 give the asymptotic distributions of ARy2n and LM2n (as

stated in Lemma 16.1 for the LM2n statistic). Note that, when WI1 = 1; we have M�a11+

WI1P�a11 = Ik; ARy21 = Z 01

�11 Z1 � �2k: The following lemma shows that QLR121 is the

asymptotic distribution of QLR12n when limn1=2��2n <1:The following lemma veri�es Assumptions B(ii), C(ii), and C(iii) for the second-step C(�)-QLR1

test.

Lemma 17.1 Suppose bgn(�1) are moment conditions, cM1n(�1) is de�ned in (??) with a > 0;

p1 < k; and p2 � 1: Let S be a null sequence (or Sm a null subsequence) that satis�es AssumptionsSS1AR; SS1LM ; and SS1QLR1: Then, for the sequence S (or subsequence Sm),

(a) ARy2n !d ARy21;

(b) when limn1=2��2n <1; (i) rk2n !d rk21; (ii)QLR12n !d QLR121; where (�a11;�

a21; D

y1;

rk21) are independent of Z1 and �aj1 has full column rank pj a.s. for j = 1; 2; and (iii) cQLR1(1�

�; rk2n;WIyn)!d c

QLR1(1� �; rk21; 1) and the convergence is joint with that in part (b)(ii),(c) when limn1=2��2n = 1; (i) rk2n !p 1; (ii) QLR12n !d LM21 � �2p2 ; and (iii) c

QLR1(1 ��; rk2n;WI

yn)!p �

2p2(1� �); and

(d) for all � 2 (0; 1); limPFn(�QLR12n (�1�n; �) > 0) = �:

Comments: (i). For the second-step C(�)-QLR1 test, Lemma 17.1(d) establishes Assumptions

B(ii) and C(ii). For a sequence S with limn1=2��2n = 1; Lemma 17.1(c) establishes AssumptionC(iii) (because the asymptotic �2p2 distribution of QLR12n is absolutely continuous on R when

p2 � 1 and the probability limit of cQLR1(1��; rk2n;WIyn) is the constant �2p2(1��)): AssumptionC(iv) holds because the conditional critical value cQLR1(1 � �; rk2n;WIyn) is nondecreasing in �since cQLR1(1� �; r; w) is the 1� � quantile of QLR1(r; w); see (??).

(ii). Under local strong and semi-strong identi�cation of �1 given �20; the terms an�1=2�1 and

an�1=2�2; which arise in the de�nition of QLR12n; do not a¤ect the asymptotic distributions in

Lemma 17.1(b) and (c).

(iii). The proof of Lemma 17.1(b)(i) and (c)(i) uses Theorem 10.4 in the SM to AG1.

44


C(�)-QLR1 test for sequences S with lim infn!1 ��n > K�U > 0:

Lemma 17.2 Suppose bgn(�1) are moment conditions, bDjn(�) is de�ned in (??) for j = 1; 2; andcM1n(�1) is de�ned in (??) with a � 0: Let S be a null sequence (or Sm a null subsequence) that

satis�es Assumptions SS1AR; SS2AR; SS1LM ; SS2LM ; and SS2QLR1: Then, under the sequence S

(or subsequence Sm), for all constants K <1;(a) sup�12B(�1�n;K=n1=2) jQLR12n(�1)�QLR12nj = op(1) and(b) sup�12B(�1�n;K=n1=2) jc

QLR1(1��; rk2n(�1);WIyn(�1))�cQLR1(1��; rk2n;WIyn)j = op(1) 8� 2(0; 1):

Comments: Lemma 17.2 does not require a > 0; but Lemma 17.1 above does.


Proof of Lemma 17.1. First, we prove part (a). By (16.5), WIn !p WI1: By (15.11), cM1n !d

M�a11and bP1n = Ik � cM1n !d Ik �M�

a11= P�a11

: Hence,

cM1n +WIn bP1n !d M�a11+WI1P�a11 : (17.10)

This, n1=2bgn !d Z1 and b�1=2n !p �1=21 give

ARy2n : = nbg0nb�1=2n

�cM1n +WIn bP1n� b�1=2n bgn!d Z

01

�1=21

�M�

a11+WI1P�a11

��1=21 Z1 =: ARy21; (17.11)

which establishes part (a).

Parts (b)(i) and (c)(i) of the lemma hold by Theorem 10.4 in the SM to AG1 (using Assump-

tion SS1QLR1 to guarantee the existence of limn1=2��2sn; C�21; and B

�21): To make this clear, the

following is the correspondence between the quantities in (17.1)�(17.7) above, which de�ne the

asymptotic distribution rk21 in part (b)(i), and those in the asymptotic distribution in Lemma

45

10.2, (10.16), and (10.17) in the SM to AG1:

ZD21 $ Dh; G2n $ EFnGi; �1=2n $WFn ; �2n $ UFn ;

(��21n; :::; ��2p2n)$ (�1Fn ; :::; �p2Fn); (h

�21;q2+1; :::; h

�21;p2)$ (h1;q+1; :::; h1;p)

B�21 $ h2; C�21 $ h3; B

�2;q2;1 $ h2;q; B

�2;p2�q2;1 $ h2;p�q;

C�21;q2 $ h3;q; C�21;k�q2 $ h3;k�q; �

�21 $ �h;

��21;q2 $ �h;q; �

�21;p2�q2 $ �h;p�q; h

��21;p2�q2 $ h�1;p�q;

�1=21 $ h71 :=W1(h7); and �21 $ h81 := U1(h8): (17.12)

Next, let

J2n := neg0n �cM1n +WIn bP1n � PDy2n

� egn: (17.13)

It follows from (??) and (??) that

ARy2n = LM2n + J2n: (17.14)

We now prove part (b)(ii). We have

QLR12n : =1

2

�ARy2n � rk2n +

q(ARy2n � rk2n)2 + 4LM2n � rk2n

�!d

1

2

�ARy21 � rk21 +

q(ARy21 � rk21)2 + 4LM21 � rk21

�= : QLR121; (17.15)

where the �rst and last equalities hold by the de�nitions of QLR12n and QLR121 in (??) and

(17.9), respectively, and the convergence holds by (17.11), rk2n !d rk21 (by part (b)(i) of the

lemma), and LM2n !d LM21 by Lemma 16.1(b). (The latter three convergence results hold

jointly because they all rely on Assumption SS1LM (ii).)

We have (�a11;�

a21; D

y1) is independent of Z1 and�

aj1 has full column rank pj a.s. for j = 1; 2

by Lemma 16.1(b). In addition, rk21 is independent of Z1 because rk21 is a deterministic function

of ZD21 by (17.7) and (17.8) and ZD21 is independent of Z1, see the discussion following (17.5).

Next, we prove part (b)(iii). First, we show that WIyn = 1 wp! 1: We have WIn !p WI1

(by (16.5)), 0 � ICS�1 = lim ��n � lim infn!1 ��2n = 0 (by (16.5), the de�nitions of ��n and ��2n

in (10.3), and limn1=2��2n < 1; which is assumed in part (b)), WI1 = 1 � s(0) = 1 (by (16.1),

ICS�1 = 0; and the de�nition of s(�) in (??)), and WIyn := 1(WIn > 0): These results combine toestablish that WIyn = 1 wp! 1: Hence, when proving part (b)(iii), we can suppose WIyn = 1 a.s.

46

To prove part (b)(iii), we need to show cQLR1(1 � �; rk2n; 1) !d cQLR1(1 � �; rk21; 1): This

holds by part (b)(i) of the lemma and the continuous mapping theorem provided cQLR1(1� �; r; 1)is continuous at all r � 0:

Now we establish the latter. For notational simplicity, let c(r) := cQLR1(1 � �; r; 1) and

QLR1(r) := QLR1(r; 1): Given any r� � 0 and sequence frn � 0 : n � 1g such that rn ! r�

(as n!1), it su¢ ces to show that for any subsequence fvng of fng there exists a subsubsequencefmng such that c(rmn) ! c(r�): We have: (i) QLR1(rn) ! QLR1(r�) a.s. (by the de�nition of

QLR1(r) in (??)), (ii) given any subsequence fvng of fng; there exists a subsubsequence fmng suchthat c1 = lim c(rmn) exists and is �nite (because part (i) implies that fc(r) : jr� r�j � "g lies in acompact set for some " > 0); and (iii) P (QLR1(r�) = c1) = 0 (because QLR1(r�) has an absolutely

continuous distribution). Results (i)�(iii) imply 1(QLR1(rmm)) � c(rmm)) ! 1(QLR1(r�) � c1)a.s. Hence, by the dominated convergence theorem,

P (QLR1(r�) � c1) = limP (QLR1(rmm)) � c(rmm)) = 1� �; (17.16)

where the last equality holds because c(rmm) is the 1 � � quantile of QLR1(rmm)) for all n � 1:

Equation (17.16) implies that c1 is the 1� � quantile of QLR1(r�); which is unique (because thedistribution function of QLR1(r�) is continuous and strictly increasing on R+): That is, c1 = c(r�);

which completes the proof that cQLR1(1� �; r; 1) is continuous at all r � 0:We now prove part (c)(ii). By part (c)(i), rk2n !p 1: By (17.14) and some algebra, we have

(ARy2n � rk2n)2 + 4LM2n � rk2n = (LM2n � J2n + rk2n)2 + 4LM2n � J2n: Therefore,

QLR12n =1

2

�LM2n + J2n � rk2n +

p(LM2n � J2n + rk2n)2 + 4LM2n � J2n

�: (17.17)

Using a mean-value expansion of the square-root expression in (17.17) about (LM2n�J2n+rk2n)2;we have

p(LM2n � J2n + rk2n)2 + 4LM2n � J2n = LM2n � J2n + rk2n + (2

p�n)

�14LM2n � J2n (17.18)

for an intermediate value �n between (LM2n�J2n+rk2n)2 and (LM2n�J2n+rk2n)2+4LM2n �J2n:It follows that

QLR12n = LM2n + op(1)!d �2p; (17.19)

where the equality holds because (p�n)

�1 = op(1) (since rk2n !p 1; LM2n = Op(1) by Lemma

16.1(b), and J2n = ARy2n �LM2n = Op(1) using part (a) of the lemma) and the convergence holds

by Lemma 16.1(b).

47

We now prove part (c)(iii). By an analogous argument to that used to prove (17.19), we obtain

QLR1(r; w)! �2p2 a.s. as r !1 (17.20)

for w = 0 or 1; where QLR1(r; w) is de�ned in (??). In consequence, cQLR1(1��; r; w)!p �2p2(1��)

as r ! 1 for w = 0 or 1: This, rk2n !p 1 (by part (c)(i)), and WIyn 2 f0; 1g imply thatcQLR1(1� �; rkn;WIyn)!p �

2p2(1� �); which establishes part (c)(iii).

Now, we prove part (d). First, consider the case when limn1=2��2n = 1: By parts (c)(ii) and(c)(iii),

QLR12n � cQLR1(1� �; rk2n;WIn)!d LM21 � �2p2(1� �): (17.21)

In consequence,

limPFn(�QLR12n (�1�n; �) > 0) = P (LM21 � �2p2(1� �) > 0) = �; (17.22)

where the �rst equality holds using P (LM21 = �2p2(1 � �)) = 0 (because LM21 � �2p2 by part

(c)(ii)) and the second equality holds because LM21 � �2p2 :Next, we prove part (d) when limn1=2��2n <1: By parts (b)(ii) and (b)(iii),

QLR12n � cQLR1(1� �; rkn;WIn)!d QLR121 � cQLR1(1� �; r21; 1): (17.23)

Thus,

limPFn(�QLR12n (�1�n; �) > 0) = P (QLR121 > cQLR1(1� �; r21; 1)) (17.24)

provided P (QLR121 = cQLR1(1 � �; r21; 1)) = 0; which holds if P (QLR121 = cQLR1(1 ��; r21; 1)j�

a11;�

a21; r21) = 0 a.s. The latter holds if, conditional on (�

a11;�

a21; rk21); QLR121

is absolutely continuous, which we now show.

As shown in the proof of part (b)(iii), WI1 = 1: This implies that ARy21 = Z 01�11 Z1 by the

de�nition of ARy21 in (17.9). In the present case where WI1 = 1; de�ne

J21 := ARy21 � LM21 = Z 01�1=21 (Ik � PDy

21)�1=21 Z1: (17.25)

Conditional on (�a11;�

a21; rk1); J21 � �2k�p2 because (i) Ik � PDy

21is a projection matrix with

rank k � p2 a.s. (since Dy21 has rank p2 a.s. by Lemma 16.1(a)), (ii) �1=21 Z1 � N(0k; Ik)

(by Assumptions SS1AR(ii) and (iii)), and (iii) �1=21 Z1 and (�

a11;�

a21; rk1) are independent

(by part (b)(ii)). In addition, conditional on (�a11;�

a21; rk21); LM21 and J21 are independent

because PDy21�1=21 Z1 and (Ik�PDy

21)

�1=21 Z1 are jointly normally distributed and uncorrelated

48

conditional on (�a11;�

a21; rk21):

In sum, conditional on (�a11;�

a21; rk1); LM21 � �2p2 ; J21 � �2k�p2 ; LM21 and J21 are inde-

pendent, and, hence, ARy21 = LM21 + J21 � �2p2 + �2k�p2 ; where �

2p2 and �

2k�p2 are independent.

Thus, the conditional distribution of QLR121 (de�ned in (17.9)) given (�a11;�

a21; rk21) with

rk21 = r is the same as that of

QLR1(r; 1) :=1

2

��2p2 + �

2k�p2 � r +

q(�2p2 + �

2k�p2 � r)

2 + 4�2p2r�

(17.26)

for all conditioning values of (�a11;�

a21); which is absolutely continuous for all r � 0: Hence,

QLR121 is absolutely continuous conditional on (�a11;�

a21; rk21) a.s., which completes the proof

of (17.24).

Using the result in (17.26), we obtain

P (QLR121 > cQLR1(1� �; r21; 1)) = E[P (QLR121 > cQLR1(1� �; r21; 1)j�a11;�

a21; r21)]

= E[P (QLR1(r21; 1) > cQLR1(1� �; r21; 1)jr21)]

= �; (17.27)

where the last equality holds because cQLR1(1 � �; r; 1) is the 1 � � quantile of the distributionof QLR1(r; 1); see (??), and QLR1(r; 1) is absolutely continuous for all r � 0: This and (17.24)

establish part (d) when limn1=2��2n <1: �

Proof of Lemma 17.2. Let �n(�1) denote LM2n(�1); b�1=2n (�1) bD2n(�1); WIn(�1); or ARy2n(�1);and let �n denote LM2n; b�1=2n

bD2n; WIn; or ARy2n: For each de�nition of �n(�1) and �n; we have:for all K <1;

sup�12B(�1�n;K=n1=2)

jj�n(�1)��njj = op(1) (17.28)

by Lemma 16.2(b) for �n(�1) = LM2n(�1); by Lemma 16.2(a) for �n(�1) = b�1=2n (�1) bD2n(�1);by (16.20) and (16.21) for �n(�1) = WIn(�1); and for �n(�1) = AR

y2n(�1) by the combination of

Lemma 15.2(c), an analogous result to Lemma 15.2(c) with bP1n(�1) in place of cM1n(�1); and the

result for �n(�1) =WIn(�1):

First, we prove part (b) of the lemma. By an analogous proof to that of condition (II) in (15.13)

with b�22sn(�1) in place of bn(�1) (using Assumptions SS2QLR1(i) and (ii) in place of AssumptionsSS2AR(x) and (xi)), we have

sup�12B(�1�n;K=n1=2)

jb�22sn(�)� b�22snj = op(1) 8s = 1; :::; p2: (17.29)

49

Using this, the de�nition of b�2n(�) in (??), and Assumptions SS2QLR1(i)�(iii) (which imply thatsup�12B(�1�n;K=n1=2) b�22sn(�1) = Op(1) and sup�12B(�1�n;K=n1=2) 1=b�2sn(�1) = Op(1)); we obtain

sup�12B(�1�n;K=n1=2)

jjb�2n(�)� b�2njj = op(1): (17.30)

Combining (17.30), (17.28) with �n(�1) = b�1=2n (�1) bD2n(�1); and the de�nition of rk2n(�1) in (??)gives

sup�12B(�1�n;K=n1=2)

jrk2n(�1)� rk2nj = op(1): (17.31)

Equation (17.31), the result of (17.28) with �n(�1) =WIn(�1); and the continuity of cQLR1(1��; r; w) in r � 0 for all � 2 (0; 1) and w = 0 combine to establish part (b) of the lemma.

The result of part (a) of the lemma follows from (17.31), (17.28) with �n(�1) = ARy2n(�1) and

�n(�1) = LM2n(�1); and the functional form of QLR12n(�1) in (??). �

18 Amalgamation of High-Level Conditions

In this section, we amalgamate results given in the preceding sections of the SM for the two-step

AR/AR, AR/LM, and AR/QLR1 tests.

18.1 Amalgamation Results for the AR/AR Test

The following theorem provides high-level (HL) su¢ cient conditions for Assumptions B and C

to hold for the two-step AR/AR test. This theorem amalgamates the results of Lemmas 12.1, 12.2,

13.1, 14.1, 15.1, and 15.2 for the two-step AR/AR test.

Assumption HL1AR=AR. For the null sequence S; for some " > 0 and 8K <1; (i) sup�12B(�1�n;")jjbgn(�1)�gn(�1)jj = op(1) for fgn(�) : n � 1g de�ned in (14.1), (ii) gn = 0k 8n � 1; (iii) �1�n ! �1�1

for some �1�1 2 �1; (iv) bgn(�1) is twice continuously di¤erentiable on B(�1�n; ") (for all samplerealizations) 8n � 1; (v) n1=2(bg0n; vec( bG1n � EFn bG1n)0)0 !d (Z

01; Z

0G11)

0 � N(0(p1+1)k; V11) for

some variance matrix V11 2 R(p1+1)k�(p1+1)k whose �rst k rows are denoted by [1 : �011] for

1 2 Rk�k and �11 2 R(p1k)�k; (vi) sup�12B(�1�n;") jj bG1n(�1) � G1n(�1)jj = op(1) for fG1n(�) :n � 1g de�ned in (14.1), (vii) sup�12B(�1�n;") jjG1n(�1)jj = O(1); (viii) sup�12B(�1�n;"n) jjG1n(�1) �G1njj = o(1) for all sequences of positive constants "n ! 0; (ix) G1n ! G11 for some matrix

G11 2 Rk�p1 ; (x) EFnb�1n = O(1); where b�1n := maxs;u�p1 sup�12B(�1�1;") jj(@2=@�1s�1u)bgn(�1)jj;(xi) sup�12B(�1�n;K=n1=2) jjbn(�1) � n(�1)jj !p 0 for fn(�) : n � 1g de�ned in (14.1), (xii)sup�12B(�1�n;K=n1=2) jjn(�1) � njj ! 0; (xiii) lim infn!1 �min(n) > 0; (xiv) n ! 1 for 1

50

as in condition (v), (xv) sup�12B(�1�n;K=n1=2) jjb�1n(�1) � �1n(�1)jj = op(1) for f�1n(�) : n � 1gde�ned in (14.2), (xvi) sup�12B(�1�n;K=n1=2) jj�1n(�1) � �1njj ! 0; (xvii) �1n ! �11 for �11 as

in condition (v), (xviii) limn1=2�1sn 2 [0;1] exists 8s � p1; (xix) C1n ! C11 for some matrix

C11 2 Rk�k; (xx) B1n ! B11 for some matrix B11 2 Rp1�p1 ; (xxi) cn ! 0 for fcn : n � 1g in(??), (xxii) ncn ! 1; (xxiii) sup�12B(�1�n;K=n1=2) jb�21sn(�1) � �21sn(�1)j !p 0 for f�21sn(�) : n � 1g8s = 1; :::; p1 de�ned in (14.3), (xxiv) sup�12B(�1�n;K=n1=2) j�

21sn(�1) � �21snj ! 0 8s = 1; :::; p1; and

(xxv) lim infn!1 �21sn > 0 8s = 1; :::; p1:

Assumption W. For the null sequence S; (i) cW1n is symmetric and psd, and (ii) cW1n !p W11

for some nonrandom nonsingular matrix W11 2 Rk�k:

Theorem 18.1 Suppose bgn(�1) are moment conditions, bD1n(�) is de�ned in (??), cM1n(�1) is

de�ned in (??) with a > 0; bQn(�) is the GMM criterion function de�ned in (??), CS1n is the

�rst-step AR CS CSAR1n ; �2n(�1; �) is the second-step C(�)-AR test �AR2n (�1; �); b�2n(�1) is de�ned

in (??)�(??), and p1 < k: Let S be a null sequence (or Sm a null subsequence) that satis�es

Assumption HL1AR=AR:

(a) Suppose, in addition, the sequence S (or subsequence Sm) is such that lim ��1n < KL (where

KL <1 appears in the de�nition of b�2n(�1) in (??)). Then, Assumption B holds for the sequenceS (or subsequence Sm).

(b) Suppose, in addition (to the conditions stated before part (a)), the sequence S (or subse-

quence Sm) is such that lim infn!1 �1n > 0 and Assumption W holds. Then, Assumption C holds

for the sequence S (or subsequence Sm).

The following lemma provides high-level su¢ cient conditions for Assumption OE to hold for a

sequence S for the two-step AR/AR test. This lemma amalgamates the results of Lemmas 12.3,

12.4, 13.2, and 14.2.

Assumption HL2AR=AR. For the null sequence S; (i) lim infn!1 ��1n > KU (for KU > 0 as in the

de�nition of b�2n(�1) in (??)), (ii) sup�12�1 n1=2jjbgn(�1)�gn(�1)jj = Op(1) for fgn(�) : n � 1g de�nedin (14.1), (iii) limn!1 inf�1 =2B(�1�n;Kn=n1=2)

n1=2jjgn(�1)jj = 1 for all sequences Kn ! 1; (iv)lim infn!1 inf�1 =2B(�1�n;") jjgn(�1)jj > 0 8" > 0; (v) sup�12�1 jjbn(�1)�n(�1)jj = op(1) for fn(�) :n � 1g de�ned in (14.1), (vi) sup�12�1 jjn(�1)jj = O(1); and (vii) lim infn!1 inf�12�1 �min(n(�1))> 0:

Lemma 18.2 Suppose bgn(�1); bD1n(�); cM1n(�1); CS1n; �2n(�1; �); and b�2n(�1) are as in Theorem18.1, a > 0; and p1 < k: Let S be a null sequence (or Sm a null subsequence) that satis�es

Assumptions HL1AR=AR; HL2AR=AR; and W. Then, Assumption OE holds for the sequence S (or

subsequence Sm).

51

18.2 Amalgamation Results for the AR/LM and AR/QLR1 Tests

The following theorem provides high-level su¢ cient conditions for Assumptions B and C to

hold for the two-step AR/LM and AR/QLR1 tests. For the two-step AR/LM test, this theorem

amalgamates the results of Lemmas 12.1, 12.2, 13.1, 14.1, 16.1, and 16.2. For the AR/QLR1 test,

it amalgamates the results of Lemmas 12.1, 12.2, 13.1, 14.1, 17.1, and 17.2.

Assumption HL1AR=LM. For the null sequence S; 8K <1; (i) bgn(�1; �2) is di¤erentiable in �2at �20 and (@=@�02)bgn(�1; �2) is di¤erentiable in �1 with both holding 8�1 2 B(�1�n; ") (for all sam-ple realizations), 8n � 1; for some " > 0; (ii) n1=2(bg0n; vec( bG1n � EFn bG1n)0; vec( bG2n � EFn bG2n)0)0!d (Z

01; Z

0G11; Z

0G21)

0 � N(0(p+1)k; V1) for some variance matrix V1 2 R(p+1)k�(p+1)k whose

�rst k rows are denoted by [1 : �011 : �021] for 1 2 Rk�k and �j1 2 R(pjk)�k for j = 1; 2;

(iii) sup�12B(�1�n;K=n1=2) jj bG2n(�1) � G2n(�1)jj !p 0 for fG2n(�) : n � 1g de�ned in (14.1), (iv)sup�12B(�1�n;K=n1=2) jjG2n(�1) � G2njj ! 0; (v) G2n ! G21 for some matrix G21 2 Rk�p2 ; (vi)sup�12B(�1�n;K=n1=2) jj(@

2=@�1s@�02)bgn(�1; �20)jj = Op(1) for s = 1; :::; p1; (vii) sup�12B(�1�n;K=n1=2)

jjb�2n(�1) � �2n(�1)jj = op(1) for f�2n(�) : n � 1g de�ned in (14.2), (viii) sup�12B(�1�n;K=n1=2)jj�2n(�1)� �2njj ! 0; (ix) �2n ! �21 for �21 as in condition (ii), (x) limn1=2�2sn 2 [0;1] exists8s � p2 (where �2sn is de�ned in the paragraph containing (10.2), (xi) C2n ! C21 for some matrix

C21 2 Rk�k; (xii) B2n ! B21 for some matrix B21 2 Rp2�p2 ; (xiii) sup�12B(�1�n;K=n1=2) jb�22sn(�1)��22sn(�1)j !p 0 for f�22sn(�) : n � 1g 8s = 1; :::; p2 de�ned in (17.1), (xiv) sup�12B(�1�n;K=n1=2) j�

22sn(�1)

� �22snj ! 0 8s = 1; :::; p2; and (xv) �2jsn ! �2js1 for some constant �2js1 2 (0;1) 8s =1; :::; p2;8j = 1; 2:

Assumption HL1AR=QLR1: For the null sequence S; (i) limn1=2��2sn 2 [0;1] exists 8s � p2

(where ��2sn := ��2sn(�1�n; �20) is de�ned in the paragraph containing (17.2), (ii) C�2n ! C�21 for

some matrix C�21 2 Rk�k; and (iii) B�2n ! B�21 for some matrix B�21 2 Rp2�p2 :

Lemma 18.3 Suppose the conditions in Theorem 18.1 hold except that �2n(�1; �) is the second-

step C(�)-LM test �LM2n (�1; �) or the second-step C(�)-QLR1 test �QLR12n (�1; �); b�2n(�1) is de�ned

accordingly in (??)�(??), and p1 < k is replaced by p2 � 1 for the C(�)-LM test and by p2 � 1 andp � k for the C(�)-QLR1 test. Let S be a null sequence (or Sm a null subsequence) that satis�es

Assumptions HL1AR=AR and HL1AR=LM and, for the second-step C(�)-QLR1 test, Assumption

HL1AR=QLR1 as well.

(a) Suppose, in addition, the sequence S (or subsequence Sm) is such that lim ��n < KL (where

KL <1 appears in the de�nition of b�2n(�1) in (??)). Then, Assumption B holds for the sequenceS (or subsequence Sm).

52


quence Sm) is such that lim infn!1 ��n > K�U and Assumption W holds. Then, Assumption C holds


Comment: When LM2n(�) is the pure C(�)-LM statistic, i.e., WIn(�) := 0; Lemma 18.3(a) holds

provided conditions (vi) and (vii) in Comment (v) to Lemma 16.1 hold, and Lemma 18.3(b) holds

with the weaker condition lim infn!1 �n > 0 in place of lim infn!1 ��n > K�U : The same is true

when the QLR12n(�) is the pure C(�)-QLR1 statistic and WIn(�) := 0 in the QLR1 critical value

function.


sequence S for the two-step AR/LM and AR/QLR1 tests. This lemma amalgamates the results of

Lemmas 12.3, 12.4, 13.2, and 14.2 for these tests.

Assumption HL2AR=LM;QLR1. Assumption HL2AR=AR holds with ��n in place of ��1n in part (i).

Lemma 18.4 Suppose bgn(�1); bD1n(�); cM1n(�1); CS1n; and fcn : n � 1g are as in Theorem 18.1,

�2n(�1; �) is the second-step C(�)-LM �LM2n (�1; �) or C(�)-QLR1 test �QLR12n (�1; �); b�2n(�1) is

de�ned accordingly in (??)�(??), a > 0; and p2 � 1: Let S be a null sequence (or Sm a null

subsequence) that satis�es Assumptions HL1AR=AR; HL1AR=LM ; HL2AR=LM;QLR1; and W. Then,

Assumption OE holds for the sequence S (or subsequence Sm).

Comment: Lemma 18.4 di¤ers from Lemma 18.2 because the second-step data-dependent signif-

icance level di¤ers between the second-step C(�)-AR test, which is considered in the latter lemma,

and the second-step C(�)-LM and C(�)-QLR1 tests, which are considered in the former lemma.

18.3 Proofs of Theorem 18.1 and Lemmas 18.2, 18.3, and 18.4

Proof of Theorem 18.1. Assumption B(i) holds by Lemma 13.1, which employs Assumption

FS1AR; because Assumption HL1AR=AR(xi)) Assumption FS1AR(i); HL1AR=AR(xiv)) FS1AR(ii);

HL1AR=AR(v) ) FS1AR(iii); and HL1AR=AR(xiii) ) FS1AR(iv).

Assumptions B(ii), C(ii), and C(iii) hold by Lemma 15.1 (and Comment (i) following it),

which employs Assumption SS1AR; because a > 0; p1 < k; Assumption HL1AR=AR(xviii) ) As-

sumption SS1AR(i); HL1AR=AR(v) ) SS1AR(ii); HL1AR=AR(xiii) & HL1AR=AR(xiv) ) SS1AR(iii);

HL1AR=AR(xv) & HL1AR=AR(xvii) ) SS1AR(iv); HL1AR=AR(xi) ) SS1AR(v); HL1AR=AR(xiv) )SS1AR(vi); HL1AR=AR(xix) ) SS1AR(vii), and HL1AR=AR(xx) ) SS1AR(viii).

Assumption B(iii) holds under the conditions of Theorem 18.1(a) by Lemma 14.1, which em-

ploys Assumption SL1AR, because the assumption lim ��1n < KL of Theorem 18.1(a) ) Assump-

tion SL1AR(i); Assumption HL1AR=AR(vi)) Assumption SL1AR(ii); HL1AR=AR(vii)) SL1AR(iii);

53

HL1AR=AR(xi) ) SL1AR(iv); HL1AR=AR(xiii) ) SL1AR(v); HL1AR=AR(xiv) ) SL1AR(vi);

HL1AR=AR(xxiii) ) SL1AR(vii); and HL1AR=AR(xxv) ) SL1AR(viii).

Assumption C(i) holds under the conditions of Theorem 18.1(b) by Lemma 12.1, which employs

Assumptions ES1 and ES2, because the result of Lemma 12.2 ) Assumption ES1 and Lemma

12.2 applies here because Assumption FOC is veri�ed below; Assumption HL1AR=AR(iv)) ES2(i);

Assumption HL1AR=AR(v)) ES2(ii); HL1AR=AR(vi) & HL1AR=AR(vii)) ES2(iii); HL1AR=AR(xxii)

) ES2(iv), and Assumption W ) ES2(v).

For the veri�cation of Assumption C(i) under the conditions of Theorem 18.1(b), it remains

to show that Assumption FOC, which is employed in Lemma 12.2, holds. We have: the assump-

tion lim infn!1 �1n > 0 of Theorem 18.1(b) ) Assumption FOC(i); Assumption HL1AR=AR(i)

) FOC(ii); HL1AR=AR(ii) ) FOC(iii); HL1AR=AR(iii) ) FOC(iv); HL1AR=AR(iv) ) FOC(v);

HL1AR=AR(v)) FOC(vi); HL1AR=AR(vi)) FOC(viii); HL1AR=AR(vii)) FOC(ix); HL1AR=AR(viii)

) FOC(x); HL1AR=AR(ix) ) FOC(xi); HL1AR=AR(xiii) ) FOC(xiii); Markov�s inequality and

HL1AR=AR(x) ) FOC(xiv); and Assumption W ) FOC(xvi). In addition, because we are consid-

ering the moment condition model here, by the paragraph following Assumption FOC, Assumptions

FOC(vii), (xii), and (xv) are implied by Assumptions FOC(v) and (xiv), which have just been ver-

i�ed, and Assumption HL1AR=AR(x).

As noted in Section 15 above, Assumption C(iv) holds automatically for the second-step AR

test provided p1 < k (which is assumed here) because its nominal level � critical value is the 1� �quantile of the �2k�p1 distribution which is nondecreasing in � for � 2 (0; 1) when p1 < k:

Assumption C(v) holds under the conditions of Theorem 18.1(b) by Lemma 15.2(d), which

employs Assumption SS2AR; because the assumption lim infn!1 �1n > 0 of Theorem 18.1(b)

) Assumption SS2AR(i); Assumption HL1AR=AR(iv) ) Assumption SS2AR(ii); HL1AR=AR(v) )SS2AR(iii); HL1AR=AR(vi) ) SS2AR(iv); HL1AR=AR(vii) ) SS2AR(v); Markov�s inequality and

HL1AR=AR(x) ) SS2AR(vi); HL1AR=AR(xv) ) SS2AR(vii); HL1AR=AR(xvi) ) SS2AR(viii);

HL1AR=AR(xvii) ) SS2AR(ix); HL1AR=AR(xi) ) SS2AR(x); HL1AR=AR(xii) ) SS2AR(xi);

HL1AR=AR(xiii) ) SS2AR(xii); and HL1AR=AR(xiv) ) SS2AR(xiii). �

Proof of Lemma 18.2. As required by Assumption OE, Assumption C holds for the se-

quence S by Theorem 18.1(b), using Assumptions HL1AR=AR and HL2AR=AR(i). Note that the

condition lim infn!1 �1n > 0 of Theorem 18.1(b) is implied by Assumption HL2AR=AR(i) (i.e.,

lim infn!1 ��1n > KU for KU > 0) plus Assumption HL1AR(xxv), which implies that lim supn!1

�max(�1n) <1; where �1n := �1n(�1�n; �20) is de�ned in (??) and (14.3).By Lemma 12.3, the results of Lemmas 12.4 and 13.2 imply that Assumption OE(i) holds.

Hence, we need to verify the assumptions used in Lemmas 12.4 and 13.2 to verify Assumption

54

OE(i).

First, we verify Assumptions ES3 and ES4, which are imposed in Lemma 12.4. Assumptions

HL2AR=AR(i) & HL1AR=AR(xxv) ) Assumption ES3(i); HL1AR=AR(iv) ) ES3(iii); HL1AR=AR(v)

) ES3(iv); HL1AR=AR(vi) ) ES3(v); HL1AR=AR(viii) ) ES3(vi); HL1AR=AR(ix) ) ES3(vii);

HL1AR=AR(xi) ) ES3(viii); HL1AR=AR(xiii) ) ES3(ix); HL1AR=AR(xxi) ) ES3(x); and W )ES3(xi). Assumption ES3(ii) holds by Lemma 12.1 under Assumptions ES1 and ES2; it is shown

in the proof of Theorem 18.1 that Assumption HL1AR=AR ) Assumption ES2; Lemma 12.2 veri�es

Assumption ES1 using Assumption FOC; and Assumption FOC is veri�ed in Theorem 18.1 (and the

present lemma imposes Assumption HL1AR=AR; which is employed in Theorem 18.1). Assumption

HL2AR=AR(ii)) Assumption ES4(i) and HL2AR=AR(iv)) ES4(ii). This completes the veri�cation

of Assumptions ES3 and ES4.

Second, we verify Assumption FS2AR; which is used in Lemma 13.2. Assumption HL2AR=AR(ii)

) Assumption FS2AR(i); HL2AR=AR(iii) ) FS2AR(ii); HL2AR=AR(v) ) FS2AR(iii); HL2AR=AR(vi)

) FS2AR(iv); and HL2AR=AR(vii) ) FS2AR(v). This completes the veri�cation of Assumption

OE(i).

Lastly, Assumption OE(ii) holds under Assumption SL2AR by Lemma 14.2. Hence, we need to

verify Assumption SL2AR. Assumption HL2AR=AR(i) ) Assumption SL2AR(i); HL1AR=AR(vi) )SL2AR(ii); HL1AR=AR(viii)) SL2AR(iii); HL1AR=AR(ix)) SL2AR(iv); HL1AR=AR(xi)) SL2AR(v);

HL1AR=AR(xii) ) SL2AR(vi); HL1AR=AR(xiii) ) SL2AR(vii); HL1AR=AR(xxiii) ) SL2AR(viii);

HL1AR=AR(xxiv) ) SL2AR(ix); and HL1AR=AR(xxv) ) SL2AR(x). This completes the proof of

the lemma. �

Proof of Lemma 18.3. For the AR/LM and AR/QLR1 tests, Assumptions B(i) and C(i) hold by

the same arguments as given in the proof of Theorem 18.1 (using the fact that lim infn!1 ��n > K�U

(> 0); which is assumed in Lemma 18.3(b), implies that lim infn!1 �1n > 0; which is assumed in

Theorem 18.1(b), when verifying Assumption C(i)).

For the AR/LM and AR/QLR1 tests, Assumption B(iii) holds under the conditions of Lemma

18.3(a) by Lemma 14.1, which employs Assumption SL1LM;QLR1 (for the second-step C(�)-LM

and C(�)-QLR1 tests), because the assumption lim ��n < KL of Lemma 18.3(a) ) Assump-

tion SL1LM;QLR1(i); Assumption HL1AR=AR ) Assumptions SL1AR(ii)�(viii), as shown in the

proof of Theorem 18.1, and the latter conditions constitute Assumption SL1LM;QLR1(ii); and

HL1AR=LM (xiii)�(xv) ) SL1LM;QLR1(iii) and (iv).

For the second-step C(�)-LM test, Assumptions B(ii), C(ii), and C(iii) hold by Lemma 16.1

and Comment (i) following it, which employs Assumptions SS1AR and SS1LM ; because Assump-

tion HL1AR=AR implies Assumption SS1AR (as shown in the proof of Theorem 18.1), and Assump-

55

tion HL1AR=LM implies Assumption SS1LM : The latter holds because Assumption HL1AR=LM (x)

) Assumption SS1LM (i); HL1AR=LM (ii) ) SS1LM (ii); HL1AR=LM (vii) & (ix) with �1 = �1�n

) SS1LM (iii); HL1AR=LM (xi) ) SS1LM (iv); HL1AR=LM (xii) ) SS1LM (v); HL1AR=AR(vi) & (ix)

and HL1AR=LM (iii) & (v) ) SS1LM (vi); and HL1AR=AR(xxiii) and HL1AR=LM (xiii) & (xv) )SS1LM (vii).

For the second-step C(�)-QLR1 test, Assumptions B(ii), C(ii), and C(iii) hold by Lemma 17.1

and Comment (i) following it because Lemma 17.1 relies on Assumptions SS1AR and SS1LM ;

which have just been veri�ed, as well as on Assumption SS1QLR1; which holds because Assumption

HL1AR=QLR1(i)�(iii) ) Assumption SS1QLR1:

For the second-step C(�)-LM test, Assumption C(iv) holds provided p2 � 1 (which is assumedhere) because its nominal level � critical value is the 1� � quantile of the �2p2 distribution which isnondecreasing in � for � 2 (0; 1) when p2 � 1:

For the second-step C(�)-QLR1 test, Assumption C(iv) holds because its conditional critical

value cQLR1(1 � �; rk2n(�1);WIn(�1)) is nondecreasing in � since cQLR1(1 � �; r; w) is the 1 � �quantile of QLR1(r; w); see (??).

For the second-step C(�)-LM test, Assumption C(v) holds under the conditions of Lemma

18.3 by Lemma 16.2(b), which employs Assumptions SS1AR; SS2AR; SS1LM ; and SS2LM ; because

Assumption HL1AR ) Assumption SS1AR; as shown above; HL1AR=LM ) SS1LM ; as shown above;

the condition lim infn!1 ��n > K�U in Lemma 18.3(b), HL1AR=AR(xxiii), and HL1AR=LM (xv) with

j = 1 ) lim infn!1 ��1n > K�U ) Assumption SS2AR(i); Assumption HL1AR ) all parts of

Assumption SS2AR except its part (i) (as shown in the proof of Theorem 18.1); the condition

lim infn!1 �n > 0 in Lemma 18.3(b), HL1AR=AR(xxiii), and HL1AR=LM (xiii) & (xv)) Assumption

SS2LM (i); HL1AR=LM (i) ) SS2LM (ii); and HL1AR=LM (iii)�(viii) ) SS2LM (iii)�(vii).

For the second-step C(�)-QLR1 test, Assumption C(v) holds under the conditions of Lemma

18.3(b) by Lemma 17.2 because Lemma 17.2 relies on Assumptions SS2AR and SS2LM ; which have

just been veri�ed above, and Assumption SS2QLR1; which holds by Assumption

HL1AR=LM (xiii)�(xv). �

Proof of Lemma 18.4. As required by Assumption OE, Assumption C holds for the sequence

S by Lemma 18.3(b), using Assumptions HL1AR=AR; HL1AR=LM ; HL2AR=LM;QLR1; and, for the

C(�)-QLR1 test, HL2AR=AR(i) as well. Note that the condition lim infn!1 ��n > K�U of Lemma

18.3(b) is implied by Assumption HL2AR=AR(i) (i.e., lim infn!1 ��n > KU ) and K�U < KL � KU

(which holds by the de�nition of the constant K�U following (??)).

The veri�cation of Assumption OE(i) is the same as in the proof of Lemma 18.2.

Assumption OE(ii) holds under Assumption SL2LM;QLR1 by Lemma 14.2. Hence, we need

56

to verify Assumption SL2LM;QLR1: Assumption HL2AR=LM;QLR1 ) Assumption SL2LM;QLR1(i);

HL1AR=AR ) SL2LM;QLR1(ii) (which consists of SL2AR(v)�(x)), as shown in the proof of Lemma

18.2 above; HL1AR=LM (iii)�(v)) SL2LM;QLR1(iii)�(v); and HL1AR=LM (xiii)�(xv)) SL2LM;QLR1(vi)�

(viii). �

19 Proof of Theorem 8.1

This section proves Theorem ?? using the results in Section 18, which, in turn, uses the results

in Sections 12-14.

19.1 Proof of Theorem ??

To prove Theorem ?? we �nd it useful to reparametrize (�1; F ) with parameter space FAR=AR;de�ned in (??), to a parameter � with parameter space �AR=AR: The parameter � is chosen such

that for some subvector of � convergence of a drifting subsequence of the subvector allows one to

verify Assumption HL1AR=AR; which is employed in Theorem 18.1, and Assumption HL2AR=AR;

which is employed in Lemma 18.2.

Let fhn(�) : n � 1g be a sequence of functions on a space �: The parameter � and functionhn(�) are of the following form:

(i) � = (�1; :::; �d; �d+1)0; where �j 2 R 8j � d and �d+1 belongs to some in�nite-dimensional

pseudo-metric space,4 and

(ii) hn(�) = (hn;1(�); :::; hn;J(�))0 and

hn;j(�) =

8<: n1=2�j for j = 1; :::; JR

�j for j = JR + 1; :::; J;for some JR � d: (19.1)

De�ne

H = fh 2 (R [ f�1g)J : hmn(�mn)! h for some subsequence fmng

of fng and some sequence f�mn 2 � : n � 1gg: (19.2)

The result in the following lemma is established in the proof of Theorem 2.2 in Andrews, Cheng,

and Guggenberger (2011). For completeness, we provide a proof below.

4For notational simplicity, we stack d real-valued quantities and one in�nite-dimensional quantity into the vector�:

57

Lemma 19.1 For any sequence f�n 2 � : n � 1g and any subsequence fwng of fng there exists asubsequence fmng of fwng such that hmn(�mn)! h for some h 2 H:

Comment: Lemma 19.1 is useful in establishing the correct asymptotic size of any two-step test,

not just the two-step AR/AR test.

Now, we specify � and � that are used with the two-step AR/AR test.

Let F (�) := V arF (gi(�)): We write a SVD of �1=2F (�1)EFG1i(�1) as

�1=2F (�1)EFG1i(�1) = C1F (�1)�1F (�1)B1F (�1)

0; (19.3)

where C1F (�1) 2 Rk�k and B1F (�1) 2 Rp1�p1 are orthogonal matrices and �1F (�1) 2 Rk�p1 has thesingular values �11F (�1); :::; �1p1F (�1) of

�1=2F (�1)EFG1i(�1) in nonincreasing order on its diagonal

and zeros elsewhere.

Let ��1p1F (�1) denote the smallest singular value of �1=2F (�1)EFG1i(�1)�1F (�1); where �1F (�1)

is de�ned in (??).

We de�ne the elements of � to be5

�1;�1;F := (�11F (�1); :::; �1p1F (�1))0 2 Rp1 ;

�2;�1;F := B1F (�1) 2 Rp1�p1 ;

�3;�1;F := C1F (�1) 2 Rk�k;

�4;�1;F := EFG1i(�1) 2 Rk�p1 ;

�5;�1;F := EF

0@ gi(�1)

vec(G1i(�1)� EFG1i(�1))

1A0@ gi(�1)

vec(G1i(�1)� EFG1i(�1))

1A0 2 R(p1+1)k�(p1+1)k;�6;�1;F := �1;

�7;�1;F := (�1p1F (�1); ��1p1F (�1))

0

�8;�1;F := F; and

� = ��1;F := (�1;�1;F ; :::; �8;�1;F ): (19.4)

We let �5;g;�1;F denote the upper left k� k submatrix of �5;�1;F: Thus, �5;g;�1F = EF gi(�1)gi(�1)0 =F (�1) for (�1; F ) 2 FAR=AR:

We consider the parameter space �AR=AR for � that corresponds to FAR=AR: The parameter5For simplicity, when writing � = (�1;F ; :::; �8;F ); we allow the elements to be scalars, vectors, matrices, and

distributions and likewise in similar expressions.

58

space �AR=AR and the function hn(�) are de�ned by

�AR=AR := f� : � = (�1;�1;F ; :::; �8;�1;F ) for some (�1; F ) 2 FAR=ARg and

hn(�) := (n1=2�1;�1;F ; �2;�1F ; �3;�1;F ; �4;�1;F ; �5;�1;F ; �6;�1;F ; �7;�1;F ): (19.5)

By the de�nition of FAR=AR; �AR=AR indexes distributions that satisfy the null hypothesis H0 :�2 = �20: Redundant elements in (�1;�1;F ; :::; �8;�1;F ); such as the redundant o¤-diagonal elements

of the symmetric matrix �5;�1;F ; are not needed, but do not cause any problem. The dimension J

of hn(�) equals the number of elements in (�1;F ; :::; �7;F ):

We de�ne � and hn(�) as in (19.4) and (19.5) because, as shown below, verifying Assump-

tion HL1AR=AR; which is employed in Theorem 18.1, and Assumptions HL2AR=AR; which is em-

ployed in Lemma 18.2, for subsequences requires convergence of the corresponding subsequences of

n1=2�1;�1n;Fn and �j;�1n;Fn for j = 2; :::; 7:

For notational convenience,

f�n;h : n � 1g denotes a sequence f�n 2 �AR=AR : n � 1g for which hn(�n)! h 2 H (19.6)

for H de�ned in (19.2) with � equal to �AR=AR:6 By the de�nitions of �AR=AR and FAR=AR;f�n;h : n � 1g is a sequence of distributions that satis�es the null hypothesis H0 : �2 = �20: Below,�all sequences f�wn;h : n � 1g�means �all sequences f�wn;h : n � 1g for any h 2 H;�where His de�ned with � equal to �AR=AR; and likewise with n in place of wn: To maintain the notation

employed above that �1�n denotes the true value of �1; we let f(�1�n; Fn) : n � 1g denote thesequence of (�1; F ) values in FAR=AR that corresponds to f�n;h : n � 1g:

We decompose h (de�ned by (19.2), (19.4), and (19.5)) analogously to the decomposition of

�: h = (h1; :::; h7); where �j;�1;F and hj have the same dimensions for j = 1; :::; 7: We further

decompose the vector h1 as h1 = (h1;1; :::; h1;minfk;p1g)0; where elements of h1 could equal 1: In

addition, we let h5;g denote the upper left k� k submatrix of h5: In consequence, under a sequencef�n;h : n � 1g; we have

n1=2�1sFn(�1�n) ! h1;s � 0 8s � p1;

�j;�1�n;Fn ! hj 8j = 2; :::; 7; and

�5;g;�1�n;Fn = Fn(�1�n) = V arFn(gi(�1�n))! h5;g: (19.7)

6Analogously, for any subsequence fwn : n � 1g; f�wn;h : n � 1g denotes a sequence f�wn 2 �AR=AR : n � 1g forwhich hwn(�wn)! h 2 H:

59

By the conditions in FAR=AR; h5;g is pd.The following lemma veri�es Assumptions HL1AR=AR; which is employed in Theorem 18.1 and

Lemma 18.2, for all subsequences f�wn;h : n � 1g:

Lemma 19.2 Suppose bgn(�1) are the moment functions de�ned in (??), gi(�) satis�es the di¤er-entiability condition in Theorem ??, fcn : n � 1g are as in Theorem ??, a > 0; p1 < k; � is open,

�1� is bounded, and B(�1�; ") � �1 for some " > 0: Let the null parameter space be FAR=AR: Then,for all subsequences f�wn;h : n � 1g; Assumption HL1AR=AR holds and lim �1n and lim ��1n exist.

The following lemma veri�es Assumption HL2AR=AR:

Lemma 19.3 Suppose bgn(�1) are the moment functions de�ned in (??), gi(�) satis�es the di¤eren-tiability condition in Theorem ??, and the null parameter space is FAR=AR: Let S be a null sequence(or Sm a null subsequence) for which Assumption SI holds. Then, Assumption HL2AR=AR holds

for the sequence S (or the subsequence Sm).

Now, we prove Theorem ?? using Theorems ?? and 18.1 and Lemmas 18.2 and 19.1-19.3.

Proof of Theorem ??. The result of Theorem ??(a) follows from the high-level result Theorem

??(a) provided Assumption CAL holds. Assumption CAL requires that for any null sequence

S and any subsequence fwng of fng; there exists a subsubsequence fmng such that Sm satis�es

Assumption B or C. Theorem 18.1 provides high-level conditions under which Assumption B or C

holds for a subsequence Sm: The condition required for Theorem 18.1 is Assumption HL1AR=AR:

By Lemma 19.1, for any null sequence S or, equivalently, any sequence f�n 2 � : n � 1g;and any subsequence fwng of fng; there exists a subsubsequence fmng such that hmn(�mn) ! h

for some h 2 H: By Lemma 19.2, for the subsequence fmng (that satis�es hmn(�mn) ! h for

some h 2 H); Assumption HL1AR=AR holds and lim �1n and lim ��1n exist. Given this, by Theorem18.1(a) and (b), the subsequence Sm satis�es Assumption B when lim ��1mn

< KL and it satis�es

Assumption C when lim infn!1 �1mn > 0 and Assumption W holds, which is assumed.

By de�nition, ��1n is the smallest singular value of �1=2n G1n�1n; see (??), and �1n :=

Diagf��111n; :::; ��11p1n

g; where �21sn := V arFn(jjG1sijj); see (14.3). Given these de�nitions and thecondition V arF (jjG1si(�1)jj) � � for (�1; F ) 2 FAR=AR; see (??), we have: lim ��1mn

� KL implieslim infn!1 �1mn > 0: Hence, every subsequence Sm with lim �

�1mn

< KL satis�es Assumption B and

every subsequence Sm with lim ��1mn� KL satis�es Assumption C. This completes the veri�cation

of Assumption CAL and the proof of Theorem ??(a).

Now we prove Theorem ??(b). The result of Theorem ??(b) that AsyNRP = � holds for

a sequence S if for any subsequence fwng of fng; there exists a subsubsequence fmng such

60

that limP��mn ;Fmn (�SV2mn

(�) > 0)) = � for the corresponding subsubsequence Sm; where ��mn =

(�01�mn; �020)

0: Take the subsubsequence fmng as above to be such that hmn(�mn) ! h for some

h 2 H: Then, by Lemma 19.2, for the subsubsequence Sm; Assumption HL1AR=AR holds.The limP��mn ;Fmn (�

SV2mn

(�) > 0)) = � result for the subsubsequence Sm follows from the high-

level result Theorem ??(c) provided Assumption OE holds for the subsubsequence Sm: Assumption

OE holds for the subsubsequence Sm by Lemma 18.2. The conditions required for Lemma 18.2

are Assumptions HL1AR=AR; HL2AR=AR; and W. Assumption HL1AR=AR is veri�ed in a previous

paragraph. In addition, Assumption W is assumed to hold. Hence, to establish Theorem ??(b),

it remains to verify Assumption HL2AR=AR: Using Assumption SI, which is imposed in Theorem

??(b), these conditions hold by Lemma 19.3. This completes the proof of Theorem ??(b).

Theorem ??(c) follows immediately from Theorem ??(a) and (b).

Theorem ??(d) and (e) hold by Theorem ??(d) and (e), respectively, because a sequence S that

satis�es Assumption SI is shown above to satisfy Assumption OE.

To establish Theorem ??(f), we use the high-level CS results given in Theorem ??(f), rather

than the high-level test results given in Theorem ??(a)�(e). This requires verifying the CS versions

of Assumptions B, C, CAL, and OE, rather than the test versions. The only di¤erence between

the CS and test versions is that �2�n appears throughout in place of �20: The veri�cation of the CS

versions these conditions is the same as given above but with some adjustments.

First, we adjust the de�nition of � that appears in (19.4) and (19.5). Speci�cally, we de�ne �

as in (19.4), but with � in place of �1 throughout. We retain the de�nition of hn(�) given in (19.5),

but with � in place of �1 in �:

Second, we adjust the parameter space �AR=AR; which appears in (19.5), to the following

parameter space:

��;AR=AR := f� : � = (�1;�;F ; :::; �9;�;F ) for some (�; F ) 2 F�;AR=ARg; (19.8)

where F�;AR=AR is de�ned in (??) using the test parameter space FAR=AR(�2) for �2 2 �2�: Notethat the moment conditions in FAR=AR(�2) hold uniformly over �2 2 �2� by the de�nitions of

F�;AR=AR: For example, EF jjgi(�1)jj2+ = EF jjgi(�1; �20)jj2+ � M (by the de�nition of FAR;ARin (??)) 8�20 2 �2� (by the de�nition of F�;AR=AR) implies that sup�22�2� EF jjgi(�1; �2)jj2+ �M: This is used in the adjusted proofs everywhere the moment conditions are employed in the

unadjusted proofs.

Third, we use the assumption that �� is bounded and B(��; ") � � for some " > 0 to ensure

that �6;�n;Fn = �n has a limit in � � Rp; call it �1; for all sequences f�n;h : n � 1g (rather

61

than a limit whose elements might equal �1): The assumption that B(��; ") � � guarantees thatthe mean-value expansions that appear in (12.3), (12.6)�(12.9), (12.19), (15.12), (15.14), (19.12),

(19.13), (19.18), (19.22, (19.37), and (19.41) also hold when �20 is replaced by �2�n:

Given the adjustments above, the results in Lemmas 12.1�12.4, 13.1, 13.2, 15.1, 15.2, 14.1, 14.2,

18.1, and 18.2 hold with the null sequence S replaced by a sequence S which has �20 replaced by

�2�n:

Furthermore, Lemmas 19.2 and 19.3 hold with the adjustments to � and �AR=AR stated imme-

diately above. In consequence, the veri�cation of Assumptions B, C, CAL, and OE given above

goes through when �2�n appears throughout in place of �20: This completes the proof of Theorem

??(f). �


Proof of Lemma 19.1. Let fwng be some subsequence of fng: Let hwn;j(�wn) denote the jthcomponent of hwn(�wn) for j = 1; :::; J: Let m1;n = wn 8n � 1: For j = 1; either (1) lim supn!1jhmj;n;j(�mj;n

)j <1 or (2) lim supn!1 jhmj;n;j(�mj;n)j =1: If (1) holds, then for some subsequence

fmj+1;ng of fmj;ng;hmj+1;n;j(�mj+1;n)! hj for some hj 2 R: (19.9)

If (2) holds, then for some subsequence fmj+1;ng of fmj;ng;

hmj+1;n;j(�mj+1;n)! hj ; where hj =1 or �1: (19.10)

Applying the same argument successively for j = 2; :::; J yields a subsequence fmng = fmJ+1;ng offwng for which hmn;j(�m�

n)! hj 8j � J; which establishes the result of the Lemma. �

Proof of Lemma 19.2. For notational simplicity, we prove the result for a sequence f�n;h : n � 1g:The same arguments go through with n replaced by wn to obtain the subsequence results that are

stated in the lemma.

We do not verify Assumptions HL1AR=AR(i), HL1AR=AR(ii), ... in numerical order because some

of these conditions are used in the veri�cation of others. For brevity, we abbreviate Assumptions

HL1AR=AR(i), HL1AR=AR(ii), ... by Assumptions (i), (ii), ....

Assumption (iv) requires that bgn(�1) is twice continuously di¤erentiable on B(�1�n; ") 8n � 1for some " > 0: Assumption (iv) holds because the present lemma imposes the di¤erentiability

condition in Theorem ?? (which states that gi(�1) is twice continuously di¤erentiable in �1 on �1

for all sample realizations), and B(�1�n; ") � �1 8n � 1 for some " > 0 because �1�n 2 �1� by the

62

de�nition of F and B(�1�; ") � �1 by an assumption of the lemma.Assumption (i) requires that sup�12B(�1�n;") jjbgn(�1) � gn(�1)jj = op(1) for some nonrandom

functions fgn(�) : n � 1g and some " > 0: We verify Assumption (i) with gn(�1) = EFnbgn(�1): Thiscondition is a uniform WLLN. Because �1�n ! �1�1 2 �1 by (19.31) below, it su¢ ces to establishthis result with B(�1�1; ") in place of B(�1�n; "): Since B(�1�1; ") is a bounded set, su¢ cient

conditions for this uniform WLLN�s are

(a) bgn(�1)� gn(�1) = op(1) 8�1 2 B(�1�1; ") and(b) sup

�a2B(�1�1;")sup

�12B(�a;"n)jjbgn(�1)� gn(�1)� (bgn(�a)� gn(�a))jj = op(1) (19.11)

for all sequences of constants f"n > 0 : n � 1g for which "n ! 0; e.g., see Theorem 1(a) of Andrews

(1991a). Conditions (a) and (b) are pointwise WLLN�s and stochastic equicontinuity, respectively.

Condition (b) of (19.11) is established as follows. In the veri�cation of condition (b) we assume

k = 1 for notational simplicity and without loss of generality (wlog) (because the veri�cation can be

done separately for each element of bgn(�)�gn(�)): Consider any �a 2 B(�1�1; ") and �1 2 B(�a; "n):Element-by-element two-term Taylor expansions of gi(�1) about �a give

gi(�1) = gi(�a) +G1i(�a)(�1 � �a) +p1Xj=1

(�1j � �aj)@

@�1jG1i(e�1i)(�1 � �a); (19.12)

where �a = (�a1; :::; �ap1)0 and e�1i lies between �1 and �a: Element-by-element mean-value expan-

sions of G1i(�a) about �1�n give

G1i(�a) = G1i +

p1Xj=1

(�aj � �1�nj)@

@�1jG1i(�1i); (19.13)

where �1�n = (�1�n1; :::; �1�np1)0 and �1i lies between �a and �1�n and may di¤er across the columns

of (@=@�1j)G1i(�1i): Equation (19.13) uses the assumption of Theorem ??, which is imposed in this

lemma, that gi(�1) is twice continuously di¤erentiable in �1 on �1 and G1i(�a) := (@=@�01)gi(�a):

Substituting (19.13) into (19.12) and taking expectations gives

sup�a2B(�1�1;")

sup�12B(�a;"n)

jjgn(�1)� gn(�a)jj

� jjG1njj � jj�1 � �ajj+ EFn�1ijj�1 � �ajj2 + EFn�1ijj�a � �1�njj � jj�1 � �ajj

= o(1); (19.14)

where gn := EFngi; G1n := EFnG1i; the inequality uses jj(@=@�1j)G1i(e�1i)jj � p1=21 �1i (when k = 1)

63

and jj(@=@�1j)G1i(�1i)jj � p1=21 �1i for �1i de�ned in (??), and the equality uses jj�1� �ajj � "n ! 0;

jj�a � �1�njj � 2" for n su¢ ciently large, and the conditions in FAR=AR that EFn�21i � M and

EFn jjvec(G1i)jj2+ := EFn jjvec(G1i(�1�n))jj2+ �M for (�1�n; Fn) 2 FAR=AR:Similarly, substituting (19.13) into (19.12) and taking averages over i = 1; :::; n gives

sup�a2B(�1�1;")

sup�12B(�a;"n)

jjbgn(�1)� bgn(�a)jj� jj bG1njj � jj�1 � �ajj+ p1n�1 nX

i=1

�1ijj�1 � �ajj2 + p1n�1nXi=1

�1ijj�a � �1�njj � jj�1 � �ajj

= op(1); (19.15)

where the equality uses jj�1 � �ajj � "n ! 0; jj�a � �1�njj � 2" for n large, and jj bG1njj = Op(1)

and n�1Pni=1 �1i = Op(1); which hold by Markov�s inequality using the same moment conditions

as used in (19.14).

Equations (19.14) and (19.15) combine to verify condition (b) in (19.11).

Condition (a) in (19.11) holds by the WLLN�s for independent L2-bounded random variables.

Again, for notational simplicity, we assume that k = 1: The L2-boundedness condition holds by

replacing �a by �1�n in (19.12), taking the inner product of the resulting expression with itself, and

then taking expectations. This yields: 8�1 2 B(�1�1; ");

EFngi(�1)0gi(�1) = EFn�

0n�n = O(1); where

�n := gi +G1i � (�1 � �1�n) +p1Xj=1

(�1j � �1�nj)@

@�1jG1i(e�1i)(�1 � �1�n) (19.16)

and the second equality holds using jj�1 � �1�njj � 2" for n large and using the moment conditionslisted after (19.14). This completes the veri�cation of Assumption (i).

Assumption (ii) requires gn = 0k 8n � 1; where gn = gn(�1�n): By the veri�cation of Assumption(i), gn(�1�n) = EFnbgn(�1�n): Hence, gn = 0k 8n � 1 holds by the condition in FAR=AR that

EFngi(�1�n) = 0k 8(�1�n; Fn) 2 FAR=AR:

Assumption (vi) requires sup�12B(�1�n;") jj bG1n(�1)�G1n(�1)jj = op(1) for some nonrandom func-tions fG1n(�) : n � 1g:We verify Assumption (vi) with G1n(�1) = EFn bG1n(�1) and we assume k = 1for notational simplicity. Assumption (vi) is a uniform WLLN. Its veri�cation is similar to, but sim-

pler than, the veri�cation of Assumption (i). To verify stochastic equicontinuity, one uses (19.13)

with �a and �1�n replaced by �1 and �a; respectively. Then, the analogues of (19.14) and (19.15)

64

are

sup�a2B(�1�1;")

sup�12B(�a;"n)

jjG1n(�1)�G1n(�a)jj � p1EFn�1ijj�1 � �ajj = o(1) and

sup�a2B(�1�1;")

sup�12B(�a;"n)

jj bG1n(�1)� bG1n(�a)jj � p1n�1

nXi=1

�1ijj�1 � �ajj = op(1); (19.17)

where the two equalities use jj�1� �ajj � "n ! 0; EFn�21i �M; and n�1

Pni=1 �1i = Op(1) as above.

This veri�es stochastic equicontinuity.

To verify the pointwise WLLN�s, i.e., bG1n(�1) � G1n(�1) = op(1) 8�1 2 B(�1�1; "); which isanalogous to condition (a) in (19.11), we use the WLLN�s for independent L2-bounded random

variables. The L2-boundedness condition holds by an analogous argument to that given in (19.16).

This completes the veri�cation of Assumption (vi).

Assumption (vii) requires sup�12B(�1�n;") jjG1n(�1)jj = O(1): By the veri�cation of Assumption(vi) above, G1n(�1) = EFnG1i(�1): Hence, Assumption (vii) holds by the moment condition on

G1i(�1) in FAR=AR:Assumption (viii) requires sup�12B(�1�n;"n) jjG1n(�1)�G1njj = o(1) for all sequences of positive

constants "n ! 0: For notational simplicity and wlog, we suppose k = p1 = 1: For �1 2 B(�1�n; "n);element-by-element mean-value expansions of G1i(�1) about �1�n give

G1i(�1) = G1i +@

@�1G1i(�1i)(�1 � �1�n); (19.18)

where �1i lies between �1 and �1�n: Taking expectations in (19.18) gives

sup�12B(�1�n;"n)

jG1n(�1)�G1nj � EFn�1i sup�12B(�1�n;"n)

j�1 � �1�nj = o(1); (19.19)

where the inequality uses j(@=@�1)G1i(�1i)j � �1i and the equality uses "n ! 0 and EFn�1i � M

for �1 2 B(�1�n; "n) and (�1�n; Fn) 2 FAR=AR: This veri�es Assumption (viii).Assumption (x) requires EFn

b�1n = O(1); where b�1n := maxs;u�p1 sup�12B(�1�1;")

jj(@2=@�1s�1u)bgn(�1)jj: By the de�nition of �1i in (??), b�1n � n�1Pni=1 �1i for " > 0 su¢ ciently

small that B(�1�1; ") � �1: Hence, Assumption (x) holds by the moment condition EF �21i � M

8(�1; F ) 2 FAR=AR:Assumption (xi) requires sup�12B(�1�n;K=n1=2) jjbn(�1) � n(�1)jj !p 0 for some nonrandom

functions fn(�) : n � 1g: We verify Assumption (xi) with n(�1) = EFngi(�1)gi(�1)0 � EFngi(�1)

65

� EFngi(�1)0: We have

sup�12B(�1�n;K=n1=2)

jjbn(�1)� n(�1)jj (19.20)

� sup�12B(�1�n;K=n1=2)

jjbn(�1)� bnjj+ sup�12B(�1�n;K=n1=2)

jjn(�1)� njj+ jjbn � njjusing the triangle inequality. Next, we have

jjbn � njj = n�1

nXi=1

gig0i � bgnbg0n � (EFngig0i � EFnbgnEFnbg0n)

(19.21)

� n�1

nXi=1

gig0i � EFngig0i

+ (jjbgnjj+ jjgnjj) n�1

nXi=1

gi � EFngi

= op(1);where the �rst equality holds by the triangle inequality and standard manipulations and the second

equality holds by the WLLN for independent L1+ =2-bounded random variables for > 0 as in

FAR=AR using the moment conditions in FAR=AR and jjbgnjj+jjgnjj = Op(1) using Markov�s inequalityand the moment conditions in FAR=AR:

To verify Assumption (xi), it remains to show that the �rst and second summands on the

rhs of (19.20) are op(1): Assumption (xii) requires sup�12B(�1�n;K=n1=2) jjn(�1)� njj ! 0: Hence,

verifying Assumption (xii) shows that the second summand on the rhs of (19.20) is op(1):

Now we verify Assumption (xii). For notational simplicity, we assume k = p1 = 1 when

verifying Assumption (xii). (The results for k; p1 � 1 hold by analogous arguments.) For �1 2B(�1�n;K=n1=2); element-by-element two-term Taylor expansions of gi(�1) about �1�n give

gi(�1) = gi +G1i � (�1 � �1�n) + (�1 � �1�n)@

@�1G1i(e�1i)(�1 � �1�n); (19.22)

where e�1i lies between �1 and �1�n: Taking expectations in (19.22) givesgn(�1) = gn +G1n � (�1 � �1�n) + (�1 � �1�n)EFn

@

@�1G1i(e�1i)(�1 � �1�n) and

sup�12B(�1�n;K=n1=2)

jgn(�1)� gnj � jG1nj � j�1 � �1�nj+ EFn�1ij�1 � �1�nj2 = o(1);

(19.23)

where gn := EFngi; G1n := EFnG1i; and the inequality uses the conditions in FAR=AR that EFn�21i �M; EFn jjvec(G1i)jj2+ := EFn jjvec(G1i(�1�n))jj2+ �M for (�1�n; Fn) 2 FAR=AR:

66

Using (19.22) and taking expectations, we have: uniformly over �1 2 B(�1�n;K=n1=2);

EFngi(�1)gi(�1) = EFngigi + EFnG21i � (�1 � �1�n)2 + EFn

�@

@�1G1i(e�1i)�2 (�1 � �1�n)4

+2EFngiG1i � (�1 � �1�n) + 2EFngi@

@�1G1i(e�1i)(�1 � �1�n)2

+2EFnG1i@

@�1G1i(e�1i)(�1 � �1�n)3

= EFngigi + o(1); (19.24)

where the second equality holds using j�1 � �1�nj � K=n1=2; the moment conditions in FAR=ARreferred to above, the inequality j(@=@�1j)G1i(e�1i)j � �1i; and the Cauchy-Bunyakovsky-Schwarz

inequality. Equations (19.23) and (19.24) yield

sup�12B(�1�n;K=n1=2)

jn(�1)� nj

= sup�12B(�1�n;K=n1=2)

jEFngi(�1)gi(�1)� EFngigi � EFngi(�1)EFngi(�1) + EFngiEFngij

= o(1); (19.25)

where the second equality uses the triangle inequality and EFngi(�1) = O(1); which holds by the

moment conditions in FAR=AR: This completes the veri�cation of Assumption (xii).Next, we show that the �rst summand on the rhs of (19.20) is op(1); which is needed to complete

the veri�cation of Assumption (xi). For notational simplicity, we assume k = p1 = 1 in this

paragraph. Using (19.22), we have: uniformly over �1 2 B(�1�n;K=n1=2);

n�1nXi=1

gi(�1)gi(�1)

= n�1nXi=1

gigi + n�1

nXi=1

G21i � (�1 � �1�n)2 + n�1nXi=1

�@

@�1G1i(e�1i)�2 (�1 � �1�n)4

+2n�1nXi=1

giG1i � (�1 � �1�n) + 2n�1nXi=1

gi@

@�1G1i(e�1i)(�1 � �1�n)2

+2n�1nXi=1

G1i@

@�1G1i(e�1i)(�1 � �1�n)3

= n�1nXi=1

gigi + op(1); (19.26)

where the second equality holds using jj�1 � �1�njj � K=n1=2; the inequality jj(@=@�1j)G1i(e�1i)jj ��1i; the WLLN for independent L1+ =2-bounded random variables for > 0 as in FAR=AR; the

67

moment conditions in FAR=AR; and the Cauchy-Bunyakovsky-Schwarz inequality. By similar, butsimpler calculations, uniformly over �1 2 B(�1�n;K=n1=2);

n�1nXi=1

gi(�1) = n�1

nXi=1

gi + op(1): (19.27)

Equations (19.26) and (19.27) imply that the �rst summand on the rhs of (19.20) is op(1): This

completes the veri�cation of Assumption (xi).

Assumption (xiii) holds by the condition in FAR=AR that �min(n) = �min(EFngi(�1�n)gi(�1�n)0)� � 8(�1�n; Fn) 2 FAR=AR:

Assumption (xv) requires sup�12B(�1�n;K=n1=2) jjb�1n(�1)��1n(�1)jj = op(1) for some nonrandomfunctions f�1n(�) : n � 1g; where b�1n(�) is de�ned in (14.2). We verify Assumption (xv) with�1n(�1) de�ned as in (14.2), i.e., �1n(�1) := EFnvec(G1i(�1) � EFnG1i(�1))gi(�1)0 using the iden-tical distribution assumption in FAR=AR: The veri�cation is quite similar to that of Assumption(xi) with vec(G1i(�1))gi(�1)0 and vec(G1i(�1)) in place of gi(�1)gi(�1)0 and gi(�1); respectively. In

consequence, we do not provide all of the details. Analogues of (19.20), (19.21), (19.26), and (19.27)

hold by analogous arguments, so it su¢ ces to show that an analogue of the second summand on the

rhs of (19.20) holds. Assumption (xvi) requires sup�12B(�1�n;K=n1=2) jj�1n(�1) � �1njj ! 0: Hence,

verifying Assumption (xvi) shows that the analogue of the second summand on the rhs of (19.20)

is op(1): We verify Assumption (xvi) below. This completes the veri�cation of Assumption (xv).

Assumption (xvi) requires sup�12B(�1�n;K=n1=2) jj�1n(�1) � �1njj ! 0; where �1n(�1) is de�ned

in the previous paragraph. For notational simplicity, we assume k = p1 = 1 in the veri�cation of

Assumption (xvi). Using (19.18) with "n = K=n1=2 and (19.22) and taking expectations, we obtain

EFnG1i(�1)gi(�1)� EFnG1igi

= EFnG21i(�1 � �1�n) + EFnG1i

@

@�1G1i(e�1i)(�1 � �1�n)2 + EFn @

@�1G1i(�1i)gi(�1 � �1�n)

+EFn@

@�1G1i(�1i)G1i(�1 � �1�n)2 + EFn

@

@�1G1i(�1i)

@

@�1G1i(e�1i)(�1 � �1�n)3

= o(1); (19.28)

where the second equality holds uniformly over �1 2 B(�1�n;K=n1=2) using sup�12B(�1�n;K=n1=2)

j�1 � �1�nj = o(1); j(@=@�1)G1i(�1i)j � �1i; j(@=@�1)G1i(e�1i)j � �1i; EFng2i + EFnG

21i + EFn�

21i

= O(1) by the moment conditions in FAR=AR; and the Cauchy-Bunyakovsky-Schwarz inequality.

68

Next, we have

sup�12B(�1�n;K=n1=2)

j�1n(�1)� �1nj

= sup�12B(�1�n;K=n1=2)

jEFnG1i(�1)gi(�1)� EFnG1igi � EFnG1i(�1)EFngi(�1) + EFnG1iEFngij

= o(1); (19.29)

where the second equality holds by (19.19) with "n = K=n1=2; (19.23), (19.28), and the moment

conditions in FAR=AR; which imply that sup�12B(�1�n;K=n1=2)(jEFnvec(G1i(�1))j + jEFngi(�1)j) =O(1) (when k = p1 = 1): This completes the veri�cation of Assumption (xvi).

Assumption (xxiii) requires sup�12B(�1�n;K=n1=2) jb�21sn(�1)��21sn(�1)j !p 0 for some nonrandom

functions f�21sn(�) : n � 1g 8s = 1; :::; p1; where b�21sn(�1) is de�ned in (??). We verify Assumption(xxiii) with �21sn(�1) := EFn jjG1si(�1)jj2 � (EFn jjG1si(�1)jj)2: Provided Assumption (xxiv) holds,Assumption (xxiii) holds by the same argument as for Assumption (xv) with jjG1si(�1)jj2 andjjG1si(�1)jj in place of vec(G1i(�1))gi(�1)0 and vec(G1i(�1)); respectively (which in turn relies onthe veri�cation of Assumption (xi)).

Now we verify Assumption (xxiv), which requires that sup�12B(�1�n;K=n1=2) j�21sn(�1)��21snj ! 0

8s = 1; :::; p1: The latter is implied by sup�12B(�1�n;K=n1=2) jEFn jjG1si(�1)jj2 � EFn jjG1sijj2j ! 0

and sup�12B(�1�n;K=n1=2) jEFn jjG1si(�1)jj � (EFn jjG1sijj)j ! 0: For notational simplicity, we suppose

k = p1 = 1: Using (19.18) with "n = K=n1=2 and taking expectations, we have

sup�1

jEFn jG1i(�1)j2 � EFn jG1ij2j

= sup�1

��EFn��G1i + @

@�1G1i(�1i)(�1 � �1�n)

��2 � EFn jG1ij2��

� sup�1

EFn

�� @@�1G1i(�1i)(�1 � �1�n)��2

+2 sup�1

EFn

�� @@�1G1i(�1i)(�1 � �1�n)G1i��

� EFn�21iK

2=n+ 2EFn(�1ijG1ij)K=n1=2

= op(1); (19.30)

where sup�1 denotes sup�12B(�1�n;K=n1=2); the �rst equality uses (19.18), the �rst inequality holds

by the triangle inequality, the second inequality uses jj(@=@�1)G1i(�1i)jj � �1i; and the last equalityuses EFn�

21i � M and EF jG1ij2+ � M (when k = p1 = 1) for (�1�n; Fn) 2 FAR=AR and the

Cauchy-Bunyakovsky-Schwarz inequality. Establishing the analogous result to that in (19.30) with

69

jG1si(�1)j2 and jG1sij2 replaced by jG1si(�1)j and jG1sij is quite similar.Assumption (xxv) requires that lim infn!1 �21sn > 0 8s = 1; :::; p1; where �21sn := EFn jjG1sijj2�

(EFn jjG1sijj)2: This holds by the condition in FAR=AR that V arF (jjG1sijj) � � > 0 8s = 1; :::; p1:By (19.6), f�n;h : n � 1g denotes a sequence f�n 2 �AR=AR : n � 1g for which hn(�n)! h 2 H:

By (19.7), we have

n1=2�1;�1�n;Fn : = n1=2(�11Fn(�1�n); :::; �1p1Fn(�1�n))0 ! h1;

�2;�1�n;Fn : = B1Fn(�1�n)! h2 =: B11;

�3;�1�n;Fn : = C1Fn(�1�n)! h3 =: C11;

�4;�1�n;Fn : = EFnG1i(�1�n)! h4 =: G11;

�5;�1�n;Fn : = EFn

0@ gi(�1�n)

vec(G1i(�1�n)� EFG1i(�1�n))

1A0@ gi(�1�n)


1A0

! h5 =: V11 :=

0@ 1 �011

�11 h5;G1G1

1A ; (19.31)

�6;�1�n;Fn : = �1�n ! h6 =: �1�1; and

�7;�1�n;Fn : = (�1p1Fn(�1�n); ��1p1Fn(�1�n))

0 := (�1n; ��1n)

0 ! h7 =: (lim �1n; lim ��1n)

0;

where 1 2 Rk�k; �11 2 R(p1k)�k; h5;G1G1 2 R(p1k)�(p1k); and the second equality in the secondlast line holds by the notation introduced in (10.1) and (10.2). The convergence results in (19.31)

verify Assumptions (iii), (ix), (xiv), (xvii), (xviii), (xix), and (xx). Note that h6 =: �1�1 lies in �1

as required by Assumption (iii) because �1� is bounded and B(�1�; ") � �1 for some " > 0 by theassumptions of the present lemma. In addition, the last convergence result in (19.31) guarantees

that lim �1n and lim ��1n; which appear in Theorem 18.1(b), exist.

Assumption (v) holds by the univariate CLT for triangular arrays of rowwise independent L2+ -

bounded random variables (where L2+ -boundedness holds by the moment conditions in FAR=AR);the convergence condition V arFn(n

1=2b0(bg0n; vec( bG1n � EFn bG1n))0) ! b0V11b 8b 2 R(p1+1)k withjjbjj > 0 (which holds by the convergence results for �5;�1�n;Fn in (19.31)), and the Cramér-Wold

device.

Assumptions (xxi) and (xxii) hold by the assumptions of the lemma on fcn : n � 1g; which arethe same as in Theorem ??, that cn ! 0 and ncn !1:

This completes the veri�cation of Assumption HL1AR=AR and of the existence of lim �1n and

lim ��1n: �

The proof of Lemma 19.3 uses the following lemma when verifying Assumption HL2AR=AR(ii).

70

Lemma 19.4 Let B be a pseudometric space with pseudometric �: Let f�n(�) : n � 1g be a se-quence of real-valued stochastic processes on B: Suppose (i) B is totally bounded, (ii) f�n(�) :n � 1g is stochastically equicontinuous under �; i.e., 8"; � > 0 9� > 0 such that lim supn!1

P (sup�1;�22B:�(�1;�2)<� j�n(�1) � �n(�2)j > �) < "; and (iii) �n(�) = Op(1) 8� 2 B. Then,

sup�2B j�n(�)j = Op(1):

Comments: (i). The result of Lemma 19.4 also holds if the stochastic equicontinuity condition

is weaken by replacing �8� > 0�to �for some � > 0:�(ii). The result of Lemma 19.4 could be obtained by establishing the weak convergence of

f�n(�) : n � 1g to some limit process and applying the continuous mapping theorem. But, condi-tion (iii) of Lemma 19.4 is noticeably weaker than the weak convergence of all �nite-dimensional

distributions of �n(�); which would be needed to establish weak convergence. Condition (iii) can beveri�ed straightforwardly using Markov�s inequality when �n(�) is a sample average for � 2 B.

Proof of Lemma 19.4. Let B(�; �) denote a closed ball in B centered at � with radius � underthe pseudometric �: Because B is totally bounded, there exists a �nite number of balls in B; say J�balls, that cover B: Let the centers of these balls be f�j� 2 B : j = 1; :::; J�g: We have

sup�2B

j�n(�)j = maxj�J�

sup�2B(�j�;�)

j�n(�)j � maxj�J�

j�n(�j�)j+ ��n �J�Xj=1

j�n(�j�)j+ ��n; where

��n := maxj�J�

sup�2B(�j�;�)

j�n(�)� �n(�j�)j (19.32)

and the �rst inequality holds by the triangle inequality.

Given any " > 0 and some � > 0; e.g., � = 1 su¢ ces, take � > 0 such that lim supn!1 P (��n >

�) < "=2: Such a value � exists by the stochastic equicontinuity condition (ii). For 0 < K <1; wehave

lim supn!1

P

sup�2B

j�n(�)j > K!� lim sup

n!1P

0@ J�Xj=1

j�n(�j�)j+ ��n > K

1A� lim sup

n!1

0@P0@ J�Xj=1

j�n(�j�)j+ ��n > K; ��n � �

1A+ P (��n > �)1A

�J�Xj=1

lim supn!1

P

�j�n(�j�)j >

K � �J�

�+ "=2

< "; (19.33)

where the �rst inequality holds by (19.32), the second and third inequalities hold by standard

71

manipulations, and the last inequality holds for K su¢ ciently large using the assumption that

�n(�j�) = Op(1) 8j � J� by condition (iii). �

Proof of Lemma 19.3. We have: Assumption SI(ii) ) Assumption HL2AR=AR(i), SI(i) )HL2AR=AR(iv), SI(iii) ) HL2AR=AR(vi), and SI(vi) ) HL2AR=AR(vii).

Next, we verify Assumption HL2AR=AR(iii). It su¢ ces to show Assumption HL2AR=AR(iii)

holds for sequences Kn ! 1 such that Kn=n1=2 ! 0 because inf�1 =2B(�1�n;K=n1=2) jjn1=2gn(�1)jj is

nonincreasing in K: Hence, we assume that Kn=n1=2 ! 0:

For n � 1; let �y1n 2 B(�1�n;Kn=n1=2) satisfy

n1=2jjgn(�y1n)jj = inf�1 =2B(�1�n;Kn=n1=2)

n1=2jjgn(�1)jj+ "n; (19.34)

where "n > 0 and "n ! 0: Such values f�y1n : n � 1g always exist. Assumption HL2AR=AR(iii) holdsi¤ n1=2jjgn(�y1n)jj ! 1:

De�ne

dyn := jj�y1n � �1�njj and syn := n1=2jjgn(�

y1n)jj: (19.35)

We want to show that syn ! 1: This holds if every subsequence fmn : n � 1g of fng has asubsubsequence fvn : n � 1g such that syvn ! 1: Given an arbitrary subsubsequence fvng eitherlim infn!1 d

yvn > 0 or lim infn!1 d

yvn = 0:

First, suppose lim infn!1 dyvn > 0: Let " 2 (0; lim infn!1 d

yvn): We have

syvn = v1=2n jjgvn(�

y1vn)jj � inf

�1 =2B(�1�vn ;")v1=2n jjgvn(�

y1vn)jj ! 1; (19.36)

as desired, where the inequality holds using the de�nitions of dyn and " and the convergence holds

by Assumption SI(i) and v1=2n !1:Second, suppose lim infn!1 d

yvn = 0: Then, there exists a subsequence frng of fvng for which

limn!1 dyrn = 0 and, in this case, we that show s

yrn !1; which completes the proof of Assumption

HL2AR=AR(iii). For notational simplicity, we replace rn by n and assume lim dyn = 0: By de�nition,

�y1n =2 B(�1�n;Kn=n1=2): Hence, Kn=n1=2 � jj�y1n��1�njj = d

yn ! 0: Element-by-element mean-value

expansions about �1�n give

gn(�y1n) = G1n(

e�1n)(�y1n � �1�n) = (G1n + o(1))(�y1n � �1�n); (19.37)

where e�1n lies between �y1n and �1�n and may di¤er across the rows ofG1n(e�1n); the �rst equality usesthe assumption that gi(�) satis�es the di¤erentiability condition in Theorem ?? and the condition in

72

FAR=AR that EFngi(�1�n) = 0k; and the second equality uses jje�1n��1�njj � jj�y1n��1�njj = dyn ! 0

and sup�12B(�1�n;"n) jjG1n(�1) = G1njj = op(1) for all sequences of positive constants "n ! 0; which

is Assumption HL1AR=AR(viii), and is veri�ed in (19.18) and (19.19).

We have

njjgn(�y1n)jj2 = njj(G1n + o(1))(�y1n � �1�n)jj2

� n inf�2Rp1 :jj�jj=1

jj(G1n + o(1))�jj2 � jj�y1n � �1�njj2

� n

�inf

�2Rp1 :jj�jj=1jjG1n�jj2 + o(1)

�(Kn=n

1=2)2

��

inf�2Rp1 :jj�jj=1

jj�1=2n G1n�jj2=�2max(�1=2n ) + o(1)

�K2n

=��21n�min(n) + o(1)

�K2n

! 1; (19.38)

where the �rst equality uses (19.37), the second inequality holds because G1n = O(1) (by the

moment condition EF jjvec(G1i(�1))jj2+ � M in FAR=AR) and because �y1n =2 B(�1�n;Kn=n1=2);the last equality uses the de�nition of �1n; and the convergence to 1 uses the assumption that

Kn ! 1; lim infn!1 �min(n) > 0 (which holds by the condition �min(F (�1)) � � in FAR=AR),and the fact that lim infn!1 �1n > 0; which we now show. We obtain lim infn!1 �1n > 0 using

Assumption SI(ii) (i.e., lim infn!1 ��1n > KU ); the de�nition of ��1n in (10.3), the de�nition of �1n

in (14.3), and the conditions V arF (jjG1si(�1)jj) � � for all s = 1; :::; p1 in FAR=AR. This completesthe veri�cation of Assumption HL2AR=AR(iii).

Now, we show that Assumptions SI(iv) and (v) imply Assumption HL2AR=AR(ii). Because

Assumption HL2AR=AR(ii) can be veri�ed element-by-element, we assume without loss of generality

that k = 1: We use Lemma 19.4 with � = �1; B = �1; and

�n(�1) := n1=2(bgn(�1)� EFnbgn(�1)): (19.39)

Condition (iii) of Lemma 19.4, i.e., �n(�1) = Op(1) 8�1 2 �1; holds because for any " > 0;

PFn(�n(�1) > K) � EFn�2n(�1)=K

2 = EFn(gi(�1)� EFn

gi(�1))2=K2 < " (19.40)

for all n � 1; where the �rst inequality holds by Markov�s inequality, the equality holds because

fgi(�1) : i � ng are i.i.d. under Fn for each n � 1; and the last inequality holds for K su¢ ciently

large using Assumption SI(iii).

73

We verify conditions (i) and (ii) of Lemma 19.4, i.e., �1 is totally bounded under � and f�n(�) :n � 1g is stochastically equicontinuous under �; using Theorem 4 in Andrews (1994, p. 2277).

Here, � is de�ned by �(�a; �b) := lim supn!1(EFn(gi(�a) � gi(�b))2)1=2 for �a; �b 2 �1: Theorem4 requires Assumptions B-D of Andrews (1994) to hold. Assumption C on p. 2269 of Andrews

(1994) holds by the independence assumption in FAR=AR: Assumption B on p. 2268 (an envelopecondition) holds by Assumption SI(iii), which states that lim supn!1EFn sup�12�1 jjgi(�1)jjr <1for some r > 2:

We verify Assumption D (Ossiander�s Lp entropy condition) in Andrews (1994) using Theorem

5 in Andrews (1994, p. 2281) with p = 2: To apply Theorem 5 it su¢ ces that the functions

fgi(�1) : �1 2 �1g are a type II class of functions (i.e., Lipschitz functions, see p. 2270 of Andrews(1994)) with �1 bounded and lim supn!1EFnB

ri < 1 for some r > 2; where Bi is a random

Lipschitz �constant.�This holds because, by the mean-value theorem, using the di¤erentiability of

gi(�1); the convexity of �1 imposed in Assumption SI(v), and the moment condition in Assumption

SI(iv), we have

jjgi(�a)� gi(�b))jj � B1ijj�a � �bjj 8�a; �b 2 �1 for B1i := sup�12�1

jj @@�1

gi(�1)jj; (19.41)

where lim supn!1EFnBr1i < 1: And, �1 is bounded by Assumption SI(v). This completes the

veri�cation of Assumption HL2AR=AR(ii).

Next, we verify Assumption HL2AR=AR(v), i.e., sup�12�1 jjbn(�1) � n(�1)jj = op(1); wherebn(�1) is de�ned in (??), and n(�1) = EFngi(�1)gi(�1)0 � EFngi(�1)EFngi(�1)0: We do so by

obtaining uniform WLLN�s over �1 for averages over i � n of gi(�1) and gi(�1)gi(�1)0: For the

average over i � n of gi(�1); a uniform WLLN�s holds (i.e., sup�12�1 jjbgn(�1) � EFnbgn(�1)jj !p 0)

by Assumption HL2AR=AR(ii), which is veri�ed above.

To obtain a uniform WLLN�s for the average over i � n of gi(�1)gi(�1)0; we use the followinggeneric uniform WLLN�s. Let fsi(�1) : i � n; n � 1g be some vector-valued random functions on

�1; where si(�1) := s(Wi; �1): Let bsn(�1) := n�1Pni=1 si(�1): Su¢ cient conditions for a uniform

WLLN�s for these random functions under fFn : n � 1g (i.e., sup�12�1 jjbsn(�1)�EFnbsn(�1)jj !p 0)

are

(a) bsn(�1)� EFnbsn(�1)!p 0 8�1 2 �1;

(b) jjsi(�a)� si(�b)jj � Bsijj�a � �bjj; 8�a; �b 2 �1; where lim supn!1

n�1nXi=1

EFnBsi <1; and

(c) �1 is bounded, (19.42)

74

e.g., see Theorem 1(a) and Lemma 2(a) of Andrews (1991a).

Now, consider si(�1) = gi(�1)gi(�1)0: A pointwise WLLN�s for n�1Pni=1 gi(�1)gi(�1)

0 holds for

each �xed �1 2 �1 under the i.i.d. condition in FAR=AR for each �xed F and Assumption SI(iii).

Hence, condition (a) of (19.42) holds. Condition (c) of (19.42) holds immediately by Assumption

SI(v).

Using (19.41), we obtain

jjgi(�a)gi(�a)0 � gi(�b)gi(�b)0jj

� 2 sup�12�1

jjgi(�1)jj � jjgi(�a)� gi(�b)jj

� 2 sup�12�1

jjgi(�1)jjBijj�a � �bjj; 8�a; �b 2 �1; (19.43)

where for matrix arguments jj � jj denotes the Frobenious norm and the �rst inequality uses

the triangle inequality. Combining (19.43) with lim supn!1EFn sup�12�1 jjgi(�1)jj2 < 1 and

lim supn!1EFnB2i < 1; which hold by Assumptions SI(iii) and (iv), and using the Cauchy-

Bunyakovsky-Schwarz inequality veri�es condition (b) of (19.42). This completes the veri�cation

of a uniform WLLN�s for the average over i � n of gi(�1)gi(�1)0; which completes the veri�cationof Assumption HL2AR=AR(v). �


The proof of Theorem ?? is similar to that of Theorem ?? given in Section 19. But, it uses an

adjusted de�nition of ��1;F in (19.4) and a parameter space �AR=LM;QLR1 (de�ned below) in place

of �AR=AR:

As above, F (�) := V arF (gi(�)): We write a SVD of �1=2F (�1)EFG2i(�1) as

�1=2F (�1)EFG2i(�1) = C2F (�1)�2F (�1)B2F (�1)

0; (20.1)




We write a SVD of �1=2F (�1)EFG2i(�1)�2F (�1) as

�1=2F (�1)EFG2i(�1)�2F (�1) = C

�2F (�1)�

�2F (�1)B

�2F (�1)

0; (20.2)

where C�2F (�1) 2 Rk�k and B�2F (�1) 2 Rp2�p2 are orthogonal matrices and ��2F (�1) 2 Rk�p2 has

75

the singular values ��21F (�1); :::; ��2p2F

(�1) of �1=2F (�1)EFG2i(�1)�2F (�1) in nonincreasing order on

its diagonal and zeros elsewhere.

Let �pF (�1) denote the smallest singular value of �1=2F (�1)EFGi(�1) and let ��pF (�1) denote

the smallest singular value of �1=2F (�1)EFGi(�1)�F (�1); where �F (�1) is de�ned in (??).

The adjusted de�nition of ��1;F is as follows. First, (�21F (�1); :::; �2p2F (�1))0 2 Rp2 and

(��21F (�1); :::; ��2p2F

(�1))0 2 Rp2 are added onto �1;�1;F so that �1;�1;F 2 Rp1+2p2 : Second, Gi(�1)

appears in place of G1i(�1) in �4;�1;F and �5;�1;F : Third, the following six elements are added onto

��1;F : �9;�1;F := B2F (�1) 2 Rp2�p2 ; �10;�1;F := C2F (�1) 2 Rk�k; �11;�1;F := (�211F (�1); :::; �21p1F (�1);�221F (�1); :::; �

22p2F

(�1))0 2 Rp; where �2jsF (�1) := V arF (jjGjsi(�1)jj) 8s = 1; :::; pj ; 8j = 1; 2;

�12;�1;F := B�2F (�1) 2 Rp2�p2 ; �13;�1;F := C�2F (�1) 2 Rk�k; and �14;�1;F := (�pF (�1); ��pF (�1))

0:

Fourth, h5 in (19.31) is de�ned by

h5 := V1 :=

0@ 1 �01

�1 h5;GG

1A ; (20.3)

where 1 2 Rk�k; �1 2 R(pk)�k; h5;GG 2 R(pk)�(pk): Fifth, hn;j(�) in (19.5) is de�ned to equal �jfor j = 9; :::; 14:

The parameter space �AR=LM;QLR1 (for �) that we use here is de�ned analogously to �AR=AR

in (19.5), but is based on the adjusted de�nition of � and the parameter space FAR=LM;QLR1;rather than FAR=AR:We use the same function hn(�) here as de�ned as in (19.5), but based on theadjusted de�nition of �:

The proof of Theorem ?? uses the following lemma. This lemma veri�es Assumptions

HL1AR=LM and HL1AR=QLR1; which are employed in Lemmas 18.3 and 18.4, for all subsequences

f�wn;h : n � 1g: The subsequences f�wn;h : n � 1g considered in this lemma are based on theadjusted de�nitions of ��1;F and hn;j(�) given immediately above.

Lemma 20.1 Suppose bgn(�1) are the moment functions de�ned in (??), gi(�) satis�es the di¤er-entiability conditions in Theorem ??, cn ! 0; ncn !1; a > 0; and p2 � 1: Let the null parameterspace be FAR=LM;QLR1: Then, for all subsequences f�wn;h : n � 1g; Assumptions HL1AR=LM and

HL1AR=QLR1 hold and lim �mn and lim ��mn

exist.

Comment: When LM2n(�) is the pure C(�)-LM statistic (i.e., WIn(�) := 0); the parameter

space FAR=LM;QLR1 is restricted as in (??), and the de�nition of ��1;F is augmented by �15;�1;F :=(r1n; r2n)

0; Lemma 20.1 holds and, in addition, conditions (vi) and (vii) in Comment (v) to Lemma

16.1 hold. The same is true when QLR12n(�) is the pure C(�)-QLR1 statistic and WIn(�) := 0

in the QLR1 critical value function. (These results are proved following the proof of Lemma 20.1

76

below.) These results imply that the results of Lemma 18.3(a) and (b) hold (using the Comment

to Lemma 18.3).

Proof of Theorem ??. The proof of Theorem ?? is the same as that of Theorem ?? given in

Section 19, but (i) with the de�nitions of ��1;F and hn;j(�) adjusted as in the paragraph containing

(20.3), (ii) using Lemmas 18.3 and Lemma 18.4 in place of Theorem 18.1 and Lemma 18.2, (iii)

with FAR=LM;QLR1 and F�;AR=LM;QLR1 in place of FAR=AR and F�;AR=AR; and (iv) using Lemma20.1 in addition to Lemma 19.2. Lemma 20.1 shows that lim �mn and lim �

�mn

exist, which is used

when Lemma 18.3 is employed and in showing that a subsequence Sm satis�es Assumption B when

lim ��mn< KL and Assumption C when lim infn!1 ��mn

> K�U ; as in the proof of Theorem ?? (with

lim infn!1 ��mn> K�

U in place of lim infn!1 �1mn > 0):

The proof of part (a) uses Lemma 18.3, which employs Assumptions HL1AR=AR and

HL1AR=LM ; and for the second-step C(�)-QLR1 test, Assumption HL1QLR1 as well. These assump-

tions are veri�ed for the parameter space FAR=LM;QLR1 by Lemmas 19.2 (using FAR=LM;QLR1 �FAR=AR) and 20.1.

The proof of part (b) uses Lemma 18.4, which employs Assumptions HL1AR=AR; HL1AR=LM ;

HL2AR=LM;QLR1; and W. Assumptions HL1AR=AR and HL1AR=LM are veri�ed in the proof of

part (a) of the theorem. Assumption HL2AR=LM;QLR1 is the same as Assumption HL2AR=AR ex-

cept for part (i). Hence, using FAR=LM;QLR1 � FAR=AR; Lemma 19.3 veri�es all of AssumptionHL2AR=LM;QLR1 except part (i). Part (i) of Assumption HL2AR=LM;QLR1 is implied by Assumption

SI2, which is imposed in part (b) of the theorem. Assumption W is imposed in the theorem, so it

holds by assumption.

Parts (c)�(f) of the theorem hold by the same arguments as given in the proof of these parts of

Theorem ??. �

Proof of Lemma 20.1. For notational simplicity, we consider a sequence f�n;h : n � 1g; ratherthan a subsequence f�wn;h : n � 1g:

Assumption HL1AR=LM (i) holds by the di¤erentiability conditions that are imposed in Theorem

??, but not in Theorem ??.

Assumption HL1AR=LM (ii) holds by the CLT using the moment conditions in FAR=LM;QLR1(including the condition EF jjvec(G2i(�1))0jj2+ � M; which does not appear in FAR=AR) by thesame argument as in the veri�cation of Assumption HL1AR=AR(v) given in the paragraph following

(19.31) in the proof of Lemma 19.2. Note that the variance matrix V11 in Assumption HL1AR=AR(v)

is the upper left (p1+1)k� (p1+1)k sub-matrix of V1; where V1 is the limit of �5;�1�n;Fn in (20.3).

The veri�cation of Assumptions HL1AR=LM (iii)�(v) is the same as for Assumptions HL1AR=AR

77

(vi), (viii), and (ix) in the proof of Lemma 19.2 (see (19.17), the two paragraphs that follow (19.17),

and (19.31)) with subscripts 2 in place of 1 and using EF �22i �M in place of EF �21i �M:Assumption HL1AR=LM (vi) holds by the condition EF �12i �M in FAR=LM;QLR1 and Markov�s

inequality.

The veri�cation of Assumptions HL1AR=LM (vii)�(ix) is the same as for Assumptions

HL1AR=AR(xv)�(xvii) in the proof of Lemma 19.2 (see (19.28), the paragraph preceding (19.28),

(19.29), and (19.31)) with subscripts 2 in place of 1; using the condition EF �22i � M in place of

E�21i �M; and using Gi(�1) in place of G1i(�1) in the de�nitions of �4;�1;F and �5;�1;F as speci�edin the paragraph that contains (20.3).

The veri�cation of Assumptions HL1AR=LM (x)�(xii) is the same as for Assumptions

HL1AR=AR(xviii)�(xx) in the proof of Lemma 19.2 (see (19.31)) using the adjusted de�nition of

�1;�1;F to include (�21F (�1); :::; �2p2F (�1))0 and the addition of �9;�1;F and �10;�1;F to ��1;F ; as

speci�ed in the paragraph that contains (20.3).

The veri�cation of Assumptions HL1AR=LM (xiii) and (xiv) is the same as for Assumptions

HL1AR=AR(xxiii) and (xxiv) in the proof of Lemma 19.2 with subscripts 2 in place of 1 (see (19.30))

using the EF jjvec(G2i(�1))jj2+ �M and EF �22i �M conditions in the de�nition of FAR=LM;QLR1:By (19.6), f�n;h : n � 1g denotes a sequence f�n 2 �AR=LM;QLR1 : n � 1g for which hn(�n)!

h 2 H: For the sequence f�n;h : n � 1g; the convergence result of Assumption HL1AR=LM (xv) (i.e.,�2jsn ! �2js1 8s = 1; :::; pj ; 8j = 1; 2) holds by the addition of �11;�1;F to ��1;F ; as speci�ed in theparagraph that contains (20.3) because �2jsn := V arFn(jjGjsijj) 8s = 1; :::; pj ; 8j = 1; 2; see (17.1).The result of Assumption HL1AR=LM (xv) that �2js1 2 (0;1) 8s = 1; :::; pj ; 8j = 1; 2 holds by

the V arF (jjGjsijj) � � and EF jjvec(Gji(�1))jj2+ � M conditions for j = 1; 2 in the de�nitions of

FAR=AR and FAR=LM;QLR1: This completes the veri�cation of Assumption HL1AR=LM :By (19.5) and the adjusted de�nition of �1;�1�n;Fn ; for the sequence f�n;h : n � 1g; n1=2��2sn

converges to some value in [0;1] 8s � p2: Hence, Assumption HL1AR=QLR1(i) holds. The conver-gence results of Assumptions HL1AR=QLR1(ii) and (iii) (i.e., C�2n ! C�21 and B�2n ! B�21) hold

by the addition of �12;�1;F and �13;�1;F to ��1;F ; as speci�ed in the paragraph that contains (20.3).

This completes the veri�cation of Assumption HL1AR=QLR1:

The limits lim �n and lim ��n exist by the addition of �14;�1;F to ��1;F : �

Now we prove the Comment to Lemma 20.1, which requires that we verify conditions (vi) and

(vii) stated in Comment (v) to Lemma 16.1. Condition (vi) (i.e., rjn = rj1 for n su¢ ciently large)

holds by the addition of �15;�1;F to ��1;F ; which implies that lim rjn exists, and the fact that rjn can

only take on a �nite number of values. Condition (vii) holds by (??) (i.e., �min(C�F (�1)0C�F (�1)) ��):

78

The Comments to Lemmas 18.3 and 20.1 imply that Lemma 18.3 holds for the pure C(�)-LM

and pure C(�)-QLR1 tests provided FAR=LM;QLR1 is restricted as in (??) and �15;�1;F is added to��1;F :

Lastly, the proof of Comment (iii) to Theorem ?? is the same as the proof of Theorem ?? given

above using the results of Lemma 18.3, which hold when FAR=LM;QLR1 is restricted as in (??) andthe test is the pure C(�)-LM or pure C(�)-QLR1 test (for which WIn(�) := 0):


The proof of Theorem 11.1 is analogous to that of Theorems ?? and ??. In the time series case,

we de�ne �; f�n;h : n � 1g; and � as in (19.5) and (19.6) and the discussion around (19.8) forthe AR/AR test and CS, respectively, and as in the discussion around (20.3) for the AR/LM and

AR/QLR1 tests and CS�s. But, we de�ne �5;�1;F and �5;�;F di¤erently from the i.i.d. case. We

de�ne

�5;�1;F := VF (�1) and �5;�;F := VF (�); (21.1)

where VF (�) is de�ned in (11.3), rather than as in (19.4). In consequence, �5;�1n;Fn ! h5 implies

that VFn(�1n) ! h5 and the condition VFn(�1�n) ! V1 in Assumption V holds with �1�n = �1n

and V1 = h5 (and analogously for CS�s and Assumption V-CS). We let �TS;AR=AR denote the time

series version of �AR=AR: It is de�ned as in (19.5), but with FTS;AR=AR in place of FAR=AR andwith the changes described above.

The proof of Theorem 11.1 uses the CLT given in the following lemma. This lemma employs

Corollary 1 in de Jong (1997) and is analogous to Lemma 20.1 in Section 20 in the SM to AG1.

Lemma 21.1 Let fi(�) := (gi(�)0; vec(Gi(�))0)0: We have: for tests, w�1=2n

Pwni=1(fi(�1�n)�

EFnfi(�1�n))!d N(0(p+1)k; h5) under all subsequences fwng and all sequences f�wn;h 2 �TS;AR=AR :

n � 1g; and for CS�s the same result holds with ��n in place of �1�n:

We use the following stochastic equicontinuity result, which is a special case of Hansen (1996)

Theorem 3, in the proof of Theorem 11.1. The strong mixing numbers of a triangular array of

random vectors are de�ned in the usual way, e.g., see Hansen (1996).

Lemma 21.2 Suppose (i) fWni : i � n; n � 1g are row-wise identically distributed, strong mix-ing random vectors taking values in a set W, (ii) B is a bounded subset of Rd� ; (iii) FLip is aset of real-valued functions s(w; �) on W �B that satisfy js(w; �1) � s(w; �2)j � B(w)jj�1 � �2jjfor some Lipschitz function B(w) on W, (iv) lim supn!1Ejs(Wni; �)jr < 1 8� 2 B for some

79

r > 2; (v) lim supn!1EjB(Wni)jr < 1; and (vi) the strong mixing numbers of fWni : i �n; n � 1g satisfy

P1m=1 �

1=q�1=rm < 1 for some q > d� and 2 � q < r: Then, 8"; � > 0

9� > 0 such that lim supn!1 P (sup�1;�22B:jj�1��2jj<� j�n(�1) � �n(�2)j > �) < "; where �n(�) :=

n�1=2Pni=1(s(Wni; �)� Es(Wni; �)):

Comment: The same constant r appears in conditions (iv)�(vi).

Proof of Theorem 11.1. The proof is the same as the proofs of Theorems ?? and ?? and Lemmas

19.2, 19.3, and 20.1, given in Sections 19 and 20, with some modi�cations. The modi�cations a¤ect

the proofs of Lemmas 19.2, 19.3, and 20.1. No modi�cations are needed elsewhere. We describe

the modi�cations for tests. The modi�cations for CS�s are analogous with ��n in place of �1�n:

The �rst modi�cation is the change in the de�nition of �5;�1;F described in (21.1). Equation

(21.1) and the �min(�) condition in FTS;AR=AR imply that Assumptions HL1AR=AR(xiv) and (xvii)(stated in Section 18) hold.

The second modi�cation is the use of a WLLN for triangular arrays of strong mixing random

vectors, rather than i.i.d. random vectors, when verifying Assumptions HL1AR=AR(i) (in condi-

tion (a) in (19.11)), HL1AR=AR(vi) (in the paragraph following (19.17)), and HL1AR=AR(xxiii) (in

the paragraph following (19.29)), and when verifying Assumptions HL1AR=LM (iii) and (xiii) in

the proof of Lemma 20.1. For the WLLN, we use Example 4 of Andrews (1988), which shows

that for a strong mixing row-wise-stationary triangular array fWni : i � n; n � 1g we haven�1

Pni=1(�(Wni) � EFn�(Wni)) !p 0 for any real-valued function �(�) (that may depend on n)

for which supn�1EFn jj�(Wni)jj1+� <1 for some � > 0:

The third modi�cation is the use of a CLT for triangular arrays of strong mixing random vectors,

rather than i.i.d. random vectors, when verifying Assumption HL1AR=AR(v) at the end of the proof

of Lemma 19.2 and when verifying Assumption HL1AR=LM (ii) in the proof of Lemma 20.1. For the

CLT, we use Lemma 21.1.

The fourth modi�cation is to use Assumption V to verify Assumptions HL1AR=AR(xi) and (xv)

in the proof of Lemma 19.2 and Assumption HL1AR=LM (vii) in the proof of Lemma 20.1 with

n(�); �1n(�); and �2n(�) de�ned by the submatrices of VFn(�); de�ned in (11.3) and partitioned

as in (11.4), e.g., �1n(�) := (�11Fn(�)0; :::;�1p1Fn(�)

0)0:

The �fth modi�cation is the veri�cation of Assumptions HL1AR=AR(xii) and (xvi) and

HL1AR=LM (viii). Assumption HL1AR=AR(xii) requires sup�12B(�1�n;K=n1=2) jjn(�1) � njj ! 0;

where n(�1) :=P1m=�1(EF gi(�)g

0i�m(�) � EF gi(�)EF gi(�)0) is de�ned in (11.1). For notational

simplicity, we suppose that k = 1; so that n(�1) is a scalar. To verify Assumption HL1AR=AR(xii),

80

we use the two-term Taylor expansion in (19.22) and write

gi(�1) = gi + �ni(�1); where

�ni(�1) := G1i � (�1 � �1�n) + (�1 � �1�n)@

@�1G1i(e�1i)(�1 � �1�n) (21.2)

and e�1i lies between �1 and �1�n:By the standard strong mixing covariance inequality of Davydov (1968), for two function s1(�)

and s1(�) on W and some > 0 and C1 <1;

jjCovF (s1(Wi); s2(Wi�m))jj � C1jjs1(Wi)jjF;2+ jjs2(Wi)jjF;2+ � =(2+ )F (m)

� C1C =(2+ )jjs1(Wi)jjF;2+ jjs2(Wi)jjF;2+ m�d =(2+ ); where

jjs(Wi)jjF;2+ := (EF jjs(Wi)jj2+ )1=(2+ ); (21.3)

d =(2 + ) > 1; and the second inequality uses the condition on the strong mixing numbers in the

de�nition of FTS;AR=AR in (11.2).Using (21.2), we have

sup�12B(�1�n;K=n1=2)

jj�ni(�1)jjFn;2+ � jjG1ijjFn;2+ K=n1=2 + @@�1G1i(e�1i)

Fn;2+

K2=n

= O(n�1=2); (21.4)

where the equality holds using the de�nition of �1i in (??) and the moment conditions EF �2+ 1i �M

and EF jjvec(G1i(�1))jj2+ �M 8(�1; F ) 2 FTS;AR=AR:Now we bound the mth term in the doubly in�nite sum over m = �1; :::;1 that de�nes

n(�1)� n: We have

Anm := sup�12B(�1�n;K=n1=2)

jjEFngi(�1)gi�m(�1)� (EFngi(�1))2 � EFngigi�m + (EFngi)2jj

= sup�12B(�1�n;K=n1=2)

jjEFngi�ni�m(�1)� EFngiEFn�ni(�1) + EFn�ni(�1)gi�m � EFn�ni(�1)EFngi

+EFn�ni(�1)�ni�m(�1)� (EFn�ni(�1))2jj

� C1C =(2+ )(2jjgijjFn;2+ jj�ni(�1)jjFn;2+ + jj�ni(�1)jj2Fn;2+ )m

�d =(2+ )

= O(n�1=2)m�d =(2+ ); (21.5)

where the O(n�1=2) term does not depend on m; the �rst equality holds by (21.2), the inequality

holds by (21.3) applied three times, and the last equality holds by (21.4) and jjgijjFn;2+ �M:

81

We have

sup�12B(�1�n;K=n1=2)

jjn(�1)� njj �1X

m=�1Anm = O(n

�1=2)1X

m=�1m�d =(2+ ) = O(n�1=2); (21.6)

where the �rst inequality holds by the de�nition of n(�1) in (11.1), the �rst equality holds by (21.5),

and the last equality uses the condition on d in FTS;AR=AR that d > (2 + )= : This completes theveri�cation of Assumption HL1AR=AR(xii). The veri�cations of Assumptions HL1AR=AR(xvi) and

HL1AR=LM (viii) are similar and hence, for brevity, are not given.

The sixth modi�cation is in the veri�cation of Assumption HL2AR=AR(ii) in (19.39)�(19.41) in

the proof of Lemma 19.3. We verify condition (ii) of Lemma 19.4, i.e., f�n(�) : n � 1g is stochasti-cally equicontinuous for �n(�1) de�ned in (19.39), using Lemma 21.2 with s(w; �) = g(w; �1; �20);

� = �1; d� = p1; and B = �1; rather than using Theorem 4 in Andrews (1994, p. 2277). (We use

the latter result in the row-wise i.i.d. case because it yields weaker conditions than are obtained by

applying Lemma 21.2 in the i.i.d. case.) Conditions (i) and (ii) of Lemma 21.2 hold by the strong

mixing condition in FTS;AR=AR and Assumption SI(v), respectively. Condition (iii) of Lemma 21.2holds with B(Wni) = sup�12�1 jjG1i(�1)jj by a mean-value expansion using Assumption SI(v), asin (19.41). Conditions (iv), (v), and (vi) of Lemma 21.2 hold by Assumptions SI(iii), SI(iv), and

SI-TS(i) (because the conditions in Assumption SI-TS(i) imply that q > p1 and 2 � q < r);

respectively.

In addition, when verifying Assumption HL2AR=AR(ii), we verify condition (iii) of Lemma 19.4,

i.e., �n(�1) = Op(1) 8�1 2 �1 for �n(�1) de�ned in (19.39), using Markov�s inequality and thestrong mixing covariance inequality in (21.3), rather than Markov�s inequality combined with the

expression for the variance of an average of i.i.d. random variables, as in (19.40). It su¢ ces to

show that V arFn(n1=2bgn(�1)) = Op(1) 8�1 2 �1: By change of variables, we have

V arFn

n�1=2

nXi=1

gi(�1)

!=

n�1Xm=�n+1

�1� jmj

n

�CovFn(gi(�1); gi�m(�1))

� O(1)

1Xm=�1

�(r�2)=rFn

(m) = O(1); (21.7)

where the inequality holds using the �rst line of (21.3) with r in place of 2 + ; s1(Wi) =

s2(Wi) = gi(�1); and lim supn!1EFn sup�12�1 jjgi(�1)jjr < 1 (by Assumption SI(iii)), and the

second equality holds by the conditions on the strong mixing numbers in Assumption SI-TI(i) by

the following argument. Given the mixing number condition in Assumption SI-TI(i), it su¢ ces

to show that 1=q � 1=r � (r � 2)=r; or equivalently, 1=q � 1 � 1=r; or q � r=(r � 1): We have

82

q � (q + �1)=(1 + �1) � (q + �1)=(q + �1 � 1) = r=(r � 1); where the �rst inequality holds becauseq � 1 and �1 > 0 and the second inequality holds because q � 1 � 1:

The seventh modi�cation is to use Assumption SI-TS(ii) to verify Assumption HL2AR=AR(v).

This completes the proof of Theorem 11.1. �

Proof of Lemma 21.1. The proof is essentially the same as that of Lemma 20.1 in Section 20 in

the SM to AG1. For the CS case, it relies on the moment conditions EFn jjfi(��n)jj2+ � M < 1for some M < 1; 8(��n; Fn) 2 F�;TS;AR=AR (or, equivalently, 8�n 2 ��;TS;AR=AR); and on thestrong mixing numbers satisfying �Fn(m) � Cm�d for some d > (2 + )= and some C < 1;8(��n; Fn) 2 F�;TS;AR=AR; where for notational simplicity we consider the sequence fng; ratherthan a subsequence fwn : n � 1g: �

Proof of Lemma 21.2. The result of Lemma 21.2 is a special case of Theorem 3 in Hansen (1996)

where Hansen�s Lipschitz exponent � equals 1; the average over n in his equations (12) and (13)

disappear because of the assumption of row-wise identical distributions (and, hence, the square

and square root in his (12) and (13) cancel), his parameter dimension �a� is d� in our notation,

his metric �r is the Lr metric given our assumption of row-wise identical distributions, and the Lr

metric on B can be replaced by the Euclidean metric on B using the Lipschitz condition (iii) andthe moment condition (v) in the statement of the Lemma. �

22 Additional Simulation Results

This section provides additional simulation results to those given in Section ??. The details

concerning the models, tests, and simulation scenarios considered are given in Section ??.

22.1 Heteroskedastic Linear IV Model

Table SM-I provides NRP�s of the AR/QLR1 test in the heteroskedastic linear IV model of

Section ?? for sample sizes n = 50 and 500 for k = 4: Table SM-II does likewise for n = 100 and

250 for k = 8: The NRP results in Table SM-I are similar to those in Table I for n = 100 and

250: Even for n = 50; the maximum NRP is :050: On the other hand, the results in Table SM-II

for k = 8 show some over-rejection of the null with the maximum NRP probability being :064 for

n = 100 and :056 for n = 250:

83

TABLE SM-I. Null Rejection Probabilities of the Nominal :05 AR/QLR1 Test for n = 50 and

500 and k = 4; and Base Case Tuning Parameters in the Heteroskedastic Linear Instrumental

Variables Model

n = 50 n = 500

jj�2jj : 40 20 12 4 0 40 20 12 4 0

40 .043 .043 .043 .046 .050 .049 .049 .049 .047 .046

20 .040 .040 .040 .042 .048 .049 .049 .048 .046 .045

jj�1jj 12 .037 .037 .038 .040 .044 .048 .048 .047 .044 .043

4 .018 .018 .019 .022 .033 .037 .034 .030 .030 .037

0 .000 .000 .000 .000 .001 .000 .000 .001 .001 .001

TABLE SM-II. Null Rejection Probabilities of the Nominal :05 AR/QLR1 Test for n = 100

and 250 and k = 8; and Base Case Tuning Parameters in the Heteroskedastic Linear Instrumental

Variables Model

n = 100 n = 250

jj�2jj : 40 20 12 4 0 40 20 12 4 0

40 .043 .045 .049 .061 .064 .045 .045 .046 .054 .056

20 .040 .042 .046 .059 .063 .044 .044 .044 .053 .055

jj�1jj 12 .038 .039 .042 .057 .061 .042 .041 .040 .051 .054

4 .014 .014 .017 .036 .047 .019 .019 .020 .038 .044

0 .000 .000 .000 .001 .001 .000 .001 .001 .001 .001

22.2 Nonlinear IV Model: Inference on the Structural Function

Table SM-III shows little sensitivity of NRP�s of the AR/QLR1 test to �1; but some sensitivity

of power to �1 when jj�jj = 4 and �2 is positive. Table SM-III shows no sensitivity of the NRP�sand power of the AR/QLR1 test to KL; K�

L; and a: The table shows no sensitivity of NRP�s to

Krk; but some sensitivity of power to Krk: The smallest value of Krk; :25; yields noticeably lower

power than larger values.

84

TABLE SM-III. Sensitivity of NRP and Power of the Nominal :05 AR/QLR1 Test to the Tuning

Parameters �1; KL; Krk; K�L; and a for jj�jj = 50 and 4 and for Five Values of �2 for Inference on

the Structural Function at y1 = 2 in the Nonlinear Instrumental Variables Model

Tuning jj�jj = 50 jj�jj = 4Parameter �2 : .00 -.130 -.094 .105 .155 .00 -1.15 -.88 2.7 8.8

.0010 .043 .796 .497 .507 .808 .038 .798 .496 .519 .808

.0025 .044 .799 .502 .507 .807 .038 .799 .496 .514 .805

�1 .0050 .044 .801 .504 .502 .803 .037 .799 .495 .504 .800

.0100 .042 .798 .502 .494 .794 .035 .797 .487 .486 .785

.0150 .041 .795 .498 .483 .784 .032 .794 .475 .465 .766

.01 .045 .804 .510 .513 .813 .039 .803 .503 .524 .800

KL .05 .044 .801 .504 .502 .803 .037 .799 .495 .504 .800

.10 .040 .791 .491 .493 .796 .035 .792 .486 .504 .800

.25 .046 .742 .445 .430 .735 .041 .769 .451 .465 .789

.50 .045 .776 .477 .468 .770 .037 .791 .474 .482 .792

Krk 1.0 .044 .801 .504 .502 .803 .037 .799 .495 .504 .800

2.0 .045 .811 .518 .524 .821 .037 .802 .500 .538 .810

4.0 .045 .812 .521 .530 .824 .038 .800 .500 .565 .826

.001 .044 .801 .504 .502 .803 .037 .799 .495 .504 .803

K�L .005 .044 .801 .504 .502 .803 .037 .799 .495 .504 .800

.010 .044 .801 .504 .502 .803 .037 .799 .495 .504 .707

.00 .044 .801 .504 .502 .803 .037 .799 .495 .504 .800

a 10�6 .044 .801 .504 .502 .803 .037 .799 .495 .504 .800

.01 .043 .801 .504 .502 .803 .036 .799 .495 .504 .798

85

TABLE SM-IV. Sensitivity of NRP and Power of the Nominal :05 AR/QLR1 Test to the Sample

Size, n; and Number of Instruments, k; for jj�jj = 50 and 4 and for Five Values of �2 for Inferenceon the Structural Function at y1 = 2 in the Nonlinear Instrumental Variables Model

jj�jj = 50 jj�jj = 4�2 : .00 -.130 -.094 .105 .155 .00 -1.15 -.88 2.7 8.8

50 .026 .271 .211 .191 .229 .018 .125 .071 .097 .256

100 .036 .596 .383 .348 .545 .025 .350 .186 .228 .536

n 250 .040 .790 .496 .479 .764 .034 .670 .381 .411 .763

500 .044 .801 .504 .502 .803 .037 .799 .495 .504 .800

1000 .048 .775 .481 .489 .787 .042 .868 .570 .572 .755

4 .044 .801 .504 .502 .803 .037 .799 .495 .504 .800

k 8 .044 .648 .362 .338 .620 .037 .742 .415 .348 .664

12 .049 .506 .268 .237 .462 .037 .658 .348 .278 .572

22.3 Nonlinear IV Model: Inference on the Derivative of the Structural

Function

Table SM-IV provides NRP�s for the nominal .05 AR/QLR1 test for hypotheses concerning the

derivative of the structural function at y1 = 2: The table shows that the NRP�s vary between :007

and :052 over these cases. The lowest NRP�s occur for jj�jj = 0: In the base case scenario, n = 500and k = 4; the NRP�s are in [:034; :047] for jj�jj � 4:

TABLE SM-V. Null Rejection Probabilities of the Nominal :05 AR/QLR1 Test for Base Case

Tuning Parameters for Inference on the Derivative of the Structural Function at y1 = 2 in the

Nonlinear Instrumental Variables

k n Errors jj�jj : 100 75 50 35 20 14 8 4 0

4 50 Homoskedastic .033 .031 .028 .024 .024 .026 .025 .019 .002

4 100 Homoskedastic .039 .039 .039 .039 .039 .037 .033 .026 .007

4 250 Homoskedastic .042 .043 .044 .043 .040 .037 .033 .031 .017

4 500 Homoskedastic .045 .046 .047 .043 .039 .038 .037 .034 .016

8 100 Homoskedastic .051 .052 .051 .050 .050 .051 .048 .036 .007

8 250 Homoskedastic .045 .046 .047 .047 .046 .044 .040 .033 .023

4 250 Heteroskedastic .032 .031 .030 .029 .026 .023 .017 .011 .008

86

The results of Table SM-VI are very similar to those of Table SM-III. The results of Table

SM-VII are broadly similar to those of Table SM-IV.

TABLE SM-VI. Sensitivity of NRP and Power of the Nominal :05 AR/QLR1 Test to the Tuning


the Derivative of the Structural Function in the Nonlinear Instrumental Variables Model

Tuning jj�jj = 50 jj�jj = 4Parameter �2 : .00 -.085 -.061 .070 .104 .00 -.80 -.60 1.6 4.5

.0010 .046 .796 .495 .497 .792 .035 .805 .505 .520 .811

.0025 .046 .796 .495 .503 .797 .035 .806 .505 .515 .809

�1 .0050 .046 .797 .495 .504 .800 .034 .805 .503 .505 .802

.0100 .046 .797 .495 .506 .804 .032 .799 .495 .485 .787

.0150 .046 .797 .495 .506 .804 .030 .792 .482 .463 .770

.01 .046 .797 .496 .505 .801 .036 .810 .508 .523 .813

KL .05 .046 .797 .495 .504 .800 .034 .805 .503 .505 .802

.10 .044 .789 .486 .489 .788 .032 .799 .494 .505 .802

.25 .048 .734 .425 .430 .736 .038 .778 .458 .470 .788

.50 .050 .770 .466 .469 .769 .035 .796 .482 .485 .793

Krk 1.0 .046 .797 .495 .504 .800 .034 .805 .503 .505 .802

2.0 .047 .809 .508 .521 .816 .036 .811 .509 .530 .816

4.0 .048 .810 .511 .526 .819 .039 .813 .516 .552 .839

.001 .046 .797 .495 .504 .800 .034 .805 .503 .505 .802

K�L .005 .046 .797 .495 .504 .800 .034 .805 .503 .505 .802

.010 .046 .797 .495 .504 .800 .034 .800 .498 .505 .796

.00 .046 .797 .495 .504 .800 .034 .805 .503 .505 .802

a 10�6 .046 .797 .495 .504 .800 .034 .805 .503 .505 .802

.01 .046 .797 .495 .504 .800 .034 .805 .503 .504 .802

87

TABLE SM-VII. Sensitivity of NRP and Power of the Nominal :05 AR/QLR1 Test to the

Sample Size, n; and Number of Instruments, k; for jj�jj = 50 and 4 and for Five Values of �2

for Inference on the Derivative of the Structural Function in the Nonlinear Instrumental Variables

Model

jj�jj = 50 jj�jj = 4�2 : .00 -.085 -.061 .070 .104 .00 -.80 -.60 1.6 4.5

50 .030 .371 .246 .213 .327 .018 .053 .035 .064 .126

100 .040 .697 .470 .433 .660 .024 .156 .090 .177 .404

n 250 .043 .834 .544 .520 .805 .031 .492 .278 .380 .723

500 .046 .797 .495 .504 .800 .034 .805 .503 .505 .802

1000 .048 .692 .403 .431 .726 .039 .959 .727 .590 .834

4 .046 .797 .495 .504 .800 .034 .805 .503 .505 .802

k 8 .046 .603 .338 .322 .591 .030 .776 .437 .356 .669

12 .053 .458 .244 .232 .439 .042 .693 .360 .284 .565

23 Additional Second-Step C(�) Tests

23.1 C(�)-QLR2 Test

Here, we de�ne a C(�) version of the test in Andrews and Guggenberger (2015) (AG), which

we refer to as the C(�)-QLR2 test. The second-step C(�)-QLR2 test statistic is

QLR22n(�) := AR2n(�)� �min(n bQ2n(�)); wherebQ2n(�) := �egn(�); bD�2n(�)�0 cM1n(�)�egn(�); bD�2n(�)� 2 R(p2+1)�(p2+1);bD�2n(�) := b�1=2n (�) bD2n(�)bL1=22n (�) 2 Rk�p2 ;bL2n(�) := (�2; Ip2)(

b�"2n(�))�1(�2; Ip2)0 2 Rp2�p2 ; (23.1)

egn(�) is de�ned in (??), and b�"2n(�) is de�ned below.7The C(�)-QLR2 test uses a conditional critical value that depends on the k � p2 matrix

n1=2 bD�2n(�1) and the k�k projection matrix cM1n(�1): For nonrandom D2 2 Rk�p2 and nonrandomsymmetric psd M 2 Rk�k; let

QLR2k;p2(D2;M) := Z0MZ � �min((Z;D2)0M(Z;D2)); where Z � N(0k; Ik): (23.2)

7Unlike the random perturbation of b�1=2n (�) bD1n(�) by an�1=2�1 in (??) and the random perturbation ofb�1=2n (�) bD2n(�) by an�1=2�2 in (??), no random perturbation of bD�2n(�) is needed in the de�nition of QLR22n(�):

88

De�ne cQLR2k;p2(D2;M; 1 � �) to be the 1 � � quantile of the distribution of QLR2k;p2(D2;M): For

given D2 and M; cQLR2k;p2

(D2;M; 1� �) can be computed by simulation very quickly and easily.For given �1 2 �1; the nominal level � second-step C(�)-QLR2 test rejects H0 : �2 = �20 when

�QLR22n (�1; �) := QLR22n(�1; �20)� cQLR2k;p2(n1=2 bD�2n(�1; �20);cM1n(�1; �20); 1� �) > 0: (23.3)

When p2 = 1; the �min(n bQ2n(�)) term that appears in (23.1) can be solved in closed form. In

this case, the QLR22n(�) statistic can be written as

QLR22n(�) :=1

2

�AR2n(�)� rk�2n(�) +

q(AR2n(�)� rk�2n(�))2 + 4LM�

2n(�) � rk�2n(�)�; where

LM�2n(�) := negn(�)0P bD�

2n(�)egn(�);

rk�2n(�) := n bD�2n(�)0cM1n(�) bD�2n(�); (23.4)

and AR2n(�) is de�ned in (??). When p2 = 1; the C(�)-QLR2 critical value, for a nominal level �

test, is as in (??) with rk2n(�) = rk�2n(�) and WIyn(�) = 0: cQLR1(1� �; rk�2n(�); 0):

Now, we de�ne b�"2n(�): De�nebR2n(�) := �B2(�)0 Ik� bV2n(�) (B2(�) Ik) 2 R(p2+1)k�(p2+1)k; wherebV2n(�) := n�1

nXi=1

�f2i(�)� bf2n(�)��f2i(�)� bf2n(�)�0 2 R(p2+1)k�(p2+1)k; (23.5)

f2i(�) :=

0@ gi(�)

vec(G2i(�))

1A ; bf2n(�) :=0@ bgn(�)vec( bG2n(�))

1A ; and B2(�) :=0@ 1 00p2

��2 �Ip2

1A :Let bR2j`n(�) denote the (j; `) k � k submatrix of bR2n(�) for j; ` � p2 + 1:8

We de�ne b�2n(�) 2 R(p2+1)�(p2+1) to be the symmetric pd matrix whose (j; `) element isb�2j`n(�) = tr( bR2j`n(�)0b�1n (�))=k (23.6)

for j; ` � p2 + 1: AG use an eigenvalue-adjusted version of b�2n(�); denoted b�"2n(�):The eigenvalue adjustment is de�ned as follows. LetH 2 RdH�dH be any non-zero positive semi-

de�nite (psd) matrix with spectral decomposition AH�HA0H ; where �H = Diagf�H1; :::; �HdHg isthe diagonal matrix of eigenvalues of H with nonnegative nonincreasing diagonal elements and

AH is a corresponding orthogonal matrix of eigenvectors of H: For " > 0; the eigenvalue-adjusted

8That is, bR2j`n(�) contains the elements of bR2n(�) indexed by rows (j � 1)k+1 to jk and columns (`� 1)k to `k:

89

matrix H" is

H" := AH�"HA

0H ; where �

"H := Diagfmaxf�H1; �max(H)"g; :::;maxf�HdH ; �max(H)"gg; (23.7)

where �max(H) denotes the maximum eigenvalue of H: Note that H" = H whenever the condition

number of H is less than or equal to 1=" (for " � 1):9

The matrix b�"2n(�) is de�ned in (23.7) with H = b�2n(�): Based on the �nite-sample simulations,AG recommend using " = :05:


In this section we introduce a C(�) version of the CQLR test in I. Andrews and Mikusheva

(2016) (AM), which we refer to as the C(�)-QLR3 test. For given �1 2 �1; the second-step C(�)-QLR3 test statistic is

AR2n(�1; �20)� inf�22�2

AR2n(�1; �2): (23.8)

Next, we de�ne AM�s data-dependent critical value. For � := (�1; �2) 2 � and �0 := (�01; �020)0 2 �;let

h2n(�) := n1=2bgn(�)� bn(�; �0)b�1n (�0; �0)n1=2bgn(�0) andg�2n(�) := h2n(�) + bn(�; �0)b�1n (�0; �0)��2; (23.9)

where ��2 � N(0; bn(�0; �0)) given bn(�; �) and bn(�) := bn(�; �): We view h2n(�1; �2) as a sto-

chastic process indexed by �2 with �1 �xed. It is designed to be asymptotically independent of

n1=2bgn(�1; �20) when �0 = (�01; �020)0 is the true value. De�neQLR3�2n(�1) := AR�2n(�1; �20)� inf

�22�2AR�2n(�1; �2); where

AR�2n(�) := g�2n(�)0b�1=2n (�)cM1n(�)b�1=2n (�)g�2n(�): (23.10)

Let cvQLR32n (h2n; �1; �) denote the 1�� quantile of the conditional distribution of QLR3�2n(�1) givenh2n := h2n(�) and bn(�; �):

For given �1 2 �1; the �1-orthogonalized nominal � second-step CQLR3 test rejects H0 : �2 =9AG shows that the eigenvalue-adjustment procedure possesses the following desirable properties: (i) H" is

uniquely de�ned, (ii) �min(H") � �max(H)"; (iii) �max(H")=�min(H") � maxf1="; 1g; (iv) for all c > 0; (cH)" = cH";

and (v) H"n ! H" for any sequence of psd matrices fHn : n � 1g with Hn ! H:

90

�20 when

�QLR32n (�1; �) := AR2n(�1; �20)� inf�22�2

AR2n(�1; �2)� cvQLR32n (h2n; �1; �) > 0: (23.11)

The second-step CQL3 test is applicable in moment condition models, but not minimum distance

models.

91

References

Aitchison, J. and S. D. Silvey (1959): �Maximum-Likelihood Estimation of Parameters Subject

to Restraints,�Annals of Mathematical Statistics, 29, 813�828.

Andrews, D. W. K. (1988): �Laws of Large Numbers for Dependent Non-identically Distributed

Random Variables,�Econometric Theory, 4, 458�467.

� � � (1991a): �Generic Uniform Convergence,�Econometric Theory, 8, 241�257.

� � � (1991b): �Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estima-

tion,�Econometrica, 59, 817�858.

� � � (1994): �Empirical Processes in Econometrics,�Ch. 37 in Handbook of Econometrics, Vol.

IV, ed. by R. F. Engle and D. L. McFadden. New York: Elselvier.

Andrews, D. W. K., X. Cheng, and P. Guggenberger (2011): �Generic Results for Establishing

the Asymptotic Size of Con�dence Sets and Tests,�Cowles Foundation Discussion Paper No.

1813, Yale University.

Andrews, D. W. K. and P. Guggenberger (2015): �Identi�cation- and Singularity-Robust Inference

for Moment Condition Models,� Cowles Foundation Discussion Paper No. 1978, revised

December 2015, Yale University.

� � � (2017): �Asymptotic Size of Kleibergen�s LM and Conditional LR Tests for Moment Con-

dition Models,�Econometric Theory, 33, forthcoming. Also available as Cowles Foundation

Discussion Paper No. 1977, revised September 2015, Yale University.

Andrews, I. and A. Mikusheva (2016): �Conditional Inference with a Functional Nuisance Para-

meter,�Econometrica, 84, 1571�1612.

Cramér, H. (1946): Mathematical Methods of Statistics. Princeton, NJ: Princeton University

Press.

Crowder, M. J. (1976): �Maximum Likelihood Estimation for Dependent Observations,�Journal

of the Royal Statistical Society, Ser. B, 38, 45�53.

Davydov, Y. A. (1968): �Convergence of Distributions Generated by Stationary Stochastic Processes,�

Theory of Probability and Its Applications, 13, 691�696.

92

de Jong, R. M. (1997): �Central Limit Theorems for Dependent Heterogeneous Random Vari-

ables,�Econometric Theory, 13, 353�367.

Newey, W. K., and K. West (1987): �A Simple, Positive Semi-de�nite, Heteroskedasticity and

Autocorrelation Consistent Covariance Matrix,�Econometrica, 55, 703�708.

van der Vaart, A. W. (1998): Asymptotic Statistics. Cambridge, UK: Cambridge University Press.

93

10 Outline

References to sections with section numbers less than 10 refer to sections of the main pa-

per �Identi�cation-Robust Subvector Inference.�Similarly, all theorems and lemmas with section

numbers less than 10 refer to results in the main paper.

Section 11 generalizes the asymptotic results in Section 8 for the moment condition model from

i.i.d. observations to strictly stationary strong mixing time series observations.

Sections 12-14 of this Supplemental Material (SM) provide high-level su¢ cient conditions for

the parts of Assumptions B, C, and OE (stated in Section 5) that concern (i) the estimator set, (ii)

the �rst-step AR CS, and (iii) the data-dependent second-step signi�cance level, respectively, in

the moment condition model. Sections 15-17 provide high-level su¢ cient conditions for the parts

of Assumptions B, C, and OE that concern the second-step C(�)-AR, C(�)-LM, and C(�)-QLR1

tests, respectively, in the moment condition model. Section 18 amalgamates the conditions in

Sections 12-17 for the two-step AR/AR, AR/LM, and AR/QLR1 subvector tests. Sections 19 and

20 prove Theorems 8.1 and 8.2 by using primitive conditions to verify the su¢ cient conditions in

Section 18.

Section 21 proves the times series results in Theorem 11.1.

Section 22 provides some additional simulation results to those presented in the main paper.

For illustrative purposes, Section 23 de�nes C(�) versions of the CQLR tests in Andrews and

Guggenberger (2015) and I. Andrews and Mikusheva (2016), but does not verify the high-level

conditions in Section 5 for these tests.

In this SM, a null sequence S is de�ned as in (5.2), i.e., S := f(��n; Fn) : (�1�n; Fn) 2 FSV ;�2�n = �20; n � 1g; except in some sections where S is de�ned with a speci�c parameter space, suchas FAR=AR; in place of the generic parameter space FSV : For an assumption that is stated for asequence S; we say that it holds for a subsequence Sm if the subsequence version of the assumption

holds.

Throughout the SM, we use the following notational convention when considering tests of H0 :

�2 = �20: For any function A(�) of � = (�01; �02)0; we de�ne

A(�1) := A(�1; �20) and A := A(�1�n; �20); (10.1)

where �20 is the null value of �2 and �1�n is the true value of �1:

We let B(�1; ") denote a closed ball in Rp1 centered at �1 with radius " > 0:

We let � jsn for s = 1; :::; pj denote the singular values of �1=2n Gjn written in nonincreasing

order, for j = 1; 2; where Gjn 2 Rk�pj and n 2 Rk�k are (nonrandom) population matrices

2

that correspond to the sample Jacobian (wrt �j) bGjn 2 Rk�pj and sample variance matrix bn;respectively. Let Gn := [G1n : G2n] 2 Rk�p: Because they arise frequently below, for notational

simplicity, we let

� jn := � jpjn = smallest singular value of �1=2n Gjn for j = 1; 2 and

�n := smallest singular value of �1=2n Gn: (10.2)

The quantity � jn is a measure of the (local) strength of identi�cation of �j at � = (�01�n; �020)

0 and

�n is a measure of the (local) strength of identi�cation of � at � = (�01�n; �020)

0:

Similarly, we let ��jsn for s = 1; :::; pj denote the singular values of �1=2n Gjn�jn written in

nonincreasing order, for j = 1; 2; where �jn 2 Rpj�pj is the (nonrandom) population matrix thatcorresponds to b�jn 2 Rpj�pj de�ned in (7.5). For notational simplicity, we let

��jn := ��jpjn = smallest singular value of �1=2n Gjn�jn for j = 1; 2 and

��n := smallest singular value of �1=2n GnDiagf�1;�2g: (10.3)

The quantity ��jn is another (equivalent) measure of the (local) strength of identi�cation of �j at

� = (�01�n; �020)

0:

In Sections 12 and 13 in this SM, the results are designed to hold not just for the moment

condition model, but also for minimum distance models and moment condition models where the

moments may depend on n1=2-consistent and asymptotically normal preliminary estimators. But,

the de�nitions of Gjn(�) for j = 1; 2 and n(�) di¤er across these models. In consequence, for

generality, Gjn(�) and n(�) are de�ned in these sections by the conditions they must satisfy in the

various results given, rather than by explicit expressions. For sample moments bgn(�) (without anypreliminary estimators), this leads to Gjn(�) = EFn bGjn(�) and n(�) = V arFn(n

1=2bgn(�)): Thelatter de�nitions are employed in Section 15 and the sections that follow it, which consider only

the moment condition model.

Many results in this SM are stated to hold for both a sequence S and a subsequence Sm: For

brevity, we only prove these results for a sequence S: For a subsequence Sm; the proofs only require

the minor notational adjustment of changing n to mn:

11 Time Series Observations

In this section, we generalize the results of Theorems 8.1 and 8.2 from i.i.d. observations to

strictly stationary strong mixing observations. In the time series case, F denotes the distribution of

3

the stationary in�nite sequence fWi : i = :::; 0; 1; :::g: Asymptotics under drifting sequences of truedistributions fFn : n � 1g are used to establish the correct asymptotic size of the two-step AR/AR,AR/LM, and AR/QLR1 tests. Under such sequences, the observations form a triangular array of

row-wise strictly stationary observations. In the time series case, we de�ne F (�) di¤erently from

its de�nitions in (8.1) for the i.i.d. case:

F (�) :=

1Xm=�1

�EF gi(�)g

0i�m(�)� EF gi(�)EF gi(�)0

�: (11.1)

Note that F (�) = limV arF (n�1=2Pni=1 gi(�)): We let f�F (m) : m � 1g denote the strong mixing

numbers of the observations under the distribution F:

The time series analogue FTS;AR=AR of the space of distributions FAR=AR; de�ned in (8.8), is

FTS;AR=AR := f(�1; F ) : EF gi(�1) = 0k; �1 2 �1�; fWi : i = :::; 0; 1; :::g are stationary

and strong mixing under F with �F (m) � Cm�d for some d > (2 + )= ;

EF jjgi(�1)jj2+ �M; EF jjvec(G1i(�1))jj2+ �M; EF �2+ 1i �M;

�min(F (�1)) � �; V arF (jjG1si(�1)jj) � � 8s = 1; :::; p1g (11.2)

for some ; � > 0 and M < 1; where F (�) is de�ned in (11.1). For the two-step AR/LM and

AR/QLR1 tests with time series observations, we use the parameter space FTS;AR=LM;QLR1; whichis de�ned as in (8.10), but with FTS;AR=AR in place of FAR=AR:

For CS�s, we use the parameter spaces F�;TS;AR=AR and F�;TS;AR=LM;QLR1; which are de-�ned as in (8.9) and (8.11), respectively, but with FTS;AR=AR(�2) in place of FAR=AR(�2); whereFTS;AR=AR(�20) denotes FTS;AR=AR with its dependence on �20 made explicit.

Next, we de�ne the second-step C(�)-AR, C(�)-LM, and C(�)-QLR1 tests in the time series

context. To do so, we let

VF (�) := limV arF

0@n�1=2 nXi=1

0@ gi(�)

vec(Gi(�))

1A1A=

1Xm=�1

EF

0@ gi(�)� EF gi(�)vec(Gi(�)� EFGi(�))

1A0@ gi�m(�)� EF gi�m(�)vec(Gi�m(�)� EFGi�m(�))

1A0 : (11.3)The second equality holds for all (�; F ) 2 F�;TS;AR=AR:

The test statistics depend on an estimator bVn(�) of VF (�): This estimator (usually) is a het-eroskedasticity and autocorrelation consistent (HAC) variance estimator based on the observations

4

ffi(�) � bfn(�) : i � ng; where fi(�) := (gi(�)0; vec(Gi(�))0)0 and bfn(�) := (bgn(�)0; vec( bGn(�))0)0:

There are a number of HAC estimators available in the literature, e.g., see Newey and West (1987)

and Andrews (1991b). The asymptotic properties of the tests are the same for any consistent HAC

estimator. Hence, for generality, we do not specify a particular estimator bVn(�): Rather, we stateresults that hold for any estimator bVn(�) that satis�es the following consistency conditions. TheAssumptions V and V-CS that follow are applied with two-step tests and CS�s, respectively.

Assumption V. 8K <1; sup�12B(�1�n;K=n1=2) jjbVn(�1)� VFn(�1)jj !p 0 under any null sequence

f(�1�n; Fn) 2 FTS;AR=AR : n � 1g for which VFn(�1�n)! V1 for some pd matrix V1:

Assumption V-CS. 8K <1; sup�12B(�1�n;K=n1=2) jjbVn(�1; ��2n)� VFn(�1; ��2n)jj !p 0 under any

sequence f(��n; Fn) 2 F�;TS;AR=AR : n � 1g for which VFn(��n)! V1 for some pd matrix V1:

We write the (p+ 1)k � (p+ 1)k matrix bVn(�) in terms of its k � k submatrices:

bVn(�) =26666664

bn(�) b�11n(�)0 � � � b�2p2n(�)0b�11n(�) bVG11n(�) � � � bV 0Gp1n(�)...

.... . .

...b�2p2n(�) bVGp1n(�) � � � bVGppn(�)

37777775 ; (11.4)

where the subscripts on b�jsn(�) run from (j; s) = (1; 1); :::; (1; p1); (2; 1); :::; (2; p2):

In the time series case, for the two-step C(�)-AR, C(�)-LM, and C(�)-QLR1 tests, we use the

same de�nitions as in Section 3 for the moment condition model and Section 7, but with bn(�) andb�jsn(�) for j = 1; :::; pj ; j = 1; 2 de�ned as in Assumption V and (11.4), rather than as in (3.6) and(7.9). The two-step C(�)-AR, C(�)-LM, and C(�)-QLR1 CS�s in the time series case are de�ned

using (4.4), the de�nitions just given for the corresponding tests, and Assumption V-CS in place

of Assumption V.

In the time series case, we employ the following assumption in addition to Assumption SI.

Assumption SI-TS. (i) For the null sequence S; the strong mixing numbers satisfyP1m=1 �

1=q�1=rFn

(m) < 1 for some q = maxfp1 + �1; 2g and r = q + �1 for some �1 > 0; where

r is as in Assumption SI, and

(ii) sup�12�1 jjbn(�1) � Fn(�1)jj = op(1) for bn(�) and F (�) de�ned in (11.4) and (11.1),respectively.

For the time series case, the asymptotic results are as follows.

Theorem 11.1 Suppose the two-step AR/AR, AR/LM, and AR/QLR1 tests and CS�s are de�ned

as in this section and Assumption V or V-CS holds. Then, the results of Theorems 8.1 and 8.2 hold

5

with the parameter spaces FTS;AR=AR; FTS;AR=LM;QLR1; F�;TS;AR=AR; and F�;TS;AR=LM;QLR1 inplace of FAR=AR; FAR=LM;QLR1; F�;AR=AR; and F�;AR=LM;QLR1; respectively, and with AssumptionSI augmented by Assumption SI-TS (everywhere Assumption SI appears in Theorems 8.1 and 8.2).

Comment: Theorem 11.1 shows that the results of Theorems 8.1 and 8.2 for i.i.d. observations

generalize to strictly stationary strong mixing observations, provided the spaces of distributions are

adjusted suitably and the variance estimator bVn(�) of VF (�) is de�ned appropriately.12 Veri�cation of Assumptions on the Estimator Set b�1n

12.1 Estimator Set Results

The estimator set (ES), b�1n; is de�ned in (7.3). The following lemma veri�es AssumptionC(i) under high-level assumptions on b�1n including the assumption that there exist n1=2-consistentsolutions f�1n : n � 1g to the FOC�s given in (7.3) (which implies that �1 is locally stronglyidenti�ed given �20): Lemma 12.2 below provides su¢ cient conditions for the existence of such

solutions f�1ng:

Assumption ES1. For the null sequence S; there exist solutions f�1n 2 �1 : n � 1g to the FOC�sgiven in (7.3) that satisfy n1=2(�1n � �1�n) = Op(1):

Assumption ES2. For the null sequence S; (i) bgn(�1) is di¤erentiable on B(�1�n; ") for some " > 0(for all sample realizations) 8n � 1; (ii) bgn = Op(n

�1=2); (iii) sup�12B(�1�n;") jj bG1n(�1)jj = Op(1) for

some " > 0; (iv) ncn !1 for fcn : n � 1g in (7.3), and (v) cW1n is a symmetric psd k � k matrix

that satis�es cW1n = Op(1):

Lemma 12.1 Suppose b�1n is de�ned in (7.3) and bQn(�) is the criterion function de�ned in (7.2).Let S be a null sequence (or Sm a null subsequence) that satis�es Assumptions ES1 and ES2. Then,b�1n is non-empty wp!1 and Assumption C(i) holds for the sequence S (or subsequence Sm).

The following lemma provides su¢ cient conditions for Assumption ES1 for sequences S that

satisfy lim �1n > 0 (i.e., for �1-locally-strongly-identi�ed sequences). Let �1 = (�11; :::; �1p1)0:

Assumption FOC. For the null sequence S and some " > 0; (i) lim infn!1 �1n > 0; (ii)

sup�12B(�1�n;") jjbgn(�1)� gn(�1)jj = op(1) for some nonrandom Rk-valued functions fgn(�) : n � 1g;(iii) gn = 0k 8n � 1; (iv) �1�n ! �1�1 for some �1�1 2 �1; (v) bgn(�1) is twice continuously di¤er-entiable on B(�1�n; ") (for all sample realizations) 8n � 1; (vi) bgn = Op(n

�1=2); (vii) gn(�1) is twice

continuously di¤erentiable on B(�1�n; ") 8n � 1; (viii) sup�12B(�1�n;") jj bG1n(�1)�G1n(�1)jj = op(1)

for some nonrandom Rk�p1-valued functions fG1n(�) : n � 1g; (ix) sup�12B(�1�n;") jjG1n(�1)jj =

6

O(1); (x) sup�12B(�1�n;"n) jjG1n(�1)�G1njj = o(1) for all sequences of positive constants "n ! 0; (xi)

G1n ! G11 for some matrix G11 2 Rk�p1 ; (xii) G1n(�1) = (@=@�01)gn(�1) 8�1 2 B(�1�n; ");8n �1; (xiii) lim infn!1 �min(n) > 0; (xiv) b�1n = Op(1); where b�1n := maxs;u�p1 sup�12B(�1�1;")

jj(@2=@�1s@�1u)bgn(�1)jj; (xv) �1n = O(1); where �1n := maxs;u�p1 sup�12B(�1�1;")

jj(@2=@�1s@�1u)gn(�1)jj; and (xvi) cW1n is symmetric and psd and cW1n !p W11 for some non-

random nonsingular matrix W11 2 Rk�k:

In the moment condition model (where bgn(�1) = n�1Pni=1 g(Wi; �1)); we have gn(�1) =

EFnbgn(�1) in Assumption FOC(ii), G1n(�1) = EFn bG1n(�1) in Assumption FOC(viii), and As-sumptions FOC(vii), (xii), and (xv) (with " replaced by "=2 in Assumptions FOC(xii) and (xv),

which does not matter in Lemma 12.2 below) are implied by Assumptions FOC(v) and (xiv) and

EFnb�1n = O(1) (for b�1n de�ned in Assumption FOC(xiv)).13 In this case, by Assumption FOC(xii),

G1n(�1) = EFn bG1n(�1) = (@=@�01)EFnbgn(�1):Lemma 12.2 Let S be a null sequence (or Sm a null subsequence) that satis�es Assumption FOC.

Then, Assumption ES1 holds for the sequence S (or subsequence Sm).

Comments: (i). The result of Lemma 12.2 is similar to classical results in the statistical literature,

see Cramér (1946), Aitchison and Silvey (1959), and Crowder (1976). The proof of Lemma 12.2

follows that of van der Vaart (1998, Thm. 5.42, p. 69).

(ii) The estimator bn plays no role in Lemma 12.2. Nevertheless, Assumption FOC(xiii)

involves n because Assumptions FOC(i) and (xiii) imply that the limit of the smallest singular

value of G1n is positive (since �1n is the smallest singular value of �1=2n G1n):

The next lemma provides su¢ cient conditions for Assumption OE(i).

Lemma 12.3 Let S be a null sequence (or Sm a null subsequence) for which (i) dH(�1�n; b�1n) =Op(n

�1=2) and (ii) dH(�1�n; CS1n [ f�1�ng) = Op(n�1=2): Then, dH(�1�n; CS

+1n) = Op(n

�1=2) holds


Comments: 1. Lemmas 12.4 and 13.2 below provide primitive su¢ cient conditions for conditions

(i) and (ii), respectively, of Lemma 12.3.

2. Condition (i) of Lemma 12.3 requires that b�1n 6= ? wp!1 by the de�nition of dH :13To see this, suppose Assumptions FOC(v) and (xiv) hold and EFnb�1n = O(1): Taking expections under Fn in

the �rst line of (12.6) below gives Assumption FOC(vii). By a similar expansion to that in (12.6), but about a pointin B(�1�n; "=2); rather than �1�n; gives (@=@�01)gn(�1) = EFn bG1n(�1) = G1n(�1) 8�1 2 B(�1�n; "=2); which impliesAssumption FOC(xii), and (@2=@�1s@�1`)gn(�1) = EFn(@

2=@�1s@�1`)bgn(�1) 8�1 2 B(�1�n; "=2); which implies that�1n � EFnb�1n = O(1) and Assumption FOC(xv) holds.

7

Now we verify dH(�1�n; b�1n) = Op(n�1=2) for b�1n de�ned in (7.3) for sequences S with lim

infn!1 �1n > 0; i.e., for �1-locally strongly-identi�ed sequences.

Assumption ES3. For the null sequence S; (i) lim infn!1 �1n > 0; (ii) b�1n is non-empty wp!1,(iii) bgn(�1) is di¤erentiable on B(�1�n; ") for some " > 0 (for all sample realizations) 8n � 1;

(iv) bgn = Op(n�1=2); (v) sup�12B(�1�n;") jj bG1n(�1) � G1n(�1)jj = op(1) for some " > 0 for some

nonrandom Rk�p1-valued functions fG1n(�) : n � 1g; (vi) sup�12B(�1�n;"n) jjG1n(�1) � G1njj = o(1)

for all sequences of positive constants "n ! 0; (vii) G1n ! G11 for some matrix G11 2 Rk�p1 ; (viii)bn � n !p 0 for some nonrandom matrices fn 2 Rk�k : n � 1g; (ix) lim infn!1 �min(n) > 0;

(x) cn ! 0 for fcn : n � 1g in (7.3), and (xi) cW1n is symmetric and psd and cW1n !p W11 for some

nonrandom nonsingular matrix W11 2 Rk�k:

Note that Assumption ES3(ii) is implied by Assumptions ES1 and ES2 by Lemma 12.1 and,

hence, by Assumptions FOC and ES2 by Lemmas 12.1 and 12.2.

Assumption ES4. For the null sequence S; (i) sup�12�1 n1=2jjbgn(�1)� gn(�1)jj = Op(1) for some

nonrandom Rk-valued functions fgn(�) : n � 1g and (ii) lim infn!1 inf�1 =2B(�1�n;") jjgn(�1)jj > 0

8" > 0:

Lemma 12.4 Suppose b�1n is of the form in (7.3) with bQn(�) as in (7.2). Let S be a null se-

quence (or Sm a null subsequence) that satis�es Assumptions ES3 and ES4. Then, dH(�1�n; b�1n) =Op(n

�1=2) for the sequence S (or subsequence Sm).


Proof of Lemma 12.1. Let �1n be as in Assumption ES1. If �1n 2 b�1n wp!1, thend(�1�n; CS

+1n) � d(�1�n; b�1n) � d(�1�n; f�1ng) = jj�1�n � �1njj = Op(n

�1=2); (12.1)

where the �rst two inequalities hold because f�1ng � b�1n � CS+1n wp!1 using the de�nition ofCS+1n in (4.1) and the last inequality holds by Assumption ES1. Hence, Assumption C(i) holds andb�1n is non-empty wp!1.

It remains to show �1n 2 b�1n wp!1. By Assumption ES1, �1n satis�es the �rst condition inthe de�nition of b�1n in (7.3). Hence, it remains to show that the second condition in the de�nitionof b�1n in (7.3) holds for �1n: That is, we need to show

bQn(�1n) � inf�12�1

bQn(�1) + cn wp! 1. (12.2)

8

Element-by-element mean-value expansions of bgn(�1n) about �1�n givebgn(�1n) = bgn + bG1n(e�1n)(�1n � �1�n) = Op(n

�1=2) +Op(1)Op(n�1=2) = Op(n

�1=2); (12.3)

where, as de�ned above, bgn := bgn(�1�n); e�1n lies between �1n and �1�n and may di¤er across therows of bG1n(e�1n); the �rst equality uses Assumption ES2(i), and the second equality holds byAssumptions ES1, ES2(ii), and ES2(iii). Equations (7.2) and (12.3) and Assumption ES2(v) yieldbQn(�1n) = Op(n

�1): Hence, we have

bQn(�1n)� cn = Op(n�1)� cn < 0 � inf

�12�1bQn(�1); (12.4)

where the strict inequality holds wp!1 because ncn ! 1 by Assumption ES2(iv). Hence, (12.2)

holds. �

Proof of Lemma 12.2. First, we establish the existence of consistent (as opposed to n1=2-

consistent) solutions to the FOC�s. Let

bn(�1) := bG1n(�1)0cW1nbgn(�1) and n(�1) := G1n(�1)0W11gn(�1): (12.5)

We use essentially the same argument as in van der Vaart (1998, Thm. 5.42, p. 69), but withbn(�1) and n(�1) in place of van der Vaart�s n(�) and (�); respectively. The main di¤erencesare that the population quantity n(�1) here depends on n; whereas van der Vaart�s population

quantity (�) does not; bn(�1) is a product of three random matrices none of which needs to be a

sample average, whereas van der Vaart�s n(�) is a sample average; and n(�1) here is the product

of three population matrices, whereas van der Vaart�s (�) is a single population matrix.

For �1 2 B(�1�n; ") (with " > 0 as in Assumption FOC), element-by-element two-term Taylor

expansions of bgn(�1) about �1�n givebgn(�1) = bgn + bG1n � (�1 � �1�n) + 1

2

p1Xs=1

(�1s � �1�ns)@

@�1sbG1n(e�1n)(�1 � �1�n)

= op(1) +G1n � (�1 � �1�n) +Op(1)jj�1 � �1�njj2; (12.6)

where e�1n lies between �1n and �1�n and may di¤er across rows of (@=@�1s) bG1n(e�1n); �1 = (�11; :::;�1p1)

0; the op(1) and Op(1) terms (in (12.6) and below) hold uniformly over �1 2 B(�1�n; ") as

n ! 1; �1�n = (�1�n1; :::; �1�np1)0; the �rst equality uses Assumption FOC(v), and the second

equality uses Assumptions FOC(vi), (viii), and (xiv).

Similarly, for �1 2 B(�1�n; "); element-by-element two-term Taylor expansions of gn(�1) about

9

�1�n give

gn(�1) =@

@�01gn � (�1 � �1�n) +

1

2

p1Xs=1

(�1s � �1�ns)@2

@�1s@�01

gn(e�1n)(�1 � �1�n)= G1n � (�1 � �1�n) +O(1)jj�1 � �1�njj2; (12.7)

where the O(1) term (in (12.7) and below) holds uniformly over �1 2 B(�1�n; ") as n ! 1; the�rst equality uses Assumption FOC(vii) and gn = 0k (by Assumption FOC(iii)), and the second

equality holds by Assumptions FOC(xii) and (xv).

For �1 2 B(�1�n; "); element-by-element mean-value expansions of bG1n(�1) about �1�n givebG1n(�1) = bG1n + p1X

s=1

@

@�1sbG1n(�y1n)(�1s � �1�ns) = G1n + op(1) +Op(1)jj�1 � �1�njj; (12.8)

where �y1n lies between �1 and �1�n and may di¤er across rows of (@=@�1s) bG1n(�y1n); the mean-value expansions use Assumption FOC(v), and the second equality uses Assumptions FOC(viii)

and (xiv).

For �1 2 B(�1�n; "); element-by-element mean-value expansions of G1n(�1) about �1�n give

G1n(�1) = G1n +

p1Xs=1

@

@�1sG1n(�

41n)(�1s � �1�ns) = G1n +O(1)jj�1 � �1�njj; (12.9)

where �41n lies between �1n and �1�n and may di¤er across rows of (@=@�1s)G1n(�41n); the �rst

equality uses Assumptions FOC(vii) and (xii) and the second equality holds using Assumption

FOC(xv) (since (@=@�1s)G1n(�1) = (@2=@�1s@�01)gn(�1) by Assumption FOC(xii)).

Combining (12.5), (12.6), and (12.8) gives: For �1 2 B(�1�n; ");

bn(�1) = (G1n + op(1) +Op(1)jj�1 � �1�njj)0(W11 + op(1))

�(op(1) +G1n � (�1 � �1�n) +Op(1)jj�1 � �1�njj2) (12.10)

= G011W11G11 � (�1 � �1�n) + op(1) + op(1)jj�1 � �1�njj+Op(1)jj�1 � �1�njj2;

where the �rst equality uses Assumption FOC(xvi) and the second equality uses Assumption

FOC(xi).

10

Combining (12.5), (12.7), and (12.9) gives: for �1 2 B(�1�n; ");

n(�1) = (G1n +O(1)jj�1 � �1�njj)0W11(G1n � (�1 � �1�n) +O(1)jj�1 � �1�njj2)

= G01nW11G1n � (�1 � �1�n) +O(1)jj�1 � �1�njj2

= G011W11G11 � (�1 � �1�n) + o(1)jj�1 � �1�njj+O(1)jj�1 � �1�njj2; (12.11)

where the third equality uses Assumption FOC(xi).

Di¤erentiability of gn(�1) and Gn(�1) on B(�1�n; ") holds by Assumptions FOC(vii) and (xii).

This implies di¤erentiability of n(�1) on B(�1�n; "): The derivative matrix of n(�1) is

@

@�01n(�1) = G1n(�1)

0W11G1n(�1) (12.12)

+

��@

@�11G1n(�1)

�0W11g1n(�1); :::;

�@

@�1p1G1n(�1)

�0W11g1n(�1)

�:

This matrix is uniformly continuous on B(�1�n; ") by Assumption FOC(vii).

Now we show that for some "1 2 (0; "];

lim infn!1

inf�12B(�1�n;"1)

�min

�@

@�01n(�1)

�> 0: (12.13)

We have lim infn!1 �min(G01nW11G1n) > 0 (by Assumptions FOC(i), (xiii), and (xvi) because �1n

is the smallest singular value of �1=2n G1n): Using this and (12.9), we obtain: for some "2 2 (0; "];

lim infn!1

inf�12B(�1�n;"2)

�min(G1n(�1)0W11G1n(�1)) > 0: (12.14)

Next, by (12.7), gn(�1) = O(1)jj�1 � �1�njj 8�1 2 B(�1�n; "): Hence,

maxs�p1

� @

@�1sG1n(�1)

�0W11g1n(�1)

= O(1)jj�1 � �1�njj 8�1 2 B(�1�n; ") (12.15)

using Assumptions FOC(xii) and (xv). This and (12.14) imply (12.13) for some su¢ ciently small

"1 > 0:

Now, by the inverse function theorem applied to n(�1); for every su¢ ciently small � > 0; there

exists an open neighborhoodMn� of �1�n such thatMn� � B(�1�n; "1) and the map n : cl(Mn�)!B(0p1 ; �) is a homeomorphism, where cl(Mn�) denotes the closure ofMn� and B(0p1 ; �) is the closed

ball at 0p1 with radius �: The diameter of cl(Mn�) is bounded by a multiple of � that does not

11

depend on n by the following argument:

sup�12Mn�

jj�1 � �1�njj = sup�2B(0p1 ;�)

jj�1n (�)��1n (0p1)jj = sup�2B(0p1 ;�)

jj(@=@�0)�1n (e�n)�jj= sup

�2B(0p1 ;�)jj[(@=@�01)n(�1)j�1=�1n (e�n)]�1�jj � �= inf

�12B(�1�n;"1)�min((@=@�

01)n(�1)) = O(1)�

(12.16)

for some e�n 2 B(0p1 ; �) that may di¤er across the rows of (@=@�0)�1n (e�n); where the �rst equalityholds because n : cl(Mn�)! B(0p1 ; �) is a homeomorphism and �1n (0

p1) = �1�n by Assumption

FOC(iii), the second equality holds by element-by-element mean-value expansions of �1n (�) about

0p1 ; the third equality holds by the standard formula for the derivative matrix of an inverse function,

and the last equality holds by (12.13).

Combining (12.10), (12.11), and (12.16) gives

sup�12cl(Mn�)

jjbn(�1)�n(�1)jj = op(1) + op(1)� +Op(1)�2; (12.17)

where the op(1) and Op(1) terms are uniform in � for � small. (This equation is analogous to the

second displayed equation on p. 69 of van der Vaart (1998). The remainder of the proof of the

existence of consistent solutions to the FOC�s is the same as in van der Vaart (1998), although for

completeness we provide more details here.)

Because PFn(op(1) + op(1)� > �=2) ! 0 8� > 0; there exists a sequence �n # 0 such thatPFn(op(1) + op(1)�n > �n=2) ! 0: Let Kn� := fsup�12cl(Mn�)

jjbn(�1) � n(�1)jj < �g: Then, wehave

PFn(Kn�n) := PFn( sup�12cl(Mn�n )

jjbn(�1)�n(�1)jj < �n)

= PFn(op(1) + op(1)�n +Op(1)�2n < �n)

= PFn(op(1) + op(1)�n +Op(1)�2n < �n; B

cn) + o(1)

� PFn(�n=2 +Op(1)�2n < �n; B

cn) + o(1)

! 1; (12.18)

where the second equality uses (12.17), the third equality holds by writing PFn(An) = PFn(An \Bcn)+PFn(An\Bn) for Bn = fop(1)+op(1)�n > �n=2g and PFn(Bn)! 0; the inequality holds using

the condition in Bcn; and the convergence holds because PFn(Bcn) ! 1 and PFn(�n=2 + Op(1)�

2n <

�n)! 1 using �n ! 0:

12

On the event Kn�; the map � ! � � bn(�1n (�)) maps B(0p1 ; �) into itself (because 8� 2B(0p1 ; �); jj�� bn(�1n (�))jj � jj��n(�1n (�))jj+sup�2B(0p1 ;�) jjbn(�1n (�))�n(�1n (�))jj � �;

where the second inequality uses the de�nition of Kn� and the fact that � 2 B(0p1 ; �) implies that�1n (�) 2 cl(Mn�)): This map is continuous. Hence, by Brouwer�s �xed point theorem, it possesses

a �xed point in B(0p1 ; �): That is, there exists �n 2 B(0p1 ; �) such that �n � bn(�1n (�n)) = �n:

For �1n := �1n (�n) 2 cl(Mn�n); this gives bn(�1n) = 0p1 : Because the set Mn�n contains �1�n; the

diameter of Mn�n is bounded by a multiple (that does not depend on n) of �n; and �n # 0; we have�1n � �1�n !p 0

p1 : Hence, �1n is a consistent solution to the FOC�s bn(�1) = 0p1 :Given the consistency of the solutions f�1n : n � 1g to the FOC�s in (7.3), we now establish

the n1=2-consistency of f�1n : n � 1g: The FOC�s, mean-value expansions around �1�n; and �1n ��1�n !p 0 give

0p1 = bG1n(�1n)0cW1nbgn(�1n) = bG1n(�1n)0cW1n

�bgn + bG1n(e�1n)(�1n � �1�n)� and�1n � �1�n = �

� bG1n(�1n)0cW1nbG1n(e�1n)��1 bG1n(�1n)0cW1nbgn = Op(n

�1=2); (12.19)

where e�1n lies between �1n and �1�n and may di¤er across the rows of bG1n(e�1n) and the last equalityholds because bG1n(�1n)0cW1n

bG1n(e�1n)!p G011W11G11 (by Assumptions FOC(viii), (x), (xi), and

(xvi) and �1n� �1�n !p 0); G011W11G11 is nonsingular (since W11 is nonsingular by Assumption

FOC(xvi) and G11 has full column rank p1 by Assumptions FOC(i), (xi), and (xiii) because �1n is

the smallest singular value of �1=2n G1n); bG1n(�1n)0cW1n = Op(1) (by Assumptions FOC(viii), (x),

(xi), and (xvi) and �1n � �1�n !p 0); and bgn = Op(n�1=2) (by Assumption FOC(vi)). Equation

(12.19) completes the proof of the lemma. �


dH(�1�n; CS+1n) = dH(�1�n; CS1n [ b�1n)1(CS1n = ?) + dH(�1�n; CS1n [ b�1n)1(CS1n 6= ?)

� dH(�1�n; b�1n)[1(CS1n = ?) + 1(CS1n 6= ?)] + dH(�1�n; CS1n)1(CS1n 6= ?)� dH(�1�n; b�1n) + dH(�1�n; CS1n [ f�1�ng)= Op(n

�1=2); (12.20)

where the �rst inequality holds using straightforward manipulations, the second inequality holds

because CS1n 6= ? implies dH(�1�n; CS1n) = dH(�1�n; CS1n [ f�1�ng); and the last equality holdsby conditions (i) and (ii) of the lemma. �

Proof of Lemma 12.4. Let fb�1n : n � 1g be a sequence in b�1n (for all n � 1 and all sample

13

realizations) for which jj�1�n �b�1njj = dH(�1�n; b�1n) + op(n�1=2): Such a sequence exists wp!1 byAssumption ES3(ii).

Let qn := inf�1 =2B(�1�n;") jjgn(�1)jj: By Assumption ES4(ii), lim infn!1 qn > 0: By the de�nition

of qn; b�1n =2 B(�1�n; ") implies jjgn(b�1n)jj � qn 8n � 1: Hence, we have

PFn(b�1n =2 B(�1�n; "))

� PFn(jjgn(b�1n)jj � qn)

� PFn(jjbgn(b�1n)jj+Op(n�1=2) � qn)

� PFn(bgn(b�1n)0cW1nbgn(b�1n)=�min(cW1n) � (qn �Op(n�1=2))2)

� PFn( inf�12�1

bgn(�1)0cW1nbgn(�1) + cn � �(qn �Op(n�1=2))2)

� PFn(bg0ncW1nbgn + cn � �(qn �Op(n�1=2))2)

= o(1); (12.21)

where the second inequality holds by Assumption ES4(i) and the triangle inequality, the third

inequality uses Assumption ES3(xi), the fourth inequality holds by the de�nition of b�1n in (7.3)and because Assumption ES3(xi) implies that �min(cW1n) � � wp!1 for some constant � > 0; thelast inequality holds because �1�n 2 �1; and the equality holds because bg0ncW1nbgn = op(1) using

Assumptions ES3(iv) and (xi), � > 0; cn ! 0 by Assumption ES3(x), and lim infn!1 qn > 0 by

Assumption ES4(ii).

Equation (12.21) implies that b�1n � �1�n !p 0:

Next, the FOC�s in (7.3), mean-value expansions around �1�n; and b�1n � �1�n !p 0 give

0p1 = bG1n(b�1n)0cW1nbgn(b�1n) = bG1n(b�1n)0cW1n

�bgn + bG1n(e�1n)(b�1n � �1�n)� and sob�1n � �1�n = �

� bG1n(b�1n)0cW1nbG1n(e�1n)��1 bG1n(b�1n)0cW1nbgn = Op(n

�1=2); (12.22)

where e�1n lies between b�1n and �1�n and may di¤er across the rows of bG1n(e�1n); the mean-value expansions use Assumption ES3(iii), and the second equality on the second line holds be-

cause bG1n(b�1n)0cW1nbG1n(e�1n)!p G

011W11G11; G011W11G11 is nonsingular, and bG1n(b�1n)0cW1n =

Op(1) by Assumptions ES3(i), (v)�(ix) and (xi), b�1n� �1�n !p 0 (which implies that there exists a

sequence of positive constants "n ! 0 for which PFn(jjb�1n � �1�njj > "n)! 0; so that Assumption

ES3(vi) can be applied), and bgn = Op(n�1=2) by Assumption ES3(iv). (Note that G11 has full

column rank p1 by Assumptions ES3(i) and (vii)�(ix) because �1n is the smallest singular value of

�1=2n G1n and lim infn!1 �1n > 0:)

14

Given the de�nition of b�1n; (12.22) implies that d(�1�n; b�1n) = Op(n�1=2): �

13 Veri�cation of Assumptions for the First-Step AR CS

13.1 First-Step AR CS Results

First, we provide a result that veri�es Assumption B(i) for the �rst-step (FS) AR CS, de�ned

in (7.1).

Assumption FS1AR. For the sequence S; (i) bn � n !p 0 for some variance matrices fn 2Rk�k : n � 1g; (ii) n ! 1 for some variance matrix 1 2 Rk�k; (iii) n1=2bgn !d Z1 �N(0k;1); and (iv) lim infn!1 �min(n) > 0:

Lemma 13.1 Let S be a null sequence (or Sm a null subsequence) that satis�es Assumption

FS1AR: Then, under the sequence S (or subsequence Sm), the nominal level 1 � �1 �rst-step AR

CS, CSAR1n ; has asymptotic coverage probability 1� �1 and, hence, satis�es Assumption B(i).

Next, we provide a result, Lemma 13.2, that is useful, in conjunction with Lemmas 12.3 and

12.4, for verifying Assumption OE(i) for the �rst-step AR CS, which equals

CS1n = f�1 2 �1 : nbgn(�1)0b�1n (�1)bgn(�1) � �2k(1� �1)g: (13.1)

Lemma 13.2 provides conditions under which dH(�1�n; CS1n [ f�1�ng) = Op(n�1=2) for a sequence

S and the AR CS CS1n in (13.1).

When verifying Assumption OE(i) for a sequence S with CS1n in place of CS+1n; where CS1n is as

in (13.1), we use the following global strong-identi�cation condition: For all sequences fKn : n � 1gfor which Kn !1;

limn!1

inf�1 =2B(�1�n;Kn=n1=2)

n1=2jjgn(�1)jj =1: (13.2)

Assumption FS2AR. For the sequence S; (i) sup�12�1 n1=2jjbgn(�1) � gn(�1)jj = Op(1) for some

nonrandom Rk-valued functions fgn(�) : n � 1g; (ii) (13.2) holds, (iii) sup�12�1 jjbn(�1)�n(�1)jj =op(1) for some nonrandom Rk�k-valued functions fn(�) : n � 1g; (iv) sup�12�1 jjn(�1)jj = O(1);

and (v) lim infn!1 inf�12�1 �min(n(�1)) > 0:

Lemma 13.2 Suppose CS1n is of the form in (13.1). Let S be a null sequence (or Sm a null

subsequence) that satis�es Assumption FS2AR: Then, dH(�1�n; CS1n [ f�1�ng) = Op(n�1=2) for the

sequence S (or subsequence Sm).

15



nbg0nb�1n bgn !d Z01

�11 Z1 � �2k; (13.3)

where the convergence in distribution holds because n1=2bgn !d Z1 by Assumption FS1AR(iii)

and b�1n !p �11 by Assumptions FS1AR(i), (ii), and (iv), and the �2k distribution arises because

Z1 � N(0k;1) by Assumption FS1AR(iii). Hence, we have

PFn(�1�n 2 CSAR1n ) = PFn(nbg0nb�1n bgn � �2k(1� �1))! 1� �1; (13.4)

which establishes the result of the lemma. �

Proof of Lemma 13.2. Let fb�1n : n � 1g be a sequence in CS1n [ f�1�ng (for all n � 1 and allsample realizations) for which jj�1�n�b�1njj = dH(�1�n; CS1n[f�1�ng)+op(n�1=2): Such a sequencealways exists because CS1n [ f�1�ng is non-empty for all n � 1:

We establish the result of the lemma by contradiction. Suppose dH(�1�n; CS1n [ f�1�ng) 6=Op(n

�1=2): Then, by de�nition of b�1n;jjb�1n � �1�njj 6= Op(n

�1=2): (13.5)

If a sequence of random variables f�n : n � 1g satis�es �n = Op(1); then 8" > 0; 9K" < 1 such

that lim supn!1 PFn(j�nj > K") < ": Hence, (13.5) implies: 9" > 0 such that 8K <1;

lim supn!1

PFn(jjb�1n � �1�njj > K=n1=2) � ": (13.6)

For n � 1 and 0 < K <1; de�ne

Pn(K) := PFn(jjb�1n � �1�njj > K=n1=2) and Ln(K) := inf�1 =2B(�1�n;K=n1=2)

n1=2jjgn(�1)jj: (13.7)

Let fKn : n � 1g be a sequence such that Kn ! 1 as n ! 1; e.g., Kn = ln(n): Let m0 = 0:

For a given positive integer n; let mn (< 1) be a positive integer for which Pmn(Kn) > "=2 and

mn > mn�1: Such an mn always exists because (13.6) can be rewritten as lim supm!1 Pm(K) � ":

The subsequence fmng satis�esPmn(Kn) > "=2 8n � 1: (13.8)

16

By the de�nition of Ln(K) in (13.7),

�1 =2 B(�1�n;K=n1=2) implies jjn1=2gn(�1)jj � Ln(K): (13.9)

Equation (13.9) implies that, for all n � 1;

(i) b�1mn =2 B(�1�mn ;Kn=m1=2n ) implies jjm1=2

n gmn(b�1mn)jj � Lmn(Kn) and b�1mn 2 CS1mn ;

(ii) PFmn (b�1mn =2 B(�1�mn ;Kn=m

1=2n ))

� PFmn (jjm1=2n gmn(

b�1mn)jj � Lmn(Kn) & b�1mn 2 CS1mn); and (13.10)

(iii) "=2 < Pmn (Kn) � PFmn (fjjm1=2n gmn(

b�1mn)jj � Lmn(Kn)g & b�1mn 2 CS1mn);

where we choose to take the subscript on Kn to be n throughout (rather than mn) because we

use (13.8) in the last line, the �rst line uses fb�1mn : n � 1g is a sequence in CS1mn [ f�1�mngby de�nition and dH(�1�mn ; fb�1mng) = 0 if b�1mn = �1�mn (so b�1mn =2 B(�1�mn ;Kn=m

1=2n ) impliesb�1mn 2 CS1mn); the �rst inequality on the last line holds by (13.8), and the second inequality on

the last line holds by the inequality in (ii) and the de�nition of Pn(K) in (13.7).

Given the de�nition of the AR statistic, ARn(�1); in (7.1), we have

AR1=2n (b�1n) � inf�12�1

�1=2min

�b�1n (�1)� jjn1=2bgn(b�1n)jj� �jjn1=2gn(b�1n) +Op(1)jj� �(jjn1=2gn(b�1n)jj �Op(1)); (13.11)

where bn(�1) is nonsingular 8�1 2 �1 wp!1 by Assumptions FS2AR(iii) and (v), which guar-antees that the AR statistic on the left-hand side (lhs) of the �rst line is well de�ned wp!1,the second inequality holds for some � > 0 wp! 1 by Assumptions FS2AR(i), (iii), and (iv) be-

cause inf�12�1 �min(b�1n (�1)) = 1=(sup�12�1 �max(bn(�1))) and sup�12�1 �max(bn(�1)) = sup�12�1�max(n(�1)) + op(1) = O(1) + op(1) by Assumptions FS2AR(iii) and (iv), and the third inequality

holds by the triangle inequality. Hence, for all n � 1;

PFmn (ARmn(b�1mn) > �2k(1� �1) & b�1mn 2 CS1mn)

� PFmn (�(jjm1=2n gmn(

b�1mn)jj �Op(1)) > (�2k(1� �1))1=2

& jjm1=2n gmn(


� PFmn (�(Lmn(Kn)�Op(1)) > (�2k(1� �1))1=2 & jjm1=2n gmn(

b�1mn)jj � Lmn(Kn)

& b�1mn 2 CS1mn): (13.12)

17

We show below that limLmn(Kn) =1: Note that limLmn(Kn) =1 implies that �(Lmn(Kn)�Op(1)) > (�

2k(1� �1))1=2 wp!1. This and (13.12) yield

lim infn!1

PFmn (ARmn(b�1mn) > �2k(1� �1) & b�1mn 2 CS1mn)

� lim infn!1

PFmn (jjm1=2n gmn(


> "=2; (13.13)

where the second inequality holds by the last line of (13.10). Equation (13.13) is a contradiction

because ARmn(b�1mn) > �2k(1 � �1) implies that b�1mn =2 CS1mn : That is, (13.13) asserts that

0 = lim infn!1 PFmn (b�1mn =2 CS1mn & b�1mn 2 CS1mn) > "=2 > 0:

It remains to show that limLmn(Kn) =1:Given the de�nition of Ln(K) in (13.7), the conditionin (13.2) (i.e., Assumption FS2AR(ii)) states that for all sequences fKng for which Kn ! 1;limLn(Kn) = 1: Hence, for all sequences fKmng for which Kmn ! 1; limn!1 Lmn(Kmn) = 1:Given fKng; we show that there exists a sequence fK�

mng such that K�

mn= Kn 8n � 1: Becausemn

is strictly increasing in n; n! mn is a one-to-one map. Let �(m) be the corresponding inverse map

for m 2 M := fmn : n � 1g: For any m 2 M; de�ne K�m = K�(m): Then, Lmn(Kn) = Lmn(K

�mn)

8n � 1 because �(mn) = n: In consequence, limn!1 Lmn(Kn) = limn!1 Lmn(K�mn) = 1; where

the second equality holds by (13.2). �


Data-Dependent Signi�cance Level

14.1 Data-Dependent Signi�cance Level Results

Here we verify Assumptions B(iii) and OE(ii) for the second-step signi�cance level (SL) b�2n(�1)de�ned as in (7.4)�(7.8).

The results in this section and the sections that follow apply only to moment condition models.

In consequence, in these sections,

gn(�) := EFnbgn(�) 2 Rk;Gjn(�) := EFn bGjn(�) 2 Rk�pj for j = 1; 2; andn(�) := V arFn(n

1=2bgn(�)) 2 Rk�k: (14.1)

18

We de�ne

b�jn(�) := n�1nXi=1

vec(Gji(�)� bGjn(�))gi(�)0 2 R(pjk)�k and�jn(�) := n�1

nXi=1

EFnvec(Gji(�)� EFnGji(�))gi(�)0 2 R(pjk)�k for j = 1; 2: (14.2)

Note that b�jn(�) = [b�j1n(�)0 : � � � : b�jpjn(�)0]0 for b�jsn(�) de�ned in (7.9) for s = 1; :::; p1 and

j = 1; 2:

We de�ne

Gjsi(�) :=@

@�jsgi(�) 2 Rk;

�2jsn(�) := V arFn(jjGjsi(�)jj) 8s = 1; :::; pj ; and

�jn(�) := Diagf��1j1n(�); :::; ��1jpjn

(�)g for j = 1; 2: (14.3)

First, we provide a lemma that veri�es Assumption B(iii) under high-level conditions. The

following assumption is employed when the second-step test is the C(�)-AR test de�ned in (7.10).

Assumption SL1AR. For the null sequence S; (i) lim ��1n < KL (where KL < 1 appears in

the de�nition of b�2n(�1) in (7.8)), (ii) bG1n � G1n !p 0 for fG1n := G1n(�1�n) : n � 1g de�nedin (14.1), (iii) lim supn!1 jjG1njj < 1; (iv) bn � n !p 0 for fn := n(�1�n) : n � 1g de�nedin (14.1), (v) lim infn!1 �min(n) > 0; (vi) lim supn!1 jjnjj < 1; (vii) b�21sn � �21sn !p 0 for

f�21sn := �21sn(�1�n) : n � 1g de�ned in (14.3) 8s = 1; :::; p1; and (viii) lim infn!1 �21sn > 0

8s = 1; :::; p1:

The following assumption is employed when the second-step test is the C(�)-LM or C(�)-QLR1

test de�ned in (7.13) and (7.18), respectively.

Assumption SL1LM;QLR1. For the null sequence S; (i) lim ��n < KL; (ii) Assumptions SL1AR(ii)�

(viii) hold, (iii) b�22sn � �22sn !p 0 for f�22sn := �22sn(�1�n) : n � 1g de�ned in (14.3) 8s = 1; :::; p2;and (iv) lim infn!1 �22sn > 0 8s = 1; :::; p2:

Lemma 14.1 Suppose b�2n(�1) is de�ned in (7.4)�(7.8) for the second-step C(�)-AR, C(�)-LM,or C(�)-QLR1 test. Let S be a null sequence (or Sm a null subsequence) that satis�es Assumption


or C(�)-QLR1 test. Then, Assumption B(iii) holds for the sequence S (or subsequence Sm).

Next, we provide high-level conditions under which a sequence S satis�es Assumption OE(ii).

The following assumption is employed when the second-step test is the C(�)-AR test.

19

Assumption SL2AR. For the null sequence S; 8K < 1; (i) lim infn!1 ��1n > KU (where

KU > 0 appears in the de�nition of b�2n(�1) in (7.8)), (ii) sup�12B(�1�n;K=n1=2) jj bG1n(�1)�G1n(�1)jj!p 0 for fG1n(�) : n � 1g de�ned in (14.1), (iii) sup�12B(�1�n;K=n1=2) jjG1n(�1) � G1njj ! 0;

(iv) G1n = O(1); (v) sup�12B(�1�n;K=n1=2) jjbn(�1) � n(�1)jj !p 0 for fn(�) : n � 1g de-�ned in (14.1), (vi) sup�12B(�1�n;K=n1=2) jjn(�1) � njj ! 0; (vii) lim infn!1 �min(n) > 0; (viii)

sup�12B(�1�n;K=n1=2) jb�21sn(�1)� �21sn(�1)j !p 0 for f�21sn(�) : n � 1g de�ned in (14.3) 8s = 1; :::; p1;(ix) sup�12B(�1�n;K=n1=2) j�

21sn(�1) � �21snj ! 0 8s = 1; :::; p1; and (x) lim infn!1 �21sn > 0 8s =

1; :::; p1:

The following assumption is employed with the second-step C(�)-LM and C(�)-QLR1 tests.

Assumption SL2LM;QLR1. For the null sequence S; 8K < 1; (i) lim infn!1 ��n > KU ; (ii) As-

sumptions SL2AR(v)�(x) hold, (iii) sup�12B(�1�n;K=n1=2) jj bGn(�1)�Gn(�1)jj !p 0 for some nonran-

dom Rk�p-valued functions fGn(�) : n � 1g; (iv) sup�12B(�1�n;K=n1=2) jjGn(�1)�Gnjj ! 0; (v) Gn =

O(1); (vi) sup�12B(�1�n;K=n1=2) jb�22sn(�1)��22sn(�1)j !p 0 for some nonrandom real-valued functions

f�22sn(�) : n � 1g de�ned in (14.3) 8s = 1; :::; p2; (vii) sup�12B(�1�n;K=n1=2) j�22sn(�1) � �22snj ! 0

8s = 1; :::; p2; and (viii) lim infn!1 �22sn > 0 8s = 1; :::; p2:

Lemma 14.2 Suppose b�2n(�1) is de�ned in (7.4)�(7.8) for the second-step C(�)-AR, C(�)-LM,or C(�)-QLR1 test. Let S be a null sequence (or Sm a null subsequence) that satis�es Assumption


or C(�)-QLR1 test. Then, Assumption OE(ii) holds for the sequence S (or subsequence Sm).


Proof of Lemma 14.1. First, we prove the lemma for the second-step C(�)-AR test under

Assumption SL1AR. De�ne �1n := Diagf��111n; :::; ��11p1n

g: We write a SVD of �1=2n G1n�1n as

C�1n��1nB

�01n; where C

�1n and B

�1n are k � k and p1 � p1 orthogonal matrices, respectively, and ��1n

is a k � p1 matrix with the singular values of �1=2n G1n�1n on its main diagonal in nonincreasing

order and zeros elsewhere. The smallest singular value of �1=2n G1n�1n is ��1n; see (10.3), and it

appears as the (p1; p1) element of ��1n: Let �1n 2 Rp1 be such that jj�1njj = 1 and B�01n�1n = ep1 :=

(0; :::; 0; 1)0 2 Rp1 : Then, ��1nB�01n�1n = ep1��1n:

20

We have

ICS21n

= �min

�b�1n( bG1n �G1n +G1n)0b�1n ( bG1n �G1n +G1n)b�1n�= �min

�(�1n + op(1))(

�1=2n G1n + op(1))

0h1=2n

b�1n 1=2n i(�1=2n G1n + op(1))(�1n + op(1))

�= inf

�:jj�jj=1

��0(�1nG

01n

�1=2n + op(1)) [Ik + op(1)] (

�1=2n G1n�1n + op(1))�

�� (�01n�1nG

01n

�1=2n + op(1)) [Ik + op(1)] (

�1=2n G1n�1n�1n + op(1))

= (��1ne0p1C

�01n + op(1))

0 [Ik + op(1)] (C�1nep1�

�1n + op(1))

= (��1n)2 + op(1); (14.4)

where the second equality uses Assumptions SL1AR(ii), (v), and (vii), the third equality uses

Assumptions SL1AR(iii)�(vi) and (viii) (where Assumptions SL1AR(v) and (viii) imply that �1=2n =

O(1) and �1n = O(1); respectively), the inequality holds with �1n de�ned as above, the second

last equality holds by the calculations above concerning �1n; and the last equality holds using

Assumption SL1AR(i).

Equation (14.4) implies that ICS1n � lim ��1n + " wp!1 under the sequence S; 8" > 0: UsingAssumption SL1AR(i) this implies that ICS1n � KL wp!1 under the sequence S: By the de�nitionof b�2n(�1) in (7.7) and (7.8), this implies that b�2n = �2 wp!1 under the sequence S: That is,Assumption B(iii) holds for the sequence S under Assumption SL1AR.

Next, we prove the lemma for the second-step C(�)-LM or C(�)-QLR1 test under Assumption

SL1LM;QLR1. The proof is the same as that given above for the C(�)-AR test but with all quantities

involving bGn; Gn; b�n; �n; and ��n ; rather than bG1n; G1n; b�1n; �1n; and ��1n; respectively. Thesechanges require the use of Assumption SL1LM;QLR1(i) (i.e., lim ��n < KL); rather than Assumption

SL1AR(i) (i.e., lim ��1n < KL); and of Assumptions SL1LM;QLR1(iii) and (iv) (to obtain the analogues

of the second and third equalities in (14.4) for the C(�)-LM and C(�)-QLR1 test cases). �

Proof of Lemma 14.2. First, we prove the lemma for the second-step C(�)-AR test under As-

sumption SL2AR: Let ��1n(�) denote the smallest singular value of �1=2n (�)G1n(�)�1n(�); where

�1n(�) 2 Rp1�p1 is de�ned in (14.3). For notational simplicity, let bG1n�1 ; G1n�1 bn�1 ; n�1 ; b�1n�1 ;�1n�1 ; and �

�1n�1

denote bG1n(�1); G1n(�1); bn(�1); n(�1); b�1n(�1); �1n(�1); and ��1n(�1); respec-tively. Let inf�1 abbreviate inf �12B(�1�n;K=n1=2) and likewise with sup�1 : Let op(�1; "n); Op(�1; "n);

and o(�1; "n) denote k � p1; k � k; or p1 � p1 matrices that depend on �1 and are op("n); Op("n);

and o("n); respectively, uniformly over �1 2 B(�1�n;K=n1=2) for a sequence of positive constants

f"ng:

21

First, we show

lim infn!1

inf�1

��1n�1=��1n = 1; (14.5)

Given Assumption SL2AR(i), (14.5) holds if

lim infn!1

inf�1

�(��1n�1)

2 � (��1n)2�= 0 8K <1 (14.6)

because inf�1�(��1n�1)

2 � (��1n)2�� 0 8n � 1:

Let �1n�1 2 Rp1 be such that jj�1n�1 jj = 1 and �min(�1n�1G01n�1�1n�1G1n�1�1n�1) = �01n�1�1n�1

�G01n�1�1n�1G1n�1�1n�1�1n�1 : Let LHS denote the lhs of (14.6). We have

0 � LHS = lim infn!1

inf�1

��min(�1n�1G

01n�1

�1n�1G1n�1�1n�1)� �min(�1nG01n�1n G1n�1n)

�� lim inf

n!1inf�1

��01n�1�1n�1G

01n�1

�1n�1G1n�1�1n�1�1n�1 � �01n�1�1nG

01n

�1n G1n�1n�1n�1

�= lim inf

n!1inf�1�01n�1 [�1n�1G

01n�1

�1n�1G1n�1�1n�1 � �1nG01n�1n G1n�1n]�1n�1

= 0; (14.7)

where the �rst equality holds because the square of the smallest singular value of an k� p1 matrixA with p1 � k equals the smallest eigenvalue of A0A and the last equality holds by Assumption

SL2AR(iii), (a) sup�1 jj�1n�1

� �1n jj ! 0 8K <1; (b) sup�1 jj�1n�1 � �1njj ! 0 8K <1; and (c)all of the multiplicands �1n; G1n; and �1n are O(1): Condition (a) holds because

sup�1

jj�1n�1 � �1n jj = sup

�1

jj � �1n�1 [n�1 � n]�1n jj = o(1); (14.8)

where the last equality holds by Assumptions SL2AR(vi) and (vii) (since Assumptions

SL2AR(vi) and (vii) imply lim infn!1 inf�1 �min(n(�1)) > 0): Condition (b) holds by the same

argument as for condition (a) using Assumption SL2AR(ix) and (x). This completes the proof of

(14.6) and, in turn, (14.5).

22

Next, we have

inf�1ICS21n(�1)

= inf�1�min

�b�1n�1 � bG1n�1 �G1n�1 +G1n�1�0 b�1n�1 � bG1n�1 �G1n�1 +G1n�1� b�1n�1�= inf

�1�min

�(�1n�1 + op(�1; 1))(

�1=2n�1

G1n�1 + op(�1; 1))0h�1=2n�1

bn�1�1=2n�1

i�1�(�1=2n�1

G1n�1 + op(�1; 1))(�1n�1 + op(�1; 1))�

= inf�1�min

�(

�1=2n�1

G1n�1�1n�1 + op(�1; 1))0 [Ik + op(�1; 1)] (

�1=2n�1

G1n�1�1n�1 + op(�1; 1))�

= inf�1

inf�:jj�jj=1

(�0�1n�1G01n�1

�1n�1G1n�1�1n�1�+ �

0�1n�1G01n�1

�1=2n�1

op(�1; 1)�1=2n�1

G1n�1�1n�1�

+2�0op(�1; 1)0 [Ik + op(�1; 1)]

�1=2n�1

G1n�1�1n�1�+ �0op(�1; 1)

0 [Ik + op(�1; 1)] op(�1; 1)�)

� inf�1

inf�:jj�jj=1

�0�1n�1G01n�1

�1n�1G1n�1�1n�1�

� sup�1

sup�:jj�jj=1

j�0�1n�1G01n�1�1=2n�1

op(�1; 1)�1=2n�1

G1n�1�1n�1�)j

�2 sup�1

sup�:jj�jj=1

j�0op(�1; 1)0�1=2n�1G1n�1�1n�1�j � sup

�1

sup�:jj�jj=1

j�0op(�1; 1)�j

= inf�1(��1n�1)

2 + op(1)

= (��1n)2 + op(1); (14.9)

where the second equality holds using Assumptions SL2AR(ii) and (vi)�(x), the third equality holds

using Assumptions SL2AR(iii)�(vii), (ix), and (x), the second last equality holds using Assumptions

SL2AR(iii), (iv), (vi), (vii), (ix), and (x), the de�nition of ��1n�1 ; and the fact that the square of

the smallest singular value of a k � p1 matrix A with p1 � k equals the smallest eigenvalue of

A0A; and the last equality holds by Assumptions SL2AR(i), (iv), (vii), and (x) and (14.5) (where

Assumptions SL2AR(i), (iv), (vii), and (x) imply that f��1n : n � 1g is bounded away from 0 and

1):Equation (14.9) and Assumption SL2AR(i) imply that inf�1 ICS1n(�1) � KU wp!1. Hence,

given the de�nition of b�2n(�1) in (7.7) and (7.8) for the second-step C(�)-AR test, Assumption

OE(ii) holds for the sequence S:

Lastly, we prove the lemma for the second-step C(�)-LM and C(�)-QLR1 tests under As-

sumption SL2LM;QLR1. The proof is the same as that given above but with all quantities involvingbGn(�1); Gn(�1); b�n(�1); and �n(�1); rather than bG1n�1 ; G1n�1 ; b�1n�1 ; and �1n�1 ; respectively. Thesechanges require the use of Assumption SL2LM;QLR1(i) (i.e., lim infn!1 ��n > KU ); rather than As-

sumption SL2AR(i) (i.e., lim infn!1 ��1n > KU ) and the use of Assumptions SL2LM;QLR1(iii)�(viii)

23

(to obtain the analogues of the second, third, and last equalities of (14.9) for the C(�)-LM and

C(�)-QLR1 test cases). �

15 Veri�cation of Assumptions for the Second-Step C(�)-AR Test

15.1 Second-Step C(�)-AR Test Results

This section veri�es Assumptions B(ii) and C(ii)-C(v) for the second-step C(�)-AR test de�ned

in Section 7.4.1.

The following lemma provides conditions under which Assumptions B(ii), C(ii), and C(iii) hold

for the second-step AR test for a sequence S (whether lim �1n > 0 or lim �1n = 0): Assumption

C(iv) automatically holds for the second-step AR test provided p1 < k because its nominal level �

critical value is the 1�� quantile of the �2k�p1 distribution which is nondecreasing in � for � 2 (0; 1)when p1 < k:

For a full column rank matrix A 2 Rk�p1 ; let MA = Ik �A(A0A)�1A0:We write a singular value decomposition (SVD) of �1=2n G1n as

�1=2n G1n = C1n�1nB01n; (15.1)

where C1n 2 Rk�k and B1n 2 Rp1�p1 are orthogonal matrices and �1n 2 Rk�p1 has the singular

values �11n; :::; �1p1n of �1=2n G1n in nonincreasing order on its diagonal and zeros elsewhere. We

specify the compact SVD of �1=2n G1n given in (8.6) with � = (�01�n; �020)

0 to be the compact SVD

that is obtained from the SVD in (15.1) by deleting the non-essential rows and columns of C1n;

�1n; and B1n: Suppose limn1=2�1sn 2 [0;1] exists for s = 1; :::; p1: Let q1 (2 f0; :::; p1g) be suchthat

limn1=2�1sn =1 for 1 � s � q1 and limn1=2�1sn <1 for q1 + 1 � s � p1: (15.2)

De�ne

S1n := Diagf(n1=2�11n)�1; :::; (n1=2�1q1n)�1; 1; :::; 1g 2 Rp1�p1 and

S11 := Diagf0; :::; 0; 1; :::; 1g 2 Rp1�p1 ; (15.3)

where q1 zeros appear in S11:We have S1n ! S11: In the case of strong or semi-strong identi�cation

of �1 given �20; q1 = p1 and S11 = 0p1�p1 : In the case of weak identi�cation of �1 given �20;

S11 6= 0p1�p1 :

24

For the second-step (SS) C(�)-AR test, we use the following assumption.

Assumption SS1AR. For the null sequence S; (i) limn1=2�1sn 2 [0;1] exists 8s � p1; (ii)

n1=2(bg0n; vec( bG1n�EFn bG1n)0)0 !d (Z01; Z

0G11)

0 � N(0(p1+1)k; V11) for some variance matrix V11 2R(p1+1)k�(p1+1)k whose �rst k rows are denoted by [1 : �011] for 1 2 Rk�k and �11 2 R(p1k)�k;(iii) 1 is nonsingular, (iv) b�1n !p �11 for �11 as in condition (ii), (v) bn � n !p 0

k�k for

fn := n(�1�n; �20) : n � 1g de�ned in (14.1), (vi) n ! 1 for 1 as in condition (ii), (vii)

C1n ! C11 for some matrix C11 2 Rk�k; and (viii) B1n ! B11 for some matrix B11 2 Rp1�p1 :

Lemma 15.1 Suppose bgn(�1) are moment conditions, bD1n(�) is de�ned in (7.9), cM1n(�1) is de�ned

in (7.10) with a > 0; and p1 < k: Let S be a null sequence (or Sm a null subsequence) that satis�es

Assumption SS1AR: Then, for the sequence S (or subsequence Sm),

(a) AR2n !d AR21 := Z 01�1=21 M�

a11�1=21 Z1 � �2k�p1

for some (possibly) random k � p1 matrix �a11 that is independent of Z1; where �

a11 has full

column rank p1 a.s. and

(b) for all � 2 (0; 1); PFn(�AR2n (�1�n; �) > 0)! �:

Comments: (i). Lemma 15.1 establishes Assumptions B(ii), C(ii), and C(iii) for the second-step

AR test. It veri�es Assumption C(iii) because the �2k�p1 distribution is absolutely continuous on

R when p1 < k:

(ii). The de�nition of the limit random matrix �a11 is complicated and its form, beyond having

full column rank a.s., is not important. In consequence, for brevity, �a11 is de�ned in the proof of

Lemma 15.1 below, see (15.6), rather than in Lemma 15.1 itself.

(iii). A key result of Lemma 15.1(a) is that �a11 has full column rank. This uses the full rank

perturbation an�1=2�1 introduced in the de�nition of cM1n in (7.10).

(iv). Under strong and semi-strong identi�cation, the term an�1=2�1 in the de�nition of cM1n

has no e¤ect on the asymptotic distribution in Lemma 15.1(a).

(v). The proof of Lemma 15.1 uses Lemmas 10.2 and 10.3 and Corollary 16.2 in the SM to

Andrews and Guggenberger (2017) (AG1) to obtain the asymptotic distribution of cM1n:

The following lemma provides conditions under which Assumption C(v) holds for the second-

step C(�)-AR test for a sequence S with lim �1n > 0; where �1n := �1p1n is the smallest singular

value of �1=2n G1n: Let �1 = (�11; :::; �1p1)0:

Assumption SS2AR. For the null sequence S; 8K < 1; (i) lim infn!1 �1n > 0; (ii) bgn(�1)is twice continuously di¤erentiable on B(�1�n; ") (for all sample realizations) 8n � 1 for some

" > 0; (iii) bgn = Op(n�1=2); (iv) sup�12B(�1�n;K=n1=2) jj bG1n(�1) � G1n(�1)jj !p 0 for fG1n(�) : n �

25

1g de�ned in (14.1), (v) lim supn!1 sup�12B(�1�n;K=n1=2) jjG1n(�1)jj < 1; (vi) sup�12B(�1�n;K=n1=2)jj(@2=@�1s@�01)bgn(�1)jj = Op(1) for s = 1; :::; p1; (vii) sup�12B(�1�n;K=n1=2) jjb�1n(�1) � �1n(�1)jj =op(1) for f�1n(�) : n � 1g de�ned in (14.2), (viii) sup�12B(�1�n;K=n1=2) jj�1n(�1) � �1njj ! 0; (ix)

jj�1njj = O(1); (x) sup�12B(�1�n;K=n1=2) jjbn(�1)�n(�1)jj !p 0 for fn(�) : n � 1g de�ned in (14.1),(xi) sup�12B(�1�n;K=n1=2) jjn(�1)� njj ! 0; (xii) lim infn!1 �min(n) > 0; and (xiii) n = O(1):

Lemma 15.2 Suppose bgn(�1) are moment conditions, bD1n(�) is de�ned in (7.9), and cM1n(�1) is

de�ned in (7.10) with a � 0: Let S be a null sequence (or Sm a null subsequence) that satis�es

Assumption SS2AR: Then, under the sequence S (or subsequence Sm), for all constants K <1;(a) sup�12B(�1�n;K=n1=2) jjcM1n(�1)� cM1njj = op(1);

(b) sup�12B(�1�n;K=n1=2)

n1=2cM1n(�1)b�1=2n (�1)bgn(�1)� n1=2cM1n(�1)b�1=2n (�1)bgn = op(1);

(c) sup�12B(�1�n;K=n1=2)

n1=2cM1n(�1)b�1=2n (�1)bgn(�1)� n1=2cM1nb�1=2n bgn = op(1); and

(d) sup�12B(�1�n;K=n1=2) jAR2n(�1)�AR2nj = op(1):

Comments: (i). Lemma 15.2(d) establishes Assumption C(v) for the second-step C(�)-AR test

for a sequence S with lim infn!1 �1n > 0:



Proof of Lemma 15.1. We have b�1=2n !p �1=21 and �1=21 is nonsingular by Assumptions

SS1AR(iii), (v), and (vi).

We write

V11 =

24 1 �011

�11 G11

35 ; where 1 2 Rk�k; �11 2 R(p1k)�k; and G11 2 R(p1k)�(p1k): (15.4)

By the argument in the proof of Lemma 10.2 in Section 15 of the SM to AG1, we have

n1=2

0@ bgnvec( bD1n � EFn bG1n)

1A !d

0@ Z1

ZG11 � �11�11 Z1

1A� N

0@0(p1+1)k;0@ 1 0k�(p1k)

0(p1k)�k D11

1A1A ; where

D11 : = G11 � �11�11 �011; (15.5)

using Assumptions SS1AR(ii)�(vi).

26

We partition B11 and C11 (de�ned in Assumption SS1AR) and de�ne �11 and �a11 as follows:

B11 = [B11;q1 : B11;p1�q1 ]; C11 = [C11;q1 : C11;k�q1 ];

L�p1�q1 :=

26640q1�(p1�q1)

Diagflimn1=2�1(q1+1)n; :::; limn1=2�1p1ng0(k�p1)�(p1�q1)

37752 Rk�(p1�q1);vec(D11) := ZG11 � �11�11 Z1 for D11 2 Rk�p1 ;

�11 = [�11;q1 : �11;p1�q1 ] 2 Rk�p1 ; �11;q1 := C11;q1 ;

�11;p1�q1 := C11L�p1�q1 +

�1=21 D11B11;p1�q1 ; and

�a11 := �11 + a�1B11S11; (15.6)

where B11;q1 2 Rp1�q1 ; B11;p1�q1 2 Rp1�(p1�q1); C11;q1 2 Rk�q1 ; C11;k�q1 2 Rk�(k�q1); �11;q1 2Rk�q1 ; �11;p1�q1 2 Rk�(p1�q1); and S11 is de�ned in (15.3).14 The limits in L�p1�q1 exist by

Assumption SS1AR(i). Note that �11;q1 (:= C11;q1) has full column rank q1 because C11 is an

orthogonal matrix (since C1n ! C11 by Assumption SS1AR(vii) and C1n is orthogonal for all n by

de�nition).

Using (15.5), by the proof of Lemma 10.3 in Section 16 of the SM to AG1 with p; bDn; cWn; WFn ;bUn; UFn ; Dh; �h; h2; h3; and h�1;p�q in AG1 set equal to p1; bD1n; b�1=2n ; n; Ip1 ; Ip1 ; D11; �11;

B11; C11; and L�p1�q1 ; respectively, we have

n1=2�1=2nbD1nT1n !d �11; where T1n := B1nS1n: (15.7)

This result uses Assumptions SS1AR(i)�(viii).

We have

T1n := B1nS1n ! B11S11 (15.8)

using S1n ! S11 and Assumption SS1AR(viii).

We have b�1=2n !p �1=211 by Assumptions SS1AR(iii), (v), and (vi). This, (15.7), and (15.8)

combine to yield

n1=2(b�1=2nbD1n + an�1=2�1)T1n = n1=2b�1=2n

bD1nT1n + a�1T1n !d �11 + a�1B11S11 =: �a11:

(15.9)

14For simplicity, there is some abuse of notation here, e.g., B11;q1 and B11;p1�q1 denote di¤erent matrices even ifp1 � q1 happens to equal q1:

27

Using the notation introduced in (15.6), we can write the limit random matrix in (15.9) as

�a11 := �11 + a�1B11S11 = [�11;q1 : �11;p1�q1 + a�1B11;p1�q1 ] (15.10)

because B11S11 = [0p1�q1 : B11;p1�q1 ] by the de�nition of S11 in (15.3). As noted above, �11;q1

has full column rank q1: In addition, �1B11;p1�q1 2 Rk�(p1�q1) is a matrix of independent standardnormal random variables (because B11 is an orthogonal matrix) and �1B11;p1�q1 is independent

of �11;p1�q1 : By Corollary 16.2 of AG1, these results and a > 0 imply that �a11 has full column

rank p1 a.s.

The matrix �a11 is independent of Z1 because �

a11 is a nonrandom function of (D11; �1); �1

is independent of (Z1; D11) by de�nition, and D11 is independent of Z1 since they are jointly

normal with zero covariance (because Evec(D11)Z 01 = E(ZG11 � �11�11 Z1)Z 01 = 0(p1k)�k)

using (15.6) and Assumption SS1AR(ii)).

Given a matrix A; the projection matrix PA is invariant to the multiplication of A by any

nonzero constant and the post-multiplication of A by any nonsingular matrix. In consequence, by

the continuous mapping theorem,

cM1n := Ik � Pb�1=2nbD1n+an�1=2�1 = Ik � Pn1=2[b�1=2n

bD1n+an�1=2�1]T1n !d M�11+a�1B11S11=:M�

a11;

(15.11)

where the second equality holds for n large because T1n is nonsingular for n large (because B1n

is orthogonal and S1n is nonsingular for n large by its de�nition in (15.3) and the de�nition of

q1 in (15.2)) and the convergence uses (15.9) and the fact, established above, that �a11 has full

column rank p1 � k (which implies that the function J(�a11) = (�

a011�

a11)

�1 is well-de�ned and

continuous a.s. so the continuous mapping theorem is applicable). The convergence in (15.11)

holds jointly with n1=2bgn !d Z1 (using Assumption SS1AR(ii)).

The result of part (a) follows from (15.11), b�1=2n !p �1=21 ; and Assumption SS1AR(ii) using

the continuous mapping theorem. We have Z 01�1=21 M�

a11�1=21 Z1 � �2k�p1 conditional on �

a11

(because, as shown above, �a11 and Z1 are independent and �

a11 has full column rank p1 a.s.

and, by Assumption SS1AR(ii), �1=21 Z1 � N(0k; Ik)) and, hence, unconditionally as well.

Part (b) follows immediately from part (a) because �AR2n (�1�n; �) = AR2n � �2k�p1(1 � �) and

�2k�p1(1� �) is the 1� � quantile of the �2k�p1 distribution. �

Proof of Lemma 15.2. For any �1 2 B(�1�n;K=n1=2); element-by-element mean-value expansions

28

give

bgn(�1) = bgn + @

@�1bgn(�1)(�1 � �1�n) + � @

@�01bgn(e�1n)� @

@�01bgn(�1)� (�1 � �1�n)

= bgn + @

@�01bgn(�1)(�1 � �1�n) +Op(n�1); (15.12)

where e�1n lies between �1 and �1�n and may di¤er across the rows of (@=@�01)bgn(e�1n) and, hence,satis�es e�1n� �1�n = Op(n

�1=2) (because �1 2 B(�1�n;K=n1=2)), the �rst equality uses AssumptionSS2AR(ii), and the second equality uses mean-value expansions of (@=@�01)bgn(e�1n) and (@=@�01)bgn(�1)about �1�n and Assumption SS2AR(vi).

For part (a), given the de�nition of cM1n(�1) in (7.10), it su¢ ces to show that

(I) sup�12B(�1�n;K=n1=2)

jj bD1n(�1)� bD1njj = op(1);

(II) sup�12B(�1�n;K=n1=2)

jjbn(�1)� bnjj = op(1); (15.13)

(III) bD1n has singular values that are bounded away from 0 and1 wp!1, (IV) bn has eigenvaluesthat are bounded away from 0 and 1 wp!1, and (V) an�1=2�1 = op(1):

Condition (II) holds by Assumptions SS2AR(x) and (xi). Condition (IV) holds by Assumptions

SS2AR(x)�(xiii). Condition (V) holds because a and �1 do not depend on n: Because bD1n(�) is asimple function of bG1n(�); b�1n(�); b�1n (�); and bgn(�); see (7.9), condition (I) holds if

sup�12B(�1�n;K=n1=2)

jj bG1n(�1)� bG1njj = op(1); sup�12B(�1�n;K=n1=2)

jjb�1n(�1)� b�1njj = op(1);

sup�12B(�1�n;K=n1=2)

jjbgn(�1)� bgnjj = op(1); (15.14)

and conditions (II) and (IV) hold (because bG1n; b�1n; and bgn are Op(1)): The �rst condition in (15.14)holds by mean-value expansions of the elements of bG1n(�1) about �1�n using Assumptions SS2AR(ii)and (vi). The second condition in (15.14) holds by Assumptions SS2AR(vii) and (viii). The third

condition in (15.14) holds by (15.12) and Assumptions SS2AR(iv) and (v). Hence, condition (I)

holds.

To establish condition (III), we have

bD1n = G1n + op(1); (15.15)

by the de�nition of bD1n in (7.9) and Assumptions SS2AR(iii), (iv), (vii), (ix), (x), and (xii). The29

singular values of G1n are bounded away from 0 and 1 by Assumptions SS2AR(i) and (v) because

�1n is the smallest singular value of �1=2n G1n and the eigenvalues of

�1=2n are bounded away from

0 and 1 by Assumptions SS2AR(xii) and (xiii). This and (15.15) establish condition (III), which

completes the proof of part (a).

Part (b) is established as follows: For all �1 2 B(�1�n;K=n1=2);

n1=2cM1n(�1)b�1=2n (�1)bgn(�1)� n1=2cM1n(�1)b�1=2n (�1)bgn= n1=2cM1n(�1)b�1=2n (�1)

@

@�01bgn(�1)(�1 � �1�n) + cM1n(�1)b�1=2n (�1)Op(n

�1=2)

= n1=2cM1n(�1)(b�1=2n (�1) bD1n(�1) + an�1=2�1)(�1 � �1�n)� n1=2cM1n(�1)an�1=2�1(�1 � �1�n)

+n1=2cM1n(�1)b�1=2n (�1)[b�11n(�1) : ::: : b�1p1n(�1)]�Ip1 b�1n (�1)bgn(�1)� (�1 � �1�n)+Op(n

�1=2)

= op(1); (15.16)

where the Op(n�1=2) terms holds uniformly over �1 2 B(�1�n;K=n1=2); the �rst equality uses

(15.12), the second equality uses the de�nition of bD1n(�1) in (7.9) and the fact that

sup�12B(�1�n;K=n1=2) jjcM1n(�1)jj = Op(1) because the eigenvalues of cM1n(�1) equal zero or one (sincecM1n(�1) is a projection matrix) and sup�12B(�1�n;K=n1=2) jjb�1=2n (�1)jj = Op(1) by Assumptions

SS2AR(x)�(xii), and the third equality uses (1) cM1n(�1)[b�1=2n (�1) bD1n(�1)+an�1=2�1] = 0k�p1 (be-cause cM1n(�1) projects onto the orthogonal complement of the space spanned by b�1=2n (�1) bD1n(�1)+an�1=2�1; see (7.10)), (2) sup�12B(�1�n;K=n1=2) jjcM1n(�1)jj = Op(1) as above, (3) sup�12B(�1�n;K=n1=2)

jjb�jn (�1)jj = Op(1) for j = 1=2; 1 as above, (4) sup�12B(�1�n;K=n1=2) jjbgn(�1)jj = op(1) (by (15.12) and

Assumptions SS2AR(iii)�(v)), (5) sup�12B(�1�n;K=n1=2) jj�1 � �1�njj = O(n�1=2); and

(6) sup�12B(�1�n;K=n1=2)b�1n(�1) = sup�12B(�1�n;K=n1=2)[

b�11n(�1)0 : � � � : b�1p1n(�1)0]0 = Op(1) by As-

sumptions SS2AR(vii)�(ix).

Part (c) holds by parts (a) and (b), bgn = Op(n�1=2) (which holds by Assumption SS2AR(iii)),

and sup�12B(�1�n;K=n1=2) jjb�1=2n (�1)� b�1=2n jj = op(1) (which is implied by Assumptions SS2AR(x)�

(xii)).

Part (d) follows from part (c) and n1=2cM1nb�1=2n bgn = Op(1) (which holds by (2) and (3) above

and bgn = Op(n�1=2)) given the de�nition of AR2n(�) in (7.10). �

30

16 Veri�cation of Assumptions for the Second-Step C(�)-LM Test

16.1 Second-Step C(�)-LM Test Results

This section veri�es Assumptions B(ii) and C(ii)-C(v) for the second-step C(�)-LM test de�ned

in Section 7.4.2. The results in this section apply only to moment condition models.

We employ the same de�nitions as in Sections 8.1.1, 14.1, and 15.1. In addition, we de�ne

�21n; :::; �2p2n; q2; C2n; �2n; B2n; S2n; S21; and �a21 as �11n; :::; �1p1n; q1; C1n; �1n; B1n; S1n; S11;

and �a11 are de�ned in Section 15.1, respectively, but with subscripts 2 in place of 1 throughout.

Given the de�nitions above, C2n�2nB02n is a SVD of �1=2n G2n and its singular values are

�21n; :::; �2p2n: We choose the compact SVD of �1=2n G2n speci�ed in (8.6) with � = (�01�n; �

020)

0 to

be the compact SVD that is obtained from the SVD C2n�2nB02n by deleting the non-essential rows

and columns of C2n; �2n; and B2n: Given the de�nition of q2; we have S2n ! S21: In the case of

(local) strong or semi-strong identi�cation, q2 = p2 and S21 = 0k�p2 : In the case of (local) weak

identi�cation, S21 6= 0k�p2 :As de�ned in (10.2), �n is the smallest singular value of

�1=2n Gn 2 Rk�p; where p = p1 + p2:

We let rjn := rjFn for rjF de�ned in (8.5) for j = 1; 2 and C�n := C�Fn for C�F de�ned in (8.7).

For the second-step (SS) C(�)-LM test, we use the following assumptions.

Assumption SS1LM. For the null sequence S; (i) limn1=2�2sn 2 [0;1] exists 8s � p2; (ii)

n1=2(bg0n; vec( bG1n � EFn bG1n)0; vec( bG2n � EFn bG2n)0)0 !d (Z01; Z

0G11; Z

0G21)

0 � N(0(p+1)k; V1) for

some variance matrix V1 2 R(p+1)k�(p+1)k whose �rst k rows are denoted by [1 : �011 : �021]

for 1 2 Rk�k and �j1 2 R(pjk)�k for j = 1; 2; (iii) b�2n !p �21 for �21 as in condition (ii),

(iv) C2n ! C21 for some matrix C21 2 Rk�k; (v) B2n ! B21 for some matrix B21 2 Rp2�p2 ;

(vi) bGn � Gn !p 0 and Gn ! G1 for some matrix G1 2 Rk�p; where Gn := EFn bGn; and (vii)b�2jsn � �2jsn !p 0 for f�2jsn : n � 1g de�ned in (14.3) and �2jsn ! �2js1 for some scalars �2js1 > 0

8s = 1; :::; pj ; 8j = 1; 2:

Assumption SS2LM. For the null sequence S; 8K < 1; (i) lim infn!1 ��n > K�U for K

�U > 0

de�ned in (7.11), (ii) bgn(�1; �2) is di¤erentiable in �2 at �20 and (@=@�02)bgn(�1; �20) is di¤erentiablein �1 with both holding 8�1 2 B(�1�n; ") (for all sample realizations), 8n � 1; for some " > 0;

(iii) sup�12B(�1�n;K=n1=2) jj bG2n(�1) � G2n(�1)jj !p 0 for fG2n(�) : n � 1g de�ned in (14.1), (iv)lim supn!1 sup�12B(�1�n;K=n1=2) jjG2n(�1)jj < 1; (v) sup�12B(�1�n;K=n1=2) jj(@

2=@�1s@�02)bgn(�1; �20)jj

= Op(1) for s = 1; :::; p1; (vi) sup�12B(�1�n;K=n1=2) jjb�2n(�1) � �2n(�1)jj = op(1) for f�2n(�) : n � 1gde�ned in (14.2), and (vii) sup�12B(�1�n;K=n1=2) jj�2n(�1)� �2njj ! 0:

31

Given the quantities 1; G1; and �2js1 in Assumption SS1LM ; we de�ne

ICS�1 := �1=2min(�

01G

01

�11 G1�1); �1 := Diagf��1111; :::; �

�11p11; �

�1211; :::; �

�12p21g 2 R

p�p;

WI1 := 1� s�ICS�1 �K�

L

K�U �K�

L

�; and Dy

1 := (M�a11+WI1P�a11)�

a21: (16.1)

As de�ned, WI1 = 0 if ICS�1 � K�U ; WI1 = 1 if ICS�1 � K�

L; and WI1 2 [0; 1] otherwise.The following lemma veri�es Assumptions B(ii), C(ii), and C(iii) for the second-step C(�)-LM

test.

Lemma 16.1 Suppose bgn(�1) are moment conditions, bDjn(�) is de�ned in (7.9) for j = 1; 2;cM1n(�1) is de�ned in (7.10) with a > 0 and p2 � 1: Let S be a null sequence (or Sm a null subse-

quence) that satis�es Assumptions SS1AR and SS1LM : Then, for the sequence S (or subsequence

Sm),

(a) Dy1 has full column rank p2 a.s.,

(b) LM2n !d LM21 := Z 01�1=21 P

Dy1�1=21 Z1 � �2p2 ; where (�

a11;�

a21; D

y1) is independent

of Z1 and �aj1 has full column rank pj a.s. for j = 1; 2; and

(c) for all � 2 (0; 1); lim supn!1 PFn(�LM2n (�1�n; �) > 0) = �:

Comments: (i). For the second-step LM test, Lemma 16.1(c) establishes Assumptions B(ii)

and C(ii). Lemma 16.1(b) establishes Assumption C(iii) because the �2p2 distribution is absolutely

continuous on R when p2 � 1: Assumption C(iv) automatically holds for the second-step LM test

provided p2 � 1 because its nominal level � critical value is the 1�� quantile of the �2p2 distributionwhich is nondecreasing in � for � 2 (0; 1):

(ii). The result of Lemma 16.1(a) is key because it allows one to use the continuous mapping

theorem to obtain the asymptotic distribution of the LM2n statistic.

(iii). When lim infn!1 �n > 0 (i.e., under strong local identi�cation of �); �aj1 reduces to

�j1 for j = 1; 2; where �11 is de�ned in (15.6) and �21 is de�ned analogously, and the terms

an�1=2�1 and an�1=2�2 do not a¤ect the asymptotic distribution of LM2n:

(iv). The proof of Lemma 16.1 uses Lemmas 10.2 and 10.3 in the SM to AG1 to obtain the

asymptotic distribution of bD2n after suitable rescaling and right-hand side (rhs) rotation, but notrecentering.

(v). To prove that the result in Comment (iii) to Theorem 8.2 holds (which considers the

pure C(�)-LM test (in which case WIn(�) := 0), we establish below that Lemma 16.1 holds with

WI1 = 0 provided Assumptions SS1LM (vi) and (vii) are replaced by (vi) rjn (:= rjFn) = rj1 for

all n su¢ ciently large for some constant rj1 2 f0; :::; pjg for j = 1; 2; and (vii) �min(C 0�nC�n) � �

32

8n � 1 for some � > 0:


C(�)-LM test for sequences S with lim infn!1 ��n > K�U :

Lemma 16.2 Suppose bgn(�1) are moment conditions, bDjn(�) is de�ned in (7.9) for j = 1; 2 andcM1n(�1) is de�ned in (7.10) with a � 0: Let S be a null sequence (or Sm a null subsequence)

that satis�es Assumptions SS1AR; SS2AR; SS1LM ; SS2LM ; and SL2LM;QLR1 with SL2LM;QLR1(i)

deleted. Then, under the sequence S (or subsequence Sm), for all constants K <1;(a) sup�12B(�1�n;K=n1=2) jjb�1=2n (�1) bD2n(�1)� b�1=2n

bD2njj = op(1) and

(b) sup�12B(�1�n;K=n1=2) jLM2n(�1)� LM2nj = op(1):

Comments: (i). Lemma 16.2(b) establishes Assumption C(v) for the second-step C(�)-LM test

for a sequence S with lim infn!1 ��n > K�U (> 0):


(iii). Lemma 16.2 holds for the pure C(�)-LM test (in which caseWIn(�) := 0) with the condi-

tion lim infn!1 ��n > K�U (> 0) in Assumption SS2LM (i) replaced by the condition lim infn!1 �n >

0:


Proof of Lemma 16.1. We write

V1 =

26641 �011 �021

�11 G11 0G2G11

�21 G2G11 G21

3775 ; where 1 2 Rk�k; �j1 2 R(pjk)�k; Gj1 2 R(pjk)�(pjk);

(16.2)

and G2G11 2 R(p2k)�(p1k) for j = 1; 2: By the argument in the proof of Lemma 10.2 in Section 15of the SM to AG1, we have

n1=2

0BB@bgn

vec( bD1n � EFn bG1n)vec( bD2n � EFn bG2n)

1CCA (16.3)

!d

0BB@Z1

ZG11 � �11�11 Z1

ZG21 � �21�11 Z1

1CCA � N

0BB@0(p+1)k;0BB@

1 0k�(p1k) 0k�(p2k)

0(p1k)�k D11 0D2D11

0(p2k)�k D2D11 D21

1CCA1CCA ; where

Dj1 := Gj1 � �j1�11 �0j1 for j = 1; 2 and D2D11 := G2G11 � �21�11 �011

33

using (16.2) and Assumptions SS1AR(iii)�(vi) and SS1LM (ii).

We de�ne B21;q2 ; B21;p2�q2 ; C21;q2 ; C21;k�q2 ; L�p2�q2 ; D21; �21; �21;q2 ; �21;p2�q2 ; and

�a21 using the de�nitions in (15.6) with subscripts 2 in place of 1: The limits in L�p2�q2 exist

by Assumption SS1LM (i). We de�ne T2n as in (15.8) with the subscript 2 in place of 1: As in

(15.7)�(15.9) with subscripts 2 in place of 1; we have

n1=2(b�1=2nbD2n + an�1=2�2)T2n = n1=2b�1=2n

bD2nT2n + a�2T2n !d �21 + a�2B21S21 =: �a21

(16.4)

using Assumptions SS1AR(iii), (v), and (vi) and SS1LM (i)�(v). By the same argument as given just

below (15.10), the limit random matrix �a21 has full column rank p2 a.s., as stated in part (b).

The convergence results in (15.11) and (16.4) hold jointly (using Assumption SS1LM (ii)). In

addition, by the same argument as in the paragraph following (15.10), �a11 and �

a21 are jointly

independent of Z1; as stated in part (b).

Now, we prove part (a). We have

ICS�n := �1=2min

�b�n bG0nb�1n bGnb�n�!p �1=2min

��1G

01

�11 G1�1

�=: ICS�1 and

WIn !p WI1; (16.5)

where the �rst and last de�nitions in the �rst line are given in (7.6) and (16.1), respectively, the

convergence in probability in the �rst line holds using Assumptions SS1AR(iii), (v), and (vi) (which

yield b�1n !p �11 ), Assumptions SS1LM (vi) and (vii) (which yield b�n !p �1 and bGn !p G1),

and Slutsky�s Theorem (because the smallest eigenvalue of a matrix is a continuous function of the

matrix), and the second line holds by the �rst line, the de�nition ofWIn(�) in (7.11), and Slutsky�s

Theorem (because the function s(�) in (7.8) is assumed to be continuous).When ICS�1 � K�

L; we have WI1 = 1;

Dy1 := (M�

a11+WI1P�a11)�

a21 = �

a21; (16.6)

and �a21 has full rank p2 by the same argument as used to prove Lemma 15.1(a) in (15.10), which

uses Corollary 16.2 of AG1, with �a21 in place of �

a11:

When ICS�1 > 0; we have �1=2min(�j1G0j1

�11 Gj1�j1) � �

1=2min(�1G

01

�11 G1�1) := ICS�1 >

0 for j = 1; 2 using the de�nition of a minimum eigenvalue. Since �2js1 > 0 8s = 1; :::; pj ;

8j = 1; 2 by Assumption SS1LM (vii), this implies that �1=2min(G

0j1

�11 Gj1) > 0 for j = 1; 2:

Using Assumptions SS1AR(vi) and SS1LM (vi), this implies that � jn := �1=2min(G

0jn

�1n Gjn) !

�1=2min(G

0j1

�11 Gj1) > 0; where G1 = [G11 : G21] for Gj1 2 Rk�pj ; � jn is de�ned in (10.2),

34

and Gjn := EFn bGjn; see (14.1), for j = 1; 2: In turn, this gives qj = pj for j = 1; 2 because, by

de�nition, qj satis�es limn1=2� jsn =1 for 1 � s � qj (see (15.2) and the second paragraph of this

section). Finally, qj = pj for j = 1; 2 implies that

�aj1 = �j1 for j = 1; 2 (16.7)

using the de�nition of �a11 in (15.6) and the analogous de�nition of �

a21 speci�ed in the second

paragraph of this section.

By an analogous argument, when ICS�1 > 0; we have �n = �1=2min(G

0n

�1n Gn)! �

1=2min(G

01

�11 G1)

> 0; where �n is de�ned in (10.2).

When ICS�1 > K�L � 0; we have

Dy1 := (M�

a11+WI1P�a11)�

a21 =

�Ik � s1P�a11

��a21 =

�Ik � s1P�11

��21; where

s1 := s

�ICS�1 �K�

L

K�U �K�

L

�> 0; (16.8)

the last equality on the �rst line holds by (16.7), and s1 > 0 because s(�) is a strictly increasingcontinuous function on [0; 1] with s(0) = 0; see (7.8).

Suppose (Ik � s1P�11)�21 has rank less than p2: Then, 9� 2 Rp2 with jj�jj = 1 such that

�21� = s1P�11�21�: Because s1 > 0; this occurs only if �21� 2 col(�11); where col(�)denotes the column space of a matrix, because the right-hand side of the equation is in col(�11):

But, �21� 2 col(�11) is a contradiction because [�11 : �21] has full column rank a.s., which we

now show.

Thus, to prove part (a), it remains to show that [�11 : �21] has full column rank a.s. when

ICS�1 > K�L � 0: It su¢ ces to show [�11 : �21] has full column rank a.s. when lim �n > 0 and

qj = pj for j = 1; 2 (because it is shown above that ICS�1 > 0 implies both conditions). We have

�j1 = Cj1;pj (which is nonrandom) when qj = pj for j = 1; 2 by (15.6) and the second paragraph

of this section, where Cj1 = [Cj1;pj : Cj1;k�pj ] 2 Rk�k:We have Cjn ! Cj1 and Bjn ! Bj1 by Assumptions SS1AR(vii) and (viii) and SS1LM (iv)

and (v), where Cjn = [Cjn;pj : Cjn;k�pj ] and Bjn are the k � k and pj � pj orthogonal matrices

in the singular value decompositions �1=2n Gjn = Cjn�jnB0jn for j = 1; 2; see (15.1). The k � pj

diagonal matrix �jn of singular values of �1=2n Gjn can be written as [�jn;pj : 0

pj�(k�pj)]0; where

�jn;pj 2 Rpj�pj is a diagonal matrix with positive diagonal elements for n su¢ ciently large (since itssmallest diagonal element is � jn and � jn ! �

1=2min(G

0j1

�11 Gj1) > 0 for j = 1; 2 as shown above). In

consequence, the singular value decomposition Cjn�jnB0jn equals Cjn;pj�jn;pjB0jn; where Cjn;pj !

35

Cj1;pj and Bjn ! Bj1 for j = 1; 2: Furthermore, �1=2n Gjn ! �1=21 Gj1: Hence, �jn;pj =

C 0jn;pj (Cjn;pj�jn;pjB0jn)Bjn ! C 0j1;pj (

�1=21 Gj1)Bj1 := �j1;pj ; where �j1;pj is a pj�pj diagonal

matrix with nonnegative elements because �jn;pj has these properties n � 1: In consequence,

�1=21 Gj1 = Cj1;pj�j1;pjB0j1: (16.9)

Suppose [�11 : �21] = [C11;p1 : C21;p2 ] has column rank less than p: Then, there is a vector

� 2 Rk with jj�jj = 1 such that [C11;p1 : C21;p2 ]� = 0: Let � = (�01; �02)0 for �j 2 Rpj : Let

�j = Bj1��1j1;pj�j for j = 1; 2 and � = (�

01; �

02)0: We have

0k = [C11;p1 : C21;p2 ]� = C11;p1�1 + C21;p2�2

= C11;p1�11;p1B011B11�

�111;p1�1 + C21;p2�21;p2B

021B21�

�121;p2�2

= C11;p1�11;p1B011�1 + C21;p2�21;p2B

021�2

= �1=21 G11�1 +�1=21 G21�2

= lim�1=2n G1n�1 + lim�1=2n G2n�2

= lim�1=2n Gn�

6= 0k; (16.10)

where the fourth equality holds by the de�nition of �j ; the �fth equality holds by (16.9), the sixth

equality holds because lim�1=2n Gn = �1=21 G1 by Assumptions SS1AR and SS1LM ; Gn = [G1n :

G2n]; and G1 = [G11 : G21]; the last equality holds because Gn = [G1n : G2n]; and the inequality

holds because lim �n > 0 (shown above), �n is the smallest singular value of �1=2n Gn; and � 6= 0

(because Bj1 is an orthogonal matrix, ��1j1;pj is nonsingular, and � 6= 0): Equation (16.10) is a

contradiction, which completes the proof that [�11 : �21] has full column rank p when lim �n > 0:

This completes the proof of part (a).

Next, we prove part (a) for the case of a pure C(�)-LM test, in which case WIn(�) := 0;

WI1 := 0; and Dy1 =M�

a11�a21; when Assumptions SS1LM (vi) and (vii) are replaced by the two

conditions (vi) and (vii) in Comment (v) to Lemma 16.1. If [�a11 : �

a21] has full column rank p;

then the matrixM�a11�a21 has rank p2: This can be proved by showing that ifM�

a11�a21 has rank

less than p2; then [�a11 : �

a21] has column rank less than p: Let (�)+ denote the Moore-Penrose

generalized inverse. Suppose M�a11�a21 has rank less than p2: Then, there exists a nonzero vector

36

' 2 Rp2 such that �a21' = P�a11�a21': That is,

�a21' = �

a11�; where � := (�

a011�

a11)

+�a011�

a21'; and

[�a11 : �

a21]� = 0

k; where � := (�0;�'0)0 6= 0k: (16.11)

In this case, [�a11 : �

a21] has column rank less than p; which establishes the claim in the �rst

sentence of the paragraph.

To prove part (a) in the pure C(�)-LM case, it remains to show that [�a11 : �

a21] has full

column rank p: By its de�nition and (15.10),

�aj1 = [Cj1;qj : �j1;pj�qj + a�jBj1;pj�qj ] for j = 1; 2: (16.12)

Suppose

[C11;q1 : C21;q2 ] has full column rank q1 + q2: (16.13)

Then, by Corollary 16.2 of AG1, [�a11 : �

a21] has full column rank p1 + p2 = p a.s. conditional

on �j1;pj�qj for j = 1; 2 and, hence, unconditionally as well. This holds because, conditional on

�j1;pj�qj for j = 1; 2;

[�11;p1�q1 + a�1B11;p1�q1 : �21;p2�q2 + a�2B21;p2�q2 ] 2 Rk�(p1�q1+p2�q2) (16.14)

has a multivariate normal distribution with identity variance matrix multiplied by a constant (since

� := [�1 : �2] has a multivariate normal distribution with identity variance matrix by assumption

and Bj1;pj�qj has orthonormal columns for j = 1; 2): This is su¢ cient to verify the condition on the

variance matrix of M2�p�q��2 in Corollary 16.2 of AG1 (where M2�p�q��2 is de�ned in Corollary

16.2 of AG1).

Thus, to prove part (a) in the pure C(�)-LM case, it remains to show (16.13). First, we show

qj � rj1 for j = 1; 2: By the de�nition of qj (see (15.2)), we have n1=2� jsn ! 1 8s � qj and

lim supn!1 n1=2� jsn <1 8s > qj : Because rjn := rjFn is the rank of �1=2Fn

EFn bGjn (see (8.5)) and� jsn is the sth largest singular value of the same matrix, � jsn = 0 8s > rjn; 8n � 1: By condition(vi) in Comment (v) to Lemma 16.1, rjn = rj1 for some rj1 2 f0; :::; pjg for all n su¢ ciently large.The latter two results imply that lim supn!1 n1=2� jsn <1 8s > rj1: In consequence, qj � rj1:

Let Cjn;qj denote the �rst qj columns of Cjn for j = 1; 2: Let Cjn;rj1 denote the �rst rj1

37

columns of Cjn for j = 1; 2: Now, we have

�min([C11;q1 : C21;q2 ]0[C11;q1 : C21;q2 ])

= lim�min([C1n;q1 : C2n;q2 ]0[C1n;q1 : C2n;q2 ])

� lim�min([C1n;r11 : C2n;r11 ]0[C1n;r11 : C2n;r11 ])

= lim�min([C�1Fn : C�2Fn ]0[C�1Fn : C�2Fn ])

= lim�min(C0�FnC�Fn)

� �

> 0; (16.15)

where the �rst equality holds by Assumptions SS1AR(vii) and SS1LM (iv) (because the smallest

eigenvalue of a matrix is a continuous function of the matrix), the �rst inequality holds because

qj � rj1; the second equality holds by the argument given in the following paragraph, the third

equality holds by de�nition, see (8.7), and the last two inequalities hold by condition (vii) in

Comment (v) to Lemma 16.1 and C�n := C�Fn :

The second equality of (16.15) holds by the following argument. As stated above (see the

third paragraph of this section), we choose the compact SVD of �1=2n Gjn speci�ed in (8.6) with

� = (�01�n; �020)

0 to be the compact SVD that is obtained from the SVD Cjn�jnB0jn by deleting the

non-essential rows and columns of Cjn; �jn; and Bjn for j = 1; 2: This implies that the matrix

containing the �rst rjn columns of Cjn; which is denoted by Cjn;rjn ; equals C�jFn (de�ned in (8.6)).

Since rjn = rj1 for all n su¢ ciently large by condition (vi) in Comment (v) to Lemma 16.1,

we obtain Cjn;rj1 = C�jFn for all n su¢ ciently large for j = 1; 2; which establishes the second

equality in (16.15). This completes the proof of part (a) in the pure C(�)-LM case because (16.15)

establishes (16.13).

Next, we complete the proof of part (b) using the result of part (a). By the convergence results

in (15.11), (16.4), and (16.5) which hold jointly (using Assumption SS1LM (ii)), the continuous

mapping theorem gives

PDyn= P

(cM1n+WIn bP1n)n1=2(b�1=2nbD2n+an�1=2�2)T2n !d P(M�

a11

+WI1P�a11)�

a21= P

Dy1; (16.16)

where the equality holds by the de�nition of Dyn(�) in (7.12) and because a projection matrix PA

is invariant to the multiplication of A by any nonzero constant and the post-multiplication of A

by any nonsingular matrix and the continuous mapping theorem applies because of the a.s. full

column rank property of Dy1 established in part (a) of the lemma. The convergence in (16.16)

38

holds jointly with n1=2bgn !d Z1 by (16.3), which uses Assumption SS1LM (ii).

The convergence result of part (b) follows from (16.16), n1=2bgn !d Z1; and b�1=2n !p �1=21

using the continuous mapping theorem. The limit of the test statistic LM2n; de�ned in (7.13), is

Z 01�1=21 P

Dy1�1=21 Z1: This limit has a �2p2 distribution conditional on D

y1 and, hence, is uncon-

ditionally �2p2 as well, because (i) �1=21 Z1 � N(0k; Ik); (ii)

�1=21 Z1 and Dy

1 are independent

(because Dy1 is a deterministic function of �

aj1 for j = 1; 2); and (iii) Dy

1 has full rank p2 a.s.

This completes the proof of part (b).

Part (c) holds because

limP (�LM2n (�1�n; �) > 0)

= limP (LM2n > �2p2(1� �))

= P (Z 01�1=21 P

Dy1�1=21 Z1 > �2p2(1� �))

= �; (16.17)

where the �rst equality holds by the de�nition of �LM2n (�1�n; �) in (7.13) with � = (�01�n; �

020)

0; and

the second and third equalities hold by part (b) of the lemma. �

Proof of Lemma 16.2. Part (a) of the Lemma holds by condition (I) in (15.13) in the proof

of Lemma 15.2 with the subscripts 1 replaced by subscripts 2 (which holds using Assumptions

SS2LM (ii) and (iii)�(vii) in place of Assumptions SS2AR(ii) and (iv)�(viii)), combined with condi-

tions (II) and (IV) in (15.13) (which hold because Assumption SS2AR is imposed in the present

lemma).

Now, we prove part (b). Assumption SS2LM (i) implies that lim infn!1 �2n = lim infn!1 �2p2n >

0; where �2n is de�ned in (10.2). In consequence, q2 = p2; where q2 is de�ned as q1 is de�ned in

(15.2) with subscripts 1 replaced by subscripts 2: By de�nition T2n := B2nS2n; where B2n is or-

thogonal and S2n = Diagf(n1=2�21n)�1; :::; (n1=2�2p2n)�1g using q2 = p2; see (15.3) with the leading

subscripts 1 replaced by 2: These results give n1=2T2n = O(1): This and part (a) of the lemma give

sup�12B(�1�n;K=n1=2)

jjn1=2b�1=2n (�1) bD2n(�1)T2n � n1=2b�1=2nbD2nT2njj = op(1): (16.18)

Next, we have

ICS�n !p ICS�1 > K�

U (16.19)

using the result in (16.5), the fact that ICS�1 := �1=2min(�

01G

01

�11 G1�1) = lim ��n (using Assump-

tions SS1AR(vi) and SS1LM (vi) and (vii)), and the condition lim infn!1 ��n > K�U (i.e., Assumption

39

SS2LM (i)). Hence, we obtain

ICS�n > K�U wp! 1 and WIn := 1� s

�ICS�n �K�

L

K�U �K�

L

�= 0 wp! 1; (16.20)

where the second result uses the �rst result and the conditions on s(�) in (7.8) (which imply thats(x) = 1 for all x � 1):

We now show that

sup�12B(�1�n;K=n1=2)

jWIn(�1)j = 0 wp! 1 (16.21)

given that lim infn!1 ��n > K�U :

By the same argument as used to show (14.5) and (14.9) in the proof of Lemma 14.2 in Section

14, but with ��n�1 (:= ��n (�1)) and ��n in place of �

�1n�1

and ��1n; respectively, we have

lim infn!1

inf�1��n�1=�

�n = 1 and inf

�1ICS�2n (�1) � (��n )2 + op(1); (16.22)

where inf�1 abbreviates inf�12B(�1�n;K=n1=2) and likewise with sup�1 :

By the same argument as used to show (14.5), but with �n; Gn; �n�1 ; Gn�1 ; lim supn!1 sup�1 ;

and � in place of �1n; G1n; �1n�1 ; G1n�1 ; lim infn!1 inf�1 ; and �; respectively, in (14.6) and (14.7),and with �n in place of �1n�1 ; where �n 2 Rp is such that jj�njj = 1 and �min(�nG0n�1n Gn�n) =

�0n�nG0n

�1n Gn�n�n; we obtain lim supn!1 sup�1 �

�n�1=��n = 1: Combining this with the �rst result

in (16.22), we get

limn!1

sup�1

j��n�1 � ��n j = 0: (16.23)

Using (16.23), by the same argument as used to show (14.9), but with sup�1 ; + inf�1 ; and � inplace of inf�1 ; � sup�1 ; and �; respectively, we get sup�1 ICS�2n (�1) � (��n )2 + op(1): This and the

second result in (16.22) give

sup�1

jICS�n(�1)� ICS�nj = op(1) (16.24)

using ICS�n = ��n + op(1) by (16.19). In turn, this establishes (16.21) using the same argument as

above to show the second result in (16.20).

Now, (16.21) implies that, for �1 2 B(�1�n;K=n1=2); the C(�)-LM statistic LM2n(�1) can be

written as in (7.12) and (7.13) but with WIn(�1) = 0 wp! 1: That is, wp! 1; the C(�)-LM

40

statistic can be written as

LM2n(�1) = (n1=2egn(�1)0cM1n(�1))

�n1=2[b�1=2n (�1) bD2n(�1) + an�1=2�2]T2n�

��

n1=2[b�1=2n (�1) bD2n(�1) + an�1=2�2]T2n�0 cM1n(�1)n1=2[b�1=2n (�1) bD2n(�1) + an�1=2�2]T2n��1

��n1=2[b�1=2n (�1) bD2n(�1) + an�1=2�2]T2n�0 cM1n(�1)n

1=2egn(�1): (16.25)

Each of the multiplicands in (16.25) di¤ers from its counterpart evaluated at �1�n by op(1) uni-

formly over �1 2 B(�1�n;K=n1=2) 8K < 1 by Lemma 15.2(a) and (c) and (16.18). We have

jjn1=2egn(�1�n)jj2 = nbg0nb�1n bgn = Op(1) by (13.3). In addition, using (15.11) and (16.4),

cM1nn1=2[b�1=2n

bD2n + an�1=2�2]Tn !d M�a11�a21 (16.26)

and the latter has full column rank p2 by Lemma 16.1(a) under Assumptions SS1AR; SS2AR; SS1LM ;

and SS2LM ; sinceWI1 = 0 and Dy1 =M�

a11�a21 in the present case by (16.1), (16.5), and (16.20).

In consequence, when �1 = �1�n; the term in the second line of (16.25) that is inverted converges in

distribution to a matrix that is nonsingular a.s. (using cM21n =

cM1n): This, (16.25), and the results

immediately following (16.25) establish the result of part (b). �

17 Veri�cation of Assumptions for the Second-Step

C(�)-QLR1 Test

17.1 Second-Step C(�)-QLR1 Test Results

This section veri�es Assumptions B(ii) and C(ii)-C(v) for the second-step C(�)-QLR1 test

de�ned in Section 7.4.3. The results in this section apply only to moment condition models.

We employ the same de�nitions as in Sections 8.1.1, 14.1, 15.1, and 16.1. In particular, the

following quantities, which appear in the asymptotic distribution in Lemma 17.1 below, are de�ned

as follows: Z1 and 1 are de�ned in Assumption SS1LM (ii), the (possibly random) k� p1 matrix�a11 is de�ned in (15.6), and the (possibly random) k � p2 matrix �

a21 is de�ned in (15.6) with

subscripts 2 in place of 1 throughout. As de�ned, �1=21 Z1 � N(0k; Ik):

41

By de�nition,

n(�1) := Fn(�1) := V arFn(n1=2bgn(�1)) 2 Rk�k;

G2n(�1) := EFn bG2n(�1) 2 Rk�p2 ;G2si(�1) :=

@

@�02sgi(�1) 2 Rk 8s = 1; :::; p2;

�22sn(�1) := V arFn(jjG2si(�1)jj) 2 R 8s = 1; :::; p2; and

�2n(�1) := Diagf��121n(�1); :::; ��12p2n

(�1)g 2 Rp2�p2 : (17.1)

We write a SVD of �1=2n G2n�2n (:= �1=2n (�1�n)G2n(�1�n)�2n(�1�n)) as

�1=2n G2n�2n = C�2n��2nB

�02n; (17.2)

where C�2n 2 Rk�k and B�2n 2 Rp2�p2 are orthogonal matrices and ��2n 2 Rk�p2 has the singular

values ��21n; :::; ��2p2n

of �1=2n G2n�2n in nonincreasing order on its diagonal and zeros elsewhere.

Suppose limn1=2��2sn 2 [0;1] exists for s = 1; :::; p2: (This is Assumption SS1QLR1(i) below.) Letq�2 (2 f0; :::; p2g) be such that

limn1=2��2sn =1 for 1 � s � q�2 and limn1=2��2sn <1 for q�2 + 1 � s � p2: (17.3)

Note that q�2 = q2 under Assumption SS1QLR1 below, where q2 is de�ned in (15.2) with subscripts

2 in place of 1:15 For notational simplicity, let ��2n := ��2p2n: That is, ��2n is the smallest singular

value of �1=2n G2n�2n:

De�ne

h�21;s := limn1=2��2sn <1 8s = q2 + 1; :::; p2;

S�2n := Diagf(n1=2��21n)�1; :::; (n1=2��2q1n)�1; 1; :::; 1g 2 Rp2�p2 and

S�21 := Diagf0; :::; 0; 1; :::; 1g 2 Rp2 ; (17.4)

where q�2 zeros appear in S�21: We have S

�2n ! S�21: In the case of local strong or semi-strong

identi�cation of �2 given �1�n; q�2 = p�2 and S�21 = 0k�p2 : In the case of local weak identi�cation of

�2 given �1�n; S�21 6= 0k�p2 :For the second-step (SS) C(�)-QLR1 test, we use the following assumptions.

15This holds because f(��2sn)2 : s = 1; :::; p2g and f(�2sn)2 : s = 1; :::; p2g are the eigenvalues of �2nG02n

�1n G2n�2n

and G02n�1n G2n; respectively, and �1 � lim infn!1 �min(�2n) � lim supn!1 �sup(�2n) � �2 for some constants

�1 > 0 and �2 <1 by Assumption SS1QLR1:

42

Assumption SS1QLR1. For the null sequence S; (i) limn1=2��2sn 2 [0;1] exists 8s � p2; (ii)

C�2n ! C�21 for some matrix C�21 2 Rk�k; and (iii) B�2n ! B�21 for some matrix B�21 2 Rp2�p2 :

Assumption SS2QLR1. For the null sequence S; 8K < 1; (i) sup�12B(�1�n;K=n1=2) jb�22sn(�1) ��22sn(�1)j !p 0 for f�22sn(�) : n � 1g de�ned in (17.1) 8s = 1; :::; p2; (ii) sup�12B(�1�n;K=n1=2) j�

22sn(�1)�

�22snj ! 0 8s = 1; :::; p2; and (iii) �22sn ! �22s1 for some constant �22s1 2 (0;1) 8s = 1; :::; p2:

The following quantities appear in the expression for the asymptotic distribution of rk2n that

is speci�ed below. De�ne ZD21 2 Rk�p2 by

vec(ZD21) := ZG21 � �21�11 Z1 2 Rp2k; (17.5)

where Z1; ZG21; and �21 are de�ned in Assumption SS1LM : The matrix ZD21 has a nor-

mal distribution and is independent of Z1 because of the joint normality of ZG21 and Z1 and

Cov(vec(ZD21); Z1) = E(ZG21 � �21�11 Z1)Z 01 = EZG21Z01 � �21 = 0(p2k)�k: Partition the

(nonrandom) matrices B�21 and C�21 as

B�21 = (B�21;q2 ; B�21;p2�q2) and C

�21 = (C�21;q2 ; C

�21;k�q2); (17.6)

where q2 = q�2 ; B�21;q2 2 Rp2�q2 ; B�21;p2�q2 2 Rp2�(p2�q2); C�21;q2 2 Rk�q2 ; and C�21;k�q2 2

Rk�(k�q2): For simplicity, there is some abuse of notation here, e.g., B�21;q2 and B�21;p2�q2 denote

di¤erent matrices even if p2 � q2 happens to equal q2:Next, de�ne the (possibly random) matrix �

�21 as follows:

��21 = (�

�21;q2 ;�

�21;p2�q2) 2 R

k�p2 ; ��21;q2 := C�21;q2 2 R

k�q2 ;

��21;p2�q2 := C�21h

��21;p2�q2 +

�1=21 ZD21�21B

�21;p2�q2 2 R

k�(p2�q2);

h��21;p2�q2 :=

26640q2�(p2�q2)

Diagfh�21;q2+1; :::; h�21;p2g

0(k�p2)�(p2�q2)

37752 Rk�(p2�q2); and�21 := Diagf��1211; :::; �

�12p21g 2 R

p2�p2 : (17.7)

When limn1=2��2n < 1; the lemma below shows that the asymptotic distribution of rk2n is

given by

rk21 := �min(K��021;p2�q2C

�21;k�q2C

�021;k�q2�

�21;p2�q2): (17.8)

43

De�ne

ARy21 := Z 01�1=21 (M�

a11+WI1P�a11)

�1=21 Z1;

LM21 := Z 01�1=21 P

Dy1�1=21 Z1; and

QLR121 :=1

2

�ARy21 � rk21 +

q(ARy21 � rk21)2 + 4LM21 � rk21

�: (17.9)

The random variables ARy21 and LM21 give the asymptotic distributions of ARy2n and LM2n (as

stated in Lemma 16.1 for the LM2n statistic). Note that, when WI1 = 1; we have M�a11+

WI1P�a11 = Ik; ARy21 = Z 01

�11 Z1 � �2k: The following lemma shows that QLR121 is the

asymptotic distribution of QLR12n when limn1=2��2n <1:The following lemma veri�es Assumptions B(ii), C(ii), and C(iii) for the second-step C(�)-QLR1

test.

Lemma 17.1 Suppose bgn(�1) are moment conditions, cM1n(�1) is de�ned in (7.10) with a > 0;

p1 < k; and p2 � 1: Let S be a null sequence (or Sm a null subsequence) that satis�es AssumptionsSS1AR; SS1LM ; and SS1QLR1: Then, for the sequence S (or subsequence Sm),

(a) ARy2n !d ARy21;

(b) when limn1=2��2n <1; (i) rk2n !d rk21; (ii)QLR12n !d QLR121; where (�a11;�

a21; D

y1;

rk21) are independent of Z1 and �aj1 has full column rank pj a.s. for j = 1; 2; and (iii) cQLR1(1�

�; rk2n;WIyn)!d cQLR1(1� �; rk21; 1) and the convergence is joint with that in part (b)(ii),

(c) when limn1=2��2n = 1; (i) rk2n !p 1; (ii) QLR12n !d LM21 � �2p2 ; and (iii) cQLR1(1 �

�; rk2n;WIyn)!p �2p2(1� �); and

(d) for all � 2 (0; 1); limPFn(�QLR12n (�1�n; �) > 0) = �:

Comments: (i). For the second-step C(�)-QLR1 test, Lemma 17.1(d) establishes Assumptions

B(ii) and C(ii). For a sequence S with limn1=2��2n = 1; Lemma 17.1(c) establishes AssumptionC(iii) (because the asymptotic �2p2 distribution of QLR12n is absolutely continuous on R when

p2 � 1 and the probability limit of cQLR1(1��; rk2n;WIyn) is the constant �2p2(1��)): AssumptionC(iv) holds because the conditional critical value cQLR1(1 � �; rk2n;WIyn) is nondecreasing in �

since cQLR1(1� �; r; w) is the 1� � quantile of QLR1(r; w); see (7.16).(ii). Under local strong and semi-strong identi�cation of �1 given �20; the terms an�1=2�1 and

an�1=2�2; which arise in the de�nition of QLR12n; do not a¤ect the asymptotic distributions in

Lemma 17.1(b) and (c).

(iii). The proof of Lemma 17.1(b)(i) and (c)(i) uses Theorem 10.4 in the SM to AG1.

44


C(�)-QLR1 test for sequences S with lim infn!1 ��n > K�U > 0:

Lemma 17.2 Suppose bgn(�1) are moment conditions, bDjn(�) is de�ned in (7.9) for j = 1; 2; andcM1n(�1) is de�ned in (7.10) with a � 0: Let S be a null sequence (or Sm a null subsequence) that

satis�es Assumptions SS1AR; SS2AR; SS1LM ; SS2LM ; and SS2QLR1: Then, under the sequence S

(or subsequence Sm), for all constants K <1;(a) sup�12B(�1�n;K=n1=2) jQLR12n(�1)�QLR12nj = op(1) and

(b) sup�12B(�1�n;K=n1=2) jcQLR1(1��; rk2n(�1);WIyn(�1))�cQLR1(1��; rk2n;WIyn)j = op(1) 8� 2

(0; 1):

Comments: Lemma 17.2 does not require a > 0; but Lemma 17.1 above does.


Proof of Lemma 17.1. First, we prove part (a). By (16.5), WIn !p WI1: By (15.11), cM1n !d

M�a11and bP1n = Ik � cM1n !d Ik �M�

a11= P�a11

: Hence,

cM1n +WIn bP1n !d M�a11+WI1P�a11 : (17.10)

This, n1=2bgn !d Z1 and b�1=2n !p �1=21 give

ARy2n : = nbg0nb�1=2n

�cM1n +WIn bP1n� b�1=2n bgn!d Z

01

�1=21

�M�

a11+WI1P�a11

��1=21 Z1 =: ARy21; (17.11)

which establishes part (a).

Parts (b)(i) and (c)(i) of the lemma hold by Theorem 10.4 in the SM to AG1 (using Assump-

tion SS1QLR1 to guarantee the existence of limn1=2��2sn; C�21; and B

�21): To make this clear, the

following is the correspondence between the quantities in (17.1)�(17.7) above, which de�ne the

asymptotic distribution rk21 in part (b)(i), and those in the asymptotic distribution in Lemma

45

10.2, (10.16), and (10.17) in the SM to AG1:

ZD21 $ Dh; G2n $ EFnGi; �1=2n $WFn ; �2n $ UFn ;

(��21n; :::; ��2p2n)$ (�1Fn ; :::; �p2Fn); (h

�21;q2+1; :::; h

�21;p2)$ (h1;q+1; :::; h1;p)

B�21 $ h2; C�21 $ h3; B

�2;q2;1 $ h2;q; B

�2;p2�q2;1 $ h2;p�q;

C�21;q2 $ h3;q; C�21;k�q2 $ h3;k�q; �

�21 $ �h;

��21;q2 $ �h;q; �

�21;p2�q2 $ �h;p�q; h

��21;p2�q2 $ h�1;p�q;

�1=21 $ h71 :=W1(h7); and �21 $ h81 := U1(h8): (17.12)

Next, let

J2n := neg0n �cM1n +WIn bP1n � PDy2n

� egn: (17.13)

It follows from (7.13) and (7.14) that

ARy2n = LM2n + J2n: (17.14)

We now prove part (b)(ii). We have

QLR12n : =1

2

�ARy2n � rk2n +

q(ARy2n � rk2n)2 + 4LM2n � rk2n

�!d

1

2

�ARy21 � rk21 +

q(ARy21 � rk21)2 + 4LM21 � rk21

�= : QLR121; (17.15)

where the �rst and last equalities hold by the de�nitions of QLR12n and QLR121 in (7.14) and

(17.9), respectively, and the convergence holds by (17.11), rk2n !d rk21 (by part (b)(i) of the

lemma), and LM2n !d LM21 by Lemma 16.1(b). (The latter three convergence results hold

jointly because they all rely on Assumption SS1LM (ii).)

We have (�a11;�

a21; D

y1) is independent of Z1 and�

aj1 has full column rank pj a.s. for j = 1; 2

by Lemma 16.1(b). In addition, rk21 is independent of Z1 because rk21 is a deterministic function

of ZD21 by (17.7) and (17.8) and ZD21 is independent of Z1, see the discussion following (17.5).

Next, we prove part (b)(iii). First, we show that WIyn = 1 wp! 1: We have WIn !p WI1

(by (16.5)), 0 � ICS�1 = lim ��n � lim infn!1 ��2n = 0 (by (16.5), the de�nitions of ��n and ��2n

in (10.3), and limn1=2��2n < 1; which is assumed in part (b)), WI1 = 1 � s(0) = 1 (by (16.1),

ICS�1 = 0; and the de�nition of s(�) in (7.8)), and WIyn := 1(WIn > 0): These results combine to

establish that WIyn = 1 wp! 1: Hence, when proving part (b)(iii), we can suppose WIyn = 1 a.s.

46

To prove part (b)(iii), we need to show cQLR1(1 � �; rk2n; 1) !d cQLR1(1 � �; rk21; 1): This

holds by part (b)(i) of the lemma and the continuous mapping theorem provided cQLR1(1� �; r; 1)is continuous at all r � 0:

Now we establish the latter. For notational simplicity, let c(r) := cQLR1(1 � �; r; 1) and

QLR1(r) := QLR1(r; 1): Given any r� � 0 and sequence frn � 0 : n � 1g such that rn ! r�

(as n!1), it su¢ ces to show that for any subsequence fvng of fng there exists a subsubsequencefmng such that c(rmn) ! c(r�): We have: (i) QLR1(rn) ! QLR1(r�) a.s. (by the de�nition of

QLR1(r) in (7.16)), (ii) given any subsequence fvng of fng; there exists a subsubsequence fmng suchthat c1 = lim c(rmn) exists and is �nite (because part (i) implies that fc(r) : jr� r�j � "g lies in acompact set for some " > 0); and (iii) P (QLR1(r�) = c1) = 0 (because QLR1(r�) has an absolutely

continuous distribution). Results (i)�(iii) imply 1(QLR1(rmm)) � c(rmm)) ! 1(QLR1(r�) � c1)

a.s. Hence, by the dominated convergence theorem,

P (QLR1(r�) � c1) = limP (QLR1(rmm)) � c(rmm)) = 1� �; (17.16)

where the last equality holds because c(rmm) is the 1 � � quantile of QLR1(rmm)) for all n � 1:

Equation (17.16) implies that c1 is the 1� � quantile of QLR1(r�); which is unique (because the

distribution function of QLR1(r�) is continuous and strictly increasing on R+): That is, c1 = c(r�);

which completes the proof that cQLR1(1� �; r; 1) is continuous at all r � 0:We now prove part (c)(ii). By part (c)(i), rk2n !p 1: By (17.14) and some algebra, we have

(ARy2n � rk2n)2 + 4LM2n � rk2n = (LM2n � J2n + rk2n)2 + 4LM2n � J2n: Therefore,

QLR12n =1

2

�LM2n + J2n � rk2n +

p(LM2n � J2n + rk2n)2 + 4LM2n � J2n

�: (17.17)

Using a mean-value expansion of the square-root expression in (17.17) about (LM2n�J2n+rk2n)2;we have

p(LM2n � J2n + rk2n)2 + 4LM2n � J2n = LM2n � J2n + rk2n + (2

p�n)

�14LM2n � J2n (17.18)

for an intermediate value �n between (LM2n�J2n+rk2n)2 and (LM2n�J2n+rk2n)2+4LM2n �J2n:It follows that

QLR12n = LM2n + op(1)!d �2p; (17.19)

where the equality holds because (p�n)

�1 = op(1) (since rk2n !p 1; LM2n = Op(1) by Lemma

16.1(b), and J2n = ARy2n �LM2n = Op(1) using part (a) of the lemma) and the convergence holds

by Lemma 16.1(b).

47

We now prove part (c)(iii). By an analogous argument to that used to prove (17.19), we obtain

QLR1(r; w)! �2p2 a.s. as r !1 (17.20)

for w = 0 or 1; where QLR1(r; w) is de�ned in (7.16). In consequence, cQLR1(1 � �; r; w) !p

�2p2(1 � �) as r ! 1 for w = 0 or 1: This, rk2n !p 1 (by part (c)(i)), and WIyn 2 f0; 1g implythat cQLR1(1� �; rkn;WIyn)!p �

2p2(1� �); which establishes part (c)(iii).

Now, we prove part (d). First, consider the case when limn1=2��2n = 1: By parts (c)(ii) and(c)(iii),

QLR12n � cQLR1(1� �; rk2n;WIn)!d LM21 � �2p2(1� �): (17.21)

In consequence,

limPFn(�QLR12n (�1�n; �) > 0) = P (LM21 � �2p2(1� �) > 0) = �; (17.22)

where the �rst equality holds using P (LM21 = �2p2(1 � �)) = 0 (because LM21 � �2p2 by part

(c)(ii)) and the second equality holds because LM21 � �2p2 :

Next, we prove part (d) when limn1=2��2n <1: By parts (b)(ii) and (b)(iii),

QLR12n � cQLR1(1� �; rkn;WIn)!d QLR121 � cQLR1(1� �; r21; 1): (17.23)

Thus,

limPFn(�QLR12n (�1�n; �) > 0) = P (QLR121 > cQLR1(1� �; r21; 1)) (17.24)

provided P (QLR121 = cQLR1(1 � �; r21; 1)) = 0; which holds if P (QLR121 = cQLR1(1 ��; r21; 1)j�

a11;�

a21; r21) = 0 a.s. The latter holds if, conditional on (�

a11;�

a21; rk21); QLR121

is absolutely continuous, which we now show.

As shown in the proof of part (b)(iii), WI1 = 1: This implies that ARy21 = Z 01�11 Z1 by the

de�nition of ARy21 in (17.9). In the present case where WI1 = 1; de�ne

J21 := ARy21 � LM21 = Z 01�1=21 (Ik � PDy

21)�1=21 Z1: (17.25)

Conditional on (�a11;�

a21; rk1); J21 � �2k�p2 because (i) Ik � P

Dy21is a projection matrix with

rank k � p2 a.s. (since Dy21 has rank p2 a.s. by Lemma 16.1(a)), (ii) �1=21 Z1 � N(0k; Ik)

(by Assumptions SS1AR(ii) and (iii)), and (iii) �1=21 Z1 and (�

a11;�

a21; rk1) are independent

(by part (b)(ii)). In addition, conditional on (�a11;�

a21; rk21); LM21 and J21 are independent

because PDy21�1=21 Z1 and (Ik�PDy

21)

�1=21 Z1 are jointly normally distributed and uncorrelated

48

conditional on (�a11;�

a21; rk21):

In sum, conditional on (�a11;�

a21; rk1); LM21 � �2p2 ; J21 � �2k�p2 ; LM21 and J21 are inde-

pendent, and, hence, ARy21 = LM21 + J21 � �2p2 + �2k�p2 ; where �

2p2 and �

2k�p2 are independent.

Thus, the conditional distribution of QLR121 (de�ned in (17.9)) given (�a11;�

a21; rk21) with

rk21 = r is the same as that of

QLR1(r; 1) :=1

2

��2p2 + �

2k�p2 � r +

q(�2p2 + �

2k�p2 � r)

2 + 4�2p2r�

(17.26)

for all conditioning values of (�a11;�

a21); which is absolutely continuous for all r � 0: Hence,

QLR121 is absolutely continuous conditional on (�a11;�

a21; rk21) a.s., which completes the proof

of (17.24).

Using the result in (17.26), we obtain

P (QLR121 > cQLR1(1� �; r21; 1)) = E[P (QLR121 > cQLR1(1� �; r21; 1)j�a11;�

a21; r21)]

= E[P (QLR1(r21; 1) > cQLR1(1� �; r21; 1)jr21)]

= �; (17.27)

where the last equality holds because cQLR1(1 � �; r; 1) is the 1 � � quantile of the distribution

of QLR1(r; 1); see (7.16), and QLR1(r; 1) is absolutely continuous for all r � 0: This and (17.24)establish part (d) when limn1=2��2n <1: �

Proof of Lemma 17.2. Let �n(�1) denote LM2n(�1); b�1=2n (�1) bD2n(�1); WIn(�1); or ARy2n(�1);

and let �n denote LM2n; b�1=2nbD2n; WIn; or AR

y2n: For each de�nition of �n(�1) and �n; we have:

for all K <1;sup

�12B(�1�n;K=n1=2)jj�n(�1)��njj = op(1) (17.28)

by Lemma 16.2(b) for �n(�1) = LM2n(�1); by Lemma 16.2(a) for �n(�1) = b�1=2n (�1) bD2n(�1);by (16.20) and (16.21) for �n(�1) = WIn(�1); and for �n(�1) = ARy2n(�1) by the combination of

Lemma 15.2(c), an analogous result to Lemma 15.2(c) with bP1n(�1) in place of cM1n(�1); and the

result for �n(�1) =WIn(�1):

First, we prove part (b) of the lemma. By an analogous proof to that of condition (II) in (15.13)

with b�22sn(�1) in place of bn(�1) (using Assumptions SS2QLR1(i) and (ii) in place of AssumptionsSS2AR(x) and (xi)), we have

sup�12B(�1�n;K=n1=2)

jb�22sn(�)� b�22snj = op(1) 8s = 1; :::; p2: (17.29)

49

Using this, the de�nition of b�2n(�) in (7.15), and Assumptions SS2QLR1(i)�(iii) (which imply thatsup�12B(�1�n;K=n1=2) b�22sn(�1) = Op(1) and sup�12B(�1�n;K=n1=2) 1=b�2sn(�1) = Op(1)); we obtain

sup�12B(�1�n;K=n1=2)

jjb�2n(�)� b�2njj = op(1): (17.30)

Combining (17.30), (17.28) with �n(�1) = b�1=2n (�1) bD2n(�1); and the de�nition of rk2n(�1) in(7.15) gives

sup�12B(�1�n;K=n1=2)

jrk2n(�1)� rk2nj = op(1): (17.31)

Equation (17.31), the result of (17.28) with �n(�1) =WIn(�1); and the continuity of cQLR1(1��; r; w) in r � 0 for all � 2 (0; 1) and w = 0 combine to establish part (b) of the lemma.

The result of part (a) of the lemma follows from (17.31), (17.28) with �n(�1) = ARy2n(�1) and

�n(�1) = LM2n(�1); and the functional form of QLR12n(�1) in (7.14). �

18 Amalgamation of High-Level Conditions

In this section, we amalgamate results given in the preceding sections of the SM for the two-step

AR/AR, AR/LM, and AR/QLR1 tests.

18.1 Amalgamation Results for the AR/AR Test

The following theorem provides high-level (HL) su¢ cient conditions for Assumptions B and C

to hold for the two-step AR/AR test. This theorem amalgamates the results of Lemmas 12.1, 12.2,

13.1, 14.1, 15.1, and 15.2 for the two-step AR/AR test.

Assumption HL1AR=AR. For the null sequence S; for some " > 0 and 8K <1; (i) sup�12B(�1�n;")jjbgn(�1)�gn(�1)jj = op(1) for fgn(�) : n � 1g de�ned in (14.1), (ii) gn = 0k 8n � 1; (iii) �1�n ! �1�1

for some �1�1 2 �1; (iv) bgn(�1) is twice continuously di¤erentiable on B(�1�n; ") (for all samplerealizations) 8n � 1; (v) n1=2(bg0n; vec( bG1n � EFn bG1n)0)0 !d (Z

01; Z

0G11)

0 � N(0(p1+1)k; V11) for

some variance matrix V11 2 R(p1+1)k�(p1+1)k whose �rst k rows are denoted by [1 : �011] for

1 2 Rk�k and �11 2 R(p1k)�k; (vi) sup�12B(�1�n;") jj bG1n(�1) � G1n(�1)jj = op(1) for fG1n(�) :n � 1g de�ned in (14.1), (vii) sup�12B(�1�n;") jjG1n(�1)jj = O(1); (viii) sup�12B(�1�n;"n) jjG1n(�1) �G1njj = o(1) for all sequences of positive constants "n ! 0; (ix) G1n ! G11 for some matrix

G11 2 Rk�p1 ; (x) EFnb�1n = O(1); where b�1n := maxs;u�p1 sup�12B(�1�1;") jj(@2=@�1s�1u)bgn(�1)jj;(xi) sup�12B(�1�n;K=n1=2) jjbn(�1) � n(�1)jj !p 0 for fn(�) : n � 1g de�ned in (14.1), (xii)sup�12B(�1�n;K=n1=2) jjn(�1) � njj ! 0; (xiii) lim infn!1 �min(n) > 0; (xiv) n ! 1 for 1

50

as in condition (v), (xv) sup�12B(�1�n;K=n1=2) jjb�1n(�1) � �1n(�1)jj = op(1) for f�1n(�) : n � 1gde�ned in (14.2), (xvi) sup�12B(�1�n;K=n1=2) jj�1n(�1) � �1njj ! 0; (xvii) �1n ! �11 for �11 as

in condition (v), (xviii) limn1=2�1sn 2 [0;1] exists 8s � p1; (xix) C1n ! C11 for some matrix

C11 2 Rk�k; (xx) B1n ! B11 for some matrix B11 2 Rp1�p1 ; (xxi) cn ! 0 for fcn : n � 1g in(7.3), (xxii) ncn ! 1; (xxiii) sup�12B(�1�n;K=n1=2) jb�21sn(�1) � �21sn(�1)j !p 0 for f�21sn(�) : n � 1g8s = 1; :::; p1 de�ned in (14.3), (xxiv) sup�12B(�1�n;K=n1=2) j�

21sn(�1) � �21snj ! 0 8s = 1; :::; p1; and

(xxv) lim infn!1 �21sn > 0 8s = 1; :::; p1:

Assumption W. For the null sequence S; (i) cW1n is symmetric and psd, and (ii) cW1n !p W11

for some nonrandom nonsingular matrix W11 2 Rk�k:

Theorem 18.1 Suppose bgn(�1) are moment conditions, bD1n(�) is de�ned in (7.9), cM1n(�1) is

de�ned in (7.10) with a > 0; bQn(�) is the GMM criterion function de�ned in (7.2), CS1n is the

�rst-step AR CS CSAR1n ; �2n(�1; �) is the second-step C(�)-AR test �AR2n (�1; �); b�2n(�1) is de�ned

in (7.4)�(7.8), and p1 < k: Let S be a null sequence (or Sm a null subsequence) that satis�es

Assumption HL1AR=AR:

(a) Suppose, in addition, the sequence S (or subsequence Sm) is such that lim ��1n < KL (where

KL <1 appears in the de�nition of b�2n(�1) in (7.8)). Then, Assumption B holds for the sequenceS (or subsequence Sm).


quence Sm) is such that lim infn!1 �1n > 0 and Assumption W holds. Then, Assumption C holds



sequence S for the two-step AR/AR test. This lemma amalgamates the results of Lemmas 12.3,

12.4, 13.2, and 14.2.

Assumption HL2AR=AR. For the null sequence S; (i) lim infn!1 ��1n > KU (for KU > 0 as in the

de�nition of b�2n(�1) in (7.8)), (ii) sup�12�1 n1=2jjbgn(�1)�gn(�1)jj = Op(1) for fgn(�) : n � 1g de�nedin (14.1), (iii) limn!1 inf�1 =2B(�1�n;Kn=n1=2)

n1=2jjgn(�1)jj = 1 for all sequences Kn ! 1; (iv)lim infn!1 inf�1 =2B(�1�n;") jjgn(�1)jj > 0 8" > 0; (v) sup�12�1 jjbn(�1)�n(�1)jj = op(1) for fn(�) :n � 1g de�ned in (14.1), (vi) sup�12�1 jjn(�1)jj = O(1); and (vii) lim infn!1 inf�12�1 �min(n(�1))

> 0:

Lemma 18.2 Suppose bgn(�1); bD1n(�); cM1n(�1); CS1n; �2n(�1; �); and b�2n(�1) are as in Theorem18.1, a > 0; and p1 < k: Let S be a null sequence (or Sm a null subsequence) that satis�es

Assumptions HL1AR=AR; HL2AR=AR; and W. Then, Assumption OE holds for the sequence S (or

subsequence Sm).

51

18.2 Amalgamation Results for the AR/LM and AR/QLR1 Tests

The following theorem provides high-level su¢ cient conditions for Assumptions B and C to

hold for the two-step AR/LM and AR/QLR1 tests. For the two-step AR/LM test, this theorem

amalgamates the results of Lemmas 12.1, 12.2, 13.1, 14.1, 16.1, and 16.2. For the AR/QLR1 test,

it amalgamates the results of Lemmas 12.1, 12.2, 13.1, 14.1, 17.1, and 17.2.

Assumption HL1AR=LM. For the null sequence S; 8K <1; (i) bgn(�1; �2) is di¤erentiable in �2at �20 and (@=@�02)bgn(�1; �2) is di¤erentiable in �1 with both holding 8�1 2 B(�1�n; ") (for all sam-ple realizations), 8n � 1; for some " > 0; (ii) n1=2(bg0n; vec( bG1n � EFn bG1n)0; vec( bG2n � EFn bG2n)0)0!d (Z

01; Z

0G11; Z

0G21)

0 � N(0(p+1)k; V1) for some variance matrix V1 2 R(p+1)k�(p+1)k whose

�rst k rows are denoted by [1 : �011 : �021] for 1 2 Rk�k and �j1 2 R(pjk)�k for j = 1; 2;

(iii) sup�12B(�1�n;K=n1=2) jj bG2n(�1) � G2n(�1)jj !p 0 for fG2n(�) : n � 1g de�ned in (14.1), (iv)sup�12B(�1�n;K=n1=2) jjG2n(�1) � G2njj ! 0; (v) G2n ! G21 for some matrix G21 2 Rk�p2 ; (vi)

sup�12B(�1�n;K=n1=2) jj(@2=@�1s@�

02)bgn(�1; �20)jj = Op(1) for s = 1; :::; p1; (vii) sup�12B(�1�n;K=n1=2)

jjb�2n(�1) � �2n(�1)jj = op(1) for f�2n(�) : n � 1g de�ned in (14.2), (viii) sup�12B(�1�n;K=n1=2)jj�2n(�1)� �2njj ! 0; (ix) �2n ! �21 for �21 as in condition (ii), (x) limn1=2�2sn 2 [0;1] exists8s � p2 (where �2sn is de�ned in the paragraph containing (10.2), (xi) C2n ! C21 for some matrix

C21 2 Rk�k; (xii) B2n ! B21 for some matrix B21 2 Rp2�p2 ; (xiii) sup�12B(�1�n;K=n1=2) jb�22sn(�1)��22sn(�1)j !p 0 for f�22sn(�) : n � 1g 8s = 1; :::; p2 de�ned in (17.1), (xiv) sup�12B(�1�n;K=n1=2) j�

22sn(�1)

� �22snj ! 0 8s = 1; :::; p2; and (xv) �2jsn ! �2js1 for some constant �2js1 2 (0;1) 8s =1; :::; p2;8j = 1; 2:

Assumption HL1AR=QLR1: For the null sequence S; (i) limn1=2��2sn 2 [0;1] exists 8s � p2

(where ��2sn := ��2sn(�1�n; �20) is de�ned in the paragraph containing (17.2), (ii) C�2n ! C�21 for

some matrix C�21 2 Rk�k; and (iii) B�2n ! B�21 for some matrix B�21 2 Rp2�p2 :

Lemma 18.3 Suppose the conditions in Theorem 18.1 hold except that �2n(�1; �) is the second-

step C(�)-LM test �LM2n (�1; �) or the second-step C(�)-QLR1 test �QLR12n (�1; �); b�2n(�1) is de�ned

accordingly in (7.6)�(7.8), and p1 < k is replaced by p2 � 1 for the C(�)-LM test and by p2 � 1 andp � k for the C(�)-QLR1 test. Let S be a null sequence (or Sm a null subsequence) that satis�es

Assumptions HL1AR=AR and HL1AR=LM and, for the second-step C(�)-QLR1 test, Assumption

HL1AR=QLR1 as well.

(a) Suppose, in addition, the sequence S (or subsequence Sm) is such that lim ��n < KL (where

KL <1 appears in the de�nition of b�2n(�1) in (7.8)). Then, Assumption B holds for the sequenceS (or subsequence Sm).

52


quence Sm) is such that lim infn!1 ��n > K�U and Assumption W holds. Then, Assumption C holds


Comment: When LM2n(�) is the pure C(�)-LM statistic, i.e., WIn(�) := 0; Lemma 18.3(a) holds

provided conditions (vi) and (vii) in Comment (v) to Lemma 16.1 hold, and Lemma 18.3(b) holds

with the weaker condition lim infn!1 �n > 0 in place of lim infn!1 ��n > K�U : The same is true

when the QLR12n(�) is the pure C(�)-QLR1 statistic and WIn(�) := 0 in the QLR1 critical value

function.


sequence S for the two-step AR/LM and AR/QLR1 tests. This lemma amalgamates the results of

Lemmas 12.3, 12.4, 13.2, and 14.2 for these tests.

Assumption HL2AR=LM;QLR1. Assumption HL2AR=AR holds with ��n in place of ��1n in part (i).

Lemma 18.4 Suppose bgn(�1); bD1n(�); cM1n(�1); CS1n; and fcn : n � 1g are as in Theorem 18.1,

�2n(�1; �) is the second-step C(�)-LM �LM2n (�1; �) or C(�)-QLR1 test �QLR12n (�1; �); b�2n(�1) is de-

�ned accordingly in (7.6)�(7.8), a > 0; and p2 � 1: Let S be a null sequence (or Sm a null

subsequence) that satis�es Assumptions HL1AR=AR; HL1AR=LM ; HL2AR=LM;QLR1; and W. Then,

Assumption OE holds for the sequence S (or subsequence Sm).

Comment: Lemma 18.4 di¤ers from Lemma 18.2 because the second-step data-dependent signif-

icance level di¤ers between the second-step C(�)-AR test, which is considered in the latter lemma,

and the second-step C(�)-LM and C(�)-QLR1 tests, which are considered in the former lemma.

18.3 Proofs of Theorem 18.1 and Lemmas 18.2, 18.3, and 18.4

Proof of Theorem 18.1. Assumption B(i) holds by Lemma 13.1, which employs Assumption

FS1AR; because Assumption HL1AR=AR(xi)) Assumption FS1AR(i); HL1AR=AR(xiv)) FS1AR(ii);

HL1AR=AR(v) ) FS1AR(iii); and HL1AR=AR(xiii) ) FS1AR(iv).

Assumptions B(ii), C(ii), and C(iii) hold by Lemma 15.1 (and Comment (i) following it),

which employs Assumption SS1AR; because a > 0; p1 < k; Assumption HL1AR=AR(xviii) ) As-

sumption SS1AR(i); HL1AR=AR(v) ) SS1AR(ii); HL1AR=AR(xiii) & HL1AR=AR(xiv) ) SS1AR(iii);

HL1AR=AR(xv) & HL1AR=AR(xvii) ) SS1AR(iv); HL1AR=AR(xi) ) SS1AR(v); HL1AR=AR(xiv) )SS1AR(vi); HL1AR=AR(xix) ) SS1AR(vii), and HL1AR=AR(xx) ) SS1AR(viii).

Assumption B(iii) holds under the conditions of Theorem 18.1(a) by Lemma 14.1, which em-

ploys Assumption SL1AR, because the assumption lim ��1n < KL of Theorem 18.1(a) ) Assump-

tion SL1AR(i); Assumption HL1AR=AR(vi)) Assumption SL1AR(ii); HL1AR=AR(vii)) SL1AR(iii);

53

HL1AR=AR(xi) ) SL1AR(iv); HL1AR=AR(xiii) ) SL1AR(v); HL1AR=AR(xiv) ) SL1AR(vi);

HL1AR=AR(xxiii) ) SL1AR(vii); and HL1AR=AR(xxv) ) SL1AR(viii).

Assumption C(i) holds under the conditions of Theorem 18.1(b) by Lemma 12.1, which employs

Assumptions ES1 and ES2, because the result of Lemma 12.2 ) Assumption ES1 and Lemma

12.2 applies here because Assumption FOC is veri�ed below; Assumption HL1AR=AR(iv)) ES2(i);

Assumption HL1AR=AR(v)) ES2(ii); HL1AR=AR(vi) & HL1AR=AR(vii)) ES2(iii); HL1AR=AR(xxii)

) ES2(iv), and Assumption W ) ES2(v).

For the veri�cation of Assumption C(i) under the conditions of Theorem 18.1(b), it remains

to show that Assumption FOC, which is employed in Lemma 12.2, holds. We have: the assump-

tion lim infn!1 �1n > 0 of Theorem 18.1(b) ) Assumption FOC(i); Assumption HL1AR=AR(i)

) FOC(ii); HL1AR=AR(ii) ) FOC(iii); HL1AR=AR(iii) ) FOC(iv); HL1AR=AR(iv) ) FOC(v);

HL1AR=AR(v)) FOC(vi); HL1AR=AR(vi)) FOC(viii); HL1AR=AR(vii)) FOC(ix); HL1AR=AR(viii)

) FOC(x); HL1AR=AR(ix) ) FOC(xi); HL1AR=AR(xiii) ) FOC(xiii); Markov�s inequality and

HL1AR=AR(x) ) FOC(xiv); and Assumption W ) FOC(xvi). In addition, because we are consid-

ering the moment condition model here, by the paragraph following Assumption FOC, Assumptions

FOC(vii), (xii), and (xv) are implied by Assumptions FOC(v) and (xiv), which have just been ver-

i�ed, and Assumption HL1AR=AR(x).

As noted in Section 15 above, Assumption C(iv) holds automatically for the second-step AR

test provided p1 < k (which is assumed here) because its nominal level � critical value is the 1� �quantile of the �2k�p1 distribution which is nondecreasing in � for � 2 (0; 1) when p1 < k:

Assumption C(v) holds under the conditions of Theorem 18.1(b) by Lemma 15.2(d), which

employs Assumption SS2AR; because the assumption lim infn!1 �1n > 0 of Theorem 18.1(b)

) Assumption SS2AR(i); Assumption HL1AR=AR(iv) ) Assumption SS2AR(ii); HL1AR=AR(v) )SS2AR(iii); HL1AR=AR(vi) ) SS2AR(iv); HL1AR=AR(vii) ) SS2AR(v); Markov�s inequality and

HL1AR=AR(x) ) SS2AR(vi); HL1AR=AR(xv) ) SS2AR(vii); HL1AR=AR(xvi) ) SS2AR(viii);

HL1AR=AR(xvii) ) SS2AR(ix); HL1AR=AR(xi) ) SS2AR(x); HL1AR=AR(xii) ) SS2AR(xi);

HL1AR=AR(xiii) ) SS2AR(xii); and HL1AR=AR(xiv) ) SS2AR(xiii). �

Proof of Lemma 18.2. As required by Assumption OE, Assumption C holds for the se-

quence S by Theorem 18.1(b), using Assumptions HL1AR=AR and HL2AR=AR(i). Note that the

condition lim infn!1 �1n > 0 of Theorem 18.1(b) is implied by Assumption HL2AR=AR(i) (i.e.,

lim infn!1 ��1n > KU for KU > 0) plus Assumption HL1AR(xxv), which implies that lim supn!1

�max(�1n) <1; where �1n := �1n(�1�n; �20) is de�ned in (8.3) and (14.3).By Lemma 12.3, the results of Lemmas 12.4 and 13.2 imply that Assumption OE(i) holds.

Hence, we need to verify the assumptions used in Lemmas 12.4 and 13.2 to verify Assumption

54

OE(i).

First, we verify Assumptions ES3 and ES4, which are imposed in Lemma 12.4. Assumptions

HL2AR=AR(i) & HL1AR=AR(xxv) ) Assumption ES3(i); HL1AR=AR(iv) ) ES3(iii); HL1AR=AR(v)

) ES3(iv); HL1AR=AR(vi) ) ES3(v); HL1AR=AR(viii) ) ES3(vi); HL1AR=AR(ix) ) ES3(vii);

HL1AR=AR(xi) ) ES3(viii); HL1AR=AR(xiii) ) ES3(ix); HL1AR=AR(xxi) ) ES3(x); and W )ES3(xi). Assumption ES3(ii) holds by Lemma 12.1 under Assumptions ES1 and ES2; it is shown

in the proof of Theorem 18.1 that Assumption HL1AR=AR ) Assumption ES2; Lemma 12.2 veri�es

Assumption ES1 using Assumption FOC; and Assumption FOC is veri�ed in Theorem 18.1 (and the

present lemma imposes Assumption HL1AR=AR; which is employed in Theorem 18.1). Assumption

HL2AR=AR(ii)) Assumption ES4(i) and HL2AR=AR(iv)) ES4(ii). This completes the veri�cation

of Assumptions ES3 and ES4.

Second, we verify Assumption FS2AR; which is used in Lemma 13.2. Assumption HL2AR=AR(ii)

) Assumption FS2AR(i); HL2AR=AR(iii) ) FS2AR(ii); HL2AR=AR(v) ) FS2AR(iii); HL2AR=AR(vi)

) FS2AR(iv); and HL2AR=AR(vii) ) FS2AR(v). This completes the veri�cation of Assumption

OE(i).

Lastly, Assumption OE(ii) holds under Assumption SL2AR by Lemma 14.2. Hence, we need to

verify Assumption SL2AR. Assumption HL2AR=AR(i) ) Assumption SL2AR(i); HL1AR=AR(vi) )SL2AR(ii); HL1AR=AR(viii)) SL2AR(iii); HL1AR=AR(ix)) SL2AR(iv); HL1AR=AR(xi)) SL2AR(v);

HL1AR=AR(xii) ) SL2AR(vi); HL1AR=AR(xiii) ) SL2AR(vii); HL1AR=AR(xxiii) ) SL2AR(viii);

HL1AR=AR(xxiv) ) SL2AR(ix); and HL1AR=AR(xxv) ) SL2AR(x). This completes the proof of

the lemma. �

Proof of Lemma 18.3. For the AR/LM and AR/QLR1 tests, Assumptions B(i) and C(i) hold by

the same arguments as given in the proof of Theorem 18.1 (using the fact that lim infn!1 ��n > K�U

(> 0); which is assumed in Lemma 18.3(b), implies that lim infn!1 �1n > 0; which is assumed in

Theorem 18.1(b), when verifying Assumption C(i)).

For the AR/LM and AR/QLR1 tests, Assumption B(iii) holds under the conditions of Lemma

18.3(a) by Lemma 14.1, which employs Assumption SL1LM;QLR1 (for the second-step C(�)-LM

and C(�)-QLR1 tests), because the assumption lim ��n < KL of Lemma 18.3(a) ) Assump-

tion SL1LM;QLR1(i); Assumption HL1AR=AR ) Assumptions SL1AR(ii)�(viii), as shown in the

proof of Theorem 18.1, and the latter conditions constitute Assumption SL1LM;QLR1(ii); and

HL1AR=LM (xiii)�(xv) ) SL1LM;QLR1(iii) and (iv).

For the second-step C(�)-LM test, Assumptions B(ii), C(ii), and C(iii) hold by Lemma 16.1

and Comment (i) following it, which employs Assumptions SS1AR and SS1LM ; because Assump-

tion HL1AR=AR implies Assumption SS1AR (as shown in the proof of Theorem 18.1), and Assump-

55

tion HL1AR=LM implies Assumption SS1LM : The latter holds because Assumption HL1AR=LM (x)

) Assumption SS1LM (i); HL1AR=LM (ii) ) SS1LM (ii); HL1AR=LM (vii) & (ix) with �1 = �1�n

) SS1LM (iii); HL1AR=LM (xi) ) SS1LM (iv); HL1AR=LM (xii) ) SS1LM (v); HL1AR=AR(vi) & (ix)

and HL1AR=LM (iii) & (v) ) SS1LM (vi); and HL1AR=AR(xxiii) and HL1AR=LM (xiii) & (xv) )SS1LM (vii).

For the second-step C(�)-QLR1 test, Assumptions B(ii), C(ii), and C(iii) hold by Lemma 17.1

and Comment (i) following it because Lemma 17.1 relies on Assumptions SS1AR and SS1LM ;

which have just been veri�ed, as well as on Assumption SS1QLR1; which holds because Assumption

HL1AR=QLR1(i)�(iii) ) Assumption SS1QLR1:

For the second-step C(�)-LM test, Assumption C(iv) holds provided p2 � 1 (which is assumedhere) because its nominal level � critical value is the 1� � quantile of the �2p2 distribution which isnondecreasing in � for � 2 (0; 1) when p2 � 1:

For the second-step C(�)-QLR1 test, Assumption C(iv) holds because its conditional critical

value cQLR1(1 � �; rk2n(�1);WIn(�1)) is nondecreasing in � since cQLR1(1 � �; r; w) is the 1 � �

quantile of QLR1(r; w); see (7.16).

For the second-step C(�)-LM test, Assumption C(v) holds under the conditions of Lemma

18.3 by Lemma 16.2(b), which employs Assumptions SS1AR; SS2AR; SS1LM ; and SS2LM ; because

Assumption HL1AR ) Assumption SS1AR; as shown above; HL1AR=LM ) SS1LM ; as shown above;

the condition lim infn!1 ��n > K�U in Lemma 18.3(b), HL1AR=AR(xxiii), and HL1AR=LM (xv) with

j = 1 ) lim infn!1 ��1n > K�U ) Assumption SS2AR(i); Assumption HL1AR ) all parts of

Assumption SS2AR except its part (i) (as shown in the proof of Theorem 18.1); the condition

lim infn!1 �n > 0 in Lemma 18.3(b), HL1AR=AR(xxiii), and HL1AR=LM (xiii) & (xv)) Assumption

SS2LM (i); HL1AR=LM (i) ) SS2LM (ii); and HL1AR=LM (iii)�(viii) ) SS2LM (iii)�(vii).

For the second-step C(�)-QLR1 test, Assumption C(v) holds under the conditions of Lemma

18.3(b) by Lemma 17.2 because Lemma 17.2 relies on Assumptions SS2AR and SS2LM ; which have

just been veri�ed above, and Assumption SS2QLR1; which holds by Assumption

HL1AR=LM (xiii)�(xv). �

Proof of Lemma 18.4. As required by Assumption OE, Assumption C holds for the sequence

S by Lemma 18.3(b), using Assumptions HL1AR=AR; HL1AR=LM ; HL2AR=LM;QLR1; and, for the

C(�)-QLR1 test, HL2AR=AR(i) as well. Note that the condition lim infn!1 ��n > K�U of Lemma

18.3(b) is implied by Assumption HL2AR=AR(i) (i.e., lim infn!1 ��n > KU ) and K�U < KL � KU

(which holds by the de�nition of the constant K�U following (7.11)).

The veri�cation of Assumption OE(i) is the same as in the proof of Lemma 18.2.

Assumption OE(ii) holds under Assumption SL2LM;QLR1 by Lemma 14.2. Hence, we need

56

to verify Assumption SL2LM;QLR1: Assumption HL2AR=LM;QLR1 ) Assumption SL2LM;QLR1(i);

HL1AR=AR ) SL2LM;QLR1(ii) (which consists of SL2AR(v)�(x)), as shown in the proof of Lemma

18.2 above; HL1AR=LM (iii)�(v)) SL2LM;QLR1(iii)�(v); and HL1AR=LM (xiii)�(xv)) SL2LM;QLR1(vi)�

(viii). �


This section proves Theorem 8.1 using the results in Section 18, which, in turn, uses the results

in Sections 12-14.

19.1 Proof of Theorem 8.1

To prove Theorem 8.1 we �nd it useful to reparametrize (�1; F ) with parameter space FAR=AR;de�ned in (8.8), to a parameter � with parameter space �AR=AR: The parameter � is chosen such

that for some subvector of � convergence of a drifting subsequence of the subvector allows one to

verify Assumption HL1AR=AR; which is employed in Theorem 18.1, and Assumption HL2AR=AR;

which is employed in Lemma 18.2.

Let fhn(�) : n � 1g be a sequence of functions on a space �: The parameter � and functionhn(�) are of the following form:

(i) � = (�1; :::; �d; �d+1)0; where �j 2 R 8j � d and �d+1 belongs to some in�nite-dimensional

pseudo-metric space,16 and

(ii) hn(�) = (hn;1(�); :::; hn;J(�))0 and

hn;j(�) =

8<: n1=2�j for j = 1; :::; JR

�j for j = JR + 1; :::; J;for some JR � d: (19.1)

De�ne

H = fh 2 (R [ f�1g)J : hmn(�mn)! h for some subsequence fmng

of fng and some sequence f�mn 2 � : n � 1gg: (19.2)

The result in the following lemma is established in the proof of Theorem 2.2 in Andrews, Cheng,

and Guggenberger (2011). For completeness, we provide a proof below.

16For notational simplicity, we stack d real-valued quantities and one in�nite-dimensional quantity into the vector�:

57

Lemma 19.1 For any sequence f�n 2 � : n � 1g and any subsequence fwng of fng there exists asubsequence fmng of fwng such that hmn(�mn)! h for some h 2 H:

Comment: Lemma 19.1 is useful in establishing the correct asymptotic size of any two-step test,

not just the two-step AR/AR test.

Now, we specify � and � that are used with the two-step AR/AR test.

Let F (�) := V arF (gi(�)): We write a SVD of �1=2F (�1)EFG1i(�1) as

�1=2F (�1)EFG1i(�1) = C1F (�1)�1F (�1)B1F (�1)

0; (19.3)




Let ��1p1F (�1) denote the smallest singular value of �1=2F (�1)EFG1i(�1)�1F (�1); where �1F (�1)

is de�ned in (8.3).

We de�ne the elements of � to be17

�1;�1;F := (�11F (�1); :::; �1p1F (�1))0 2 Rp1 ;

�2;�1;F := B1F (�1) 2 Rp1�p1 ;

�3;�1;F := C1F (�1) 2 Rk�k;

�4;�1;F := EFG1i(�1) 2 Rk�p1 ;

�5;�1;F := EF

0@ gi(�1)

vec(G1i(�1)� EFG1i(�1))

1A0@ gi(�1)

vec(G1i(�1)� EFG1i(�1))

1A0 2 R(p1+1)k�(p1+1)k;�6;�1;F := �1;

�7;�1;F := (�1p1F (�1); ��1p1F (�1))

0

�8;�1;F := F; and

� = ��1;F := (�1;�1;F ; :::; �8;�1;F ): (19.4)

We let �5;g;�1;F denote the upper left k� k submatrix of �5;�1;F: Thus, �5;g;�1F = EF gi(�1)gi(�1)0 =

F (�1) for (�1; F ) 2 FAR=AR:We consider the parameter space �AR=AR for � that corresponds to FAR=AR: The parameter

17For simplicity, when writing � = (�1;F ; :::; �8;F ); we allow the elements to be scalars, vectors, matrices, anddistributions and likewise in similar expressions.

58

space �AR=AR and the function hn(�) are de�ned by

�AR=AR := f� : � = (�1;�1;F ; :::; �8;�1;F ) for some (�1; F ) 2 FAR=ARg and

hn(�) := (n1=2�1;�1;F ; �2;�1F ; �3;�1;F ; �4;�1;F ; �5;�1;F ; �6;�1;F ; �7;�1;F ): (19.5)

By the de�nition of FAR=AR; �AR=AR indexes distributions that satisfy the null hypothesis H0 :�2 = �20: Redundant elements in (�1;�1;F ; :::; �8;�1;F ); such as the redundant o¤-diagonal elements

of the symmetric matrix �5;�1;F ; are not needed, but do not cause any problem. The dimension J

of hn(�) equals the number of elements in (�1;F ; :::; �7;F ):

We de�ne � and hn(�) as in (19.4) and (19.5) because, as shown below, verifying Assump-

tion HL1AR=AR; which is employed in Theorem 18.1, and Assumptions HL2AR=AR; which is em-

ployed in Lemma 18.2, for subsequences requires convergence of the corresponding subsequences of

n1=2�1;�1n;Fn and �j;�1n;Fn for j = 2; :::; 7:

For notational convenience,

f�n;h : n � 1g denotes a sequence f�n 2 �AR=AR : n � 1g for which hn(�n)! h 2 H (19.6)

for H de�ned in (19.2) with � equal to �AR=AR:18 By the de�nitions of �AR=AR and FAR=AR;f�n;h : n � 1g is a sequence of distributions that satis�es the null hypothesis H0 : �2 = �20: Below,

�all sequences f�wn;h : n � 1g�means �all sequences f�wn;h : n � 1g for any h 2 H;�where H

is de�ned with � equal to �AR=AR; and likewise with n in place of wn: To maintain the notation

employed above that �1�n denotes the true value of �1; we let f(�1�n; Fn) : n � 1g denote thesequence of (�1; F ) values in FAR=AR that corresponds to f�n;h : n � 1g:

We decompose h (de�ned by (19.2), (19.4), and (19.5)) analogously to the decomposition of

�: h = (h1; :::; h7); where �j;�1;F and hj have the same dimensions for j = 1; :::; 7: We further

decompose the vector h1 as h1 = (h1;1; :::; h1;minfk;p1g)0; where elements of h1 could equal 1: In

addition, we let h5;g denote the upper left k� k submatrix of h5: In consequence, under a sequencef�n;h : n � 1g; we have

n1=2�1sFn(�1�n) ! h1;s � 0 8s � p1;

�j;�1�n;Fn ! hj 8j = 2; :::; 7; and

�5;g;�1�n;Fn = Fn(�1�n) = V arFn(gi(�1�n))! h5;g: (19.7)

18Analogously, for any subsequence fwn : n � 1g; f�wn;h : n � 1g denotes a sequence f�wn 2 �AR=AR : n � 1g forwhich hwn(�wn)! h 2 H:

59

By the conditions in FAR=AR; h5;g is pd.The following lemma veri�es Assumptions HL1AR=AR; which is employed in Theorem 18.1 and

Lemma 18.2, for all subsequences f�wn;h : n � 1g:

Lemma 19.2 Suppose bgn(�1) are the moment functions de�ned in (3.3), gi(�) satis�es the di¤er-entiability condition in Theorem 8.1, fcn : n � 1g are as in Theorem 8.1, a > 0; p1 < k; � is open,

�1� is bounded, and B(�1�; ") � �1 for some " > 0: Let the null parameter space be FAR=AR: Then,for all subsequences f�wn;h : n � 1g; Assumption HL1AR=AR holds and lim �1n and lim ��1n exist.

The following lemma veri�es Assumption HL2AR=AR:

Lemma 19.3 Suppose bgn(�1) are the moment functions de�ned in (3.3), gi(�) satis�es the dif-ferentiability condition in Theorem 8.1, and the null parameter space is FAR=AR: Let S be a nullsequence (or Sm a null subsequence) for which Assumption SI holds. Then, Assumption HL2AR=AR

holds for the sequence S (or the subsequence Sm).

Now, we prove Theorem 8.1 using Theorems 5.1 and 18.1 and Lemmas 18.2 and 19.1-19.3.

Proof of Theorem 8.1. The result of Theorem 8.1(a) follows from the high-level result Theorem

5.1(a) provided Assumption CAL holds. Assumption CAL requires that for any null sequence

S and any subsequence fwng of fng; there exists a subsubsequence fmng such that Sm satis�es

Assumption B or C. Theorem 18.1 provides high-level conditions under which Assumption B or C

holds for a subsequence Sm: The condition required for Theorem 18.1 is Assumption HL1AR=AR:

By Lemma 19.1, for any null sequence S or, equivalently, any sequence f�n 2 � : n � 1g;and any subsequence fwng of fng; there exists a subsubsequence fmng such that hmn(�mn) ! h

for some h 2 H: By Lemma 19.2, for the subsequence fmng (that satis�es hmn(�mn) ! h for

some h 2 H); Assumption HL1AR=AR holds and lim �1n and lim ��1n exist. Given this, by Theorem18.1(a) and (b), the subsequence Sm satis�es Assumption B when lim ��1mn

< KL and it satis�es

Assumption C when lim infn!1 �1mn > 0 and Assumption W holds, which is assumed.

By de�nition, ��1n is the smallest singular value of �1=2n G1n�1n; see (8.3), and �1n :=

Diagf��111n; :::; ��11p1n

g; where �21sn := V arFn(jjG1sijj); see (14.3). Given these de�nitions and thecondition V arF (jjG1si(�1)jj) � � for (�1; F ) 2 FAR=AR; see (8.8), we have: lim ��1mn

� KL implies

lim infn!1 �1mn > 0: Hence, every subsequence Sm with lim ��1mn

< KL satis�es Assumption B and

every subsequence Sm with lim ��1mn� KL satis�es Assumption C. This completes the veri�cation

of Assumption CAL and the proof of Theorem 8.1(a).

Now we prove Theorem 8.1(b). The result of Theorem 8.1(b) that AsyNRP = � holds

for a sequence S if for any subsequence fwng of fng; there exists a subsubsequence fmng such

60

that limP��mn ;Fmn (�SV2mn

(�) > 0)) = � for the corresponding subsubsequence Sm; where ��mn =

(�01�mn; �020)

0: Take the subsubsequence fmng as above to be such that hmn(�mn) ! h for some

h 2 H: Then, by Lemma 19.2, for the subsubsequence Sm; Assumption HL1AR=AR holds.The limP��mn ;Fmn (�

SV2mn

(�) > 0)) = � result for the subsubsequence Sm follows from the high-

level result Theorem 5.1(c) provided Assumption OE holds for the subsubsequence Sm: Assumption

OE holds for the subsubsequence Sm by Lemma 18.2. The conditions required for Lemma 18.2

are Assumptions HL1AR=AR; HL2AR=AR; and W. Assumption HL1AR=AR is veri�ed in a previous

paragraph. In addition, Assumption W is assumed to hold. Hence, to establish Theorem 8.1(b),

it remains to verify Assumption HL2AR=AR: Using Assumption SI, which is imposed in Theorem

8.1(b), these conditions hold by Lemma 19.3. This completes the proof of Theorem 8.1(b).

Theorem 8.1(c) follows immediately from Theorem 8.1(a) and (b).

Theorem 8.1(d) and (e) hold by Theorem 5.1(d) and (e), respectively, because a sequence S

that satis�es Assumption SI is shown above to satisfy Assumption OE.

To establish Theorem 8.1(f), we use the high-level CS results given in Theorem 5.1(f), rather

than the high-level test results given in Theorem 5.1(a)�(e). This requires verifying the CS versions

of Assumptions B, C, CAL, and OE, rather than the test versions. The only di¤erence between

the CS and test versions is that �2�n appears throughout in place of �20: The veri�cation of the CS

versions these conditions is the same as given above but with some adjustments.

First, we adjust the de�nition of � that appears in (19.4) and (19.5). Speci�cally, we de�ne �

as in (19.4), but with � in place of �1 throughout. We retain the de�nition of hn(�) given in (19.5),

but with � in place of �1 in �:

Second, we adjust the parameter space �AR=AR; which appears in (19.5), to the following

parameter space:

��;AR=AR := f� : � = (�1;�;F ; :::; �9;�;F ) for some (�; F ) 2 F�;AR=ARg; (19.8)

where F�;AR=AR is de�ned in (8.9) using the test parameter space FAR=AR(�2) for �2 2 �2�: Notethat the moment conditions in FAR=AR(�2) hold uniformly over �2 2 �2� by the de�nitions of

F�;AR=AR: For example, EF jjgi(�1)jj2+ = EF jjgi(�1; �20)jj2+ � M (by the de�nition of FAR;ARin (8.8)) 8�20 2 �2� (by the de�nition of F�;AR=AR) implies that sup�22�2� EF jjgi(�1; �2)jj2+ �M: This is used in the adjusted proofs everywhere the moment conditions are employed in the

unadjusted proofs.

Third, we use the assumption that �� is bounded and B(��; ") � � for some " > 0 to ensure

that �6;�n;Fn = �n has a limit in � � Rp; call it �1; for all sequences f�n;h : n � 1g (rather

61

than a limit whose elements might equal �1): The assumption that B(��; ") � � guarantees thatthe mean-value expansions that appear in (12.3), (12.6)�(12.9), (12.19), (15.12), (15.14), (19.12),

(19.13), (19.18), (19.22, (19.37), and (19.41) also hold when �20 is replaced by �2�n:

Given the adjustments above, the results in Lemmas 12.1�12.4, 13.1, 13.2, 15.1, 15.2, 14.1, 14.2,

18.1, and 18.2 hold with the null sequence S replaced by a sequence S which has �20 replaced by

�2�n:

Furthermore, Lemmas 19.2 and 19.3 hold with the adjustments to � and �AR=AR stated imme-

diately above. In consequence, the veri�cation of Assumptions B, C, CAL, and OE given above

goes through when �2�n appears throughout in place of �20: This completes the proof of Theorem

8.1(f). �


Proof of Lemma 19.1. Let fwng be some subsequence of fng: Let hwn;j(�wn) denote the jthcomponent of hwn(�wn) for j = 1; :::; J: Let m1;n = wn 8n � 1: For j = 1; either (1) lim supn!1jhmj;n;j(�mj;n

)j <1 or (2) lim supn!1 jhmj;n;j(�mj;n)j =1: If (1) holds, then for some subsequence

fmj+1;ng of fmj;ng;hmj+1;n;j(�mj+1;n)! hj for some hj 2 R: (19.9)

If (2) holds, then for some subsequence fmj+1;ng of fmj;ng;

hmj+1;n;j(�mj+1;n)! hj ; where hj =1 or �1: (19.10)

Applying the same argument successively for j = 2; :::; J yields a subsequence fmng = fmJ+1;ng offwng for which hmn;j(�m�

n)! hj 8j � J; which establishes the result of the Lemma. �

Proof of Lemma 19.2. For notational simplicity, we prove the result for a sequence f�n;h : n � 1g:The same arguments go through with n replaced by wn to obtain the subsequence results that are

stated in the lemma.

We do not verify Assumptions HL1AR=AR(i), HL1AR=AR(ii), ... in numerical order because some

of these conditions are used in the veri�cation of others. For brevity, we abbreviate Assumptions

HL1AR=AR(i), HL1AR=AR(ii), ... by Assumptions (i), (ii), ....

Assumption (iv) requires that bgn(�1) is twice continuously di¤erentiable on B(�1�n; ") 8n � 1for some " > 0: Assumption (iv) holds because the present lemma imposes the di¤erentiability

condition in Theorem 8.1 (which states that gi(�1) is twice continuously di¤erentiable in �1 on �1

for all sample realizations), and B(�1�n; ") � �1 8n � 1 for some " > 0 because �1�n 2 �1� by the

62

de�nition of F and B(�1�; ") � �1 by an assumption of the lemma.Assumption (i) requires that sup�12B(�1�n;") jjbgn(�1) � gn(�1)jj = op(1) for some nonrandom

functions fgn(�) : n � 1g and some " > 0: We verify Assumption (i) with gn(�1) = EFnbgn(�1): Thiscondition is a uniform WLLN. Because �1�n ! �1�1 2 �1 by (19.31) below, it su¢ ces to establishthis result with B(�1�1; ") in place of B(�1�n; "): Since B(�1�1; ") is a bounded set, su¢ cient

conditions for this uniform WLLN�s are

(a) bgn(�1)� gn(�1) = op(1) 8�1 2 B(�1�1; ") and

(b) sup�a2B(�1�1;")

sup�12B(�a;"n)

jjbgn(�1)� gn(�1)� (bgn(�a)� gn(�a))jj = op(1) (19.11)

for all sequences of constants f"n > 0 : n � 1g for which "n ! 0; e.g., see Theorem 1(a) of Andrews

(1991a). Conditions (a) and (b) are pointwise WLLN�s and stochastic equicontinuity, respectively.

Condition (b) of (19.11) is established as follows. In the veri�cation of condition (b) we assume

k = 1 for notational simplicity and without loss of generality (wlog) (because the veri�cation can be

done separately for each element of bgn(�)�gn(�)): Consider any �a 2 B(�1�1; ") and �1 2 B(�a; "n):Element-by-element two-term Taylor expansions of gi(�1) about �a give

gi(�1) = gi(�a) +G1i(�a)(�1 � �a) +p1Xj=1

(�1j � �aj)@

@�1jG1i(e�1i)(�1 � �a); (19.12)

where �a = (�a1; :::; �ap1)0 and e�1i lies between �1 and �a: Element-by-element mean-value expan-

sions of G1i(�a) about �1�n give

G1i(�a) = G1i +

p1Xj=1

(�aj � �1�nj)@

@�1jG1i(�1i); (19.13)

where �1�n = (�1�n1; :::; �1�np1)0 and �1i lies between �a and �1�n and may di¤er across the columns

of (@=@�1j)G1i(�1i): Equation (19.13) uses the assumption of Theorem 8.1, which is imposed in this

lemma, that gi(�1) is twice continuously di¤erentiable in �1 on �1 and G1i(�a) := (@=@�01)gi(�a):

Substituting (19.13) into (19.12) and taking expectations gives

sup�a2B(�1�1;")

sup�12B(�a;"n)

jjgn(�1)� gn(�a)jj

� jjG1njj � jj�1 � �ajj+ EFn�1ijj�1 � �ajj2 + EFn�1ijj�a � �1�njj � jj�1 � �ajj

= o(1); (19.14)

where gn := EFngi; G1n := EFnG1i; the inequality uses jj(@=@�1j)G1i(e�1i)jj � p1=21 �1i (when k = 1)

63

and jj(@=@�1j)G1i(�1i)jj � p1=21 �1i for �1i de�ned in (8.2), and the equality uses jj�1��ajj � "n ! 0;

jj�a � �1�njj � 2" for n su¢ ciently large, and the conditions in FAR=AR that EFn�21i � M and

EFn jjvec(G1i)jj2+ := EFn jjvec(G1i(�1�n))jj2+ �M for (�1�n; Fn) 2 FAR=AR:Similarly, substituting (19.13) into (19.12) and taking averages over i = 1; :::; n gives

sup�a2B(�1�1;")

sup�12B(�a;"n)

jjbgn(�1)� bgn(�a)jj� jj bG1njj � jj�1 � �ajj+ p1n�1 nX

i=1

�1ijj�1 � �ajj2 + p1n�1nXi=1

�1ijj�a � �1�njj � jj�1 � �ajj

= op(1); (19.15)

where the equality uses jj�1 � �ajj � "n ! 0; jj�a � �1�njj � 2" for n large, and jj bG1njj = Op(1)

and n�1Pni=1 �1i = Op(1); which hold by Markov�s inequality using the same moment conditions

as used in (19.14).

Equations (19.14) and (19.15) combine to verify condition (b) in (19.11).

Condition (a) in (19.11) holds by the WLLN�s for independent L2-bounded random variables.

Again, for notational simplicity, we assume that k = 1: The L2-boundedness condition holds by

replacing �a by �1�n in (19.12), taking the inner product of the resulting expression with itself, and

then taking expectations. This yields: 8�1 2 B(�1�1; ");

EFngi(�1)0gi(�1) = EFn�

0n�n = O(1); where

�n := gi +G1i � (�1 � �1�n) +p1Xj=1

(�1j � �1�nj)@

@�1jG1i(e�1i)(�1 � �1�n) (19.16)

and the second equality holds using jj�1 � �1�njj � 2" for n large and using the moment conditionslisted after (19.14). This completes the veri�cation of Assumption (i).

Assumption (ii) requires gn = 0k 8n � 1; where gn = gn(�1�n): By the veri�cation of Assumption

(i), gn(�1�n) = EFnbgn(�1�n): Hence, gn = 0k 8n � 1 holds by the condition in FAR=AR that

EFngi(�1�n) = 0k 8(�1�n; Fn) 2 FAR=AR:

Assumption (vi) requires sup�12B(�1�n;") jj bG1n(�1)�G1n(�1)jj = op(1) for some nonrandom func-

tions fG1n(�) : n � 1g:We verify Assumption (vi) with G1n(�1) = EFn bG1n(�1) and we assume k = 1for notational simplicity. Assumption (vi) is a uniform WLLN. Its veri�cation is similar to, but sim-

pler than, the veri�cation of Assumption (i). To verify stochastic equicontinuity, one uses (19.13)

with �a and �1�n replaced by �1 and �a; respectively. Then, the analogues of (19.14) and (19.15)

64

are

sup�a2B(�1�1;")

sup�12B(�a;"n)

jjG1n(�1)�G1n(�a)jj � p1EFn�1ijj�1 � �ajj = o(1) and

sup�a2B(�1�1;")

sup�12B(�a;"n)

jj bG1n(�1)� bG1n(�a)jj � p1n�1

nXi=1

�1ijj�1 � �ajj = op(1); (19.17)

where the two equalities use jj�1� �ajj � "n ! 0; EFn�21i �M; and n�1

Pni=1 �1i = Op(1) as above.

This veri�es stochastic equicontinuity.

To verify the pointwise WLLN�s, i.e., bG1n(�1) � G1n(�1) = op(1) 8�1 2 B(�1�1; "); which is

analogous to condition (a) in (19.11), we use the WLLN�s for independent L2-bounded random

variables. The L2-boundedness condition holds by an analogous argument to that given in (19.16).

This completes the veri�cation of Assumption (vi).

Assumption (vii) requires sup�12B(�1�n;") jjG1n(�1)jj = O(1): By the veri�cation of Assumption

(vi) above, G1n(�1) = EFnG1i(�1): Hence, Assumption (vii) holds by the moment condition on

G1i(�1) in FAR=AR:Assumption (viii) requires sup�12B(�1�n;"n) jjG1n(�1)�G1njj = o(1) for all sequences of positive

constants "n ! 0: For notational simplicity and wlog, we suppose k = p1 = 1: For �1 2 B(�1�n; "n);element-by-element mean-value expansions of G1i(�1) about �1�n give

G1i(�1) = G1i +@

@�1G1i(�1i)(�1 � �1�n); (19.18)

where �1i lies between �1 and �1�n: Taking expectations in (19.18) gives

sup�12B(�1�n;"n)

jG1n(�1)�G1nj � EFn�1i sup�12B(�1�n;"n)

j�1 � �1�nj = o(1); (19.19)

where the inequality uses j(@=@�1)G1i(�1i)j � �1i and the equality uses "n ! 0 and EFn�1i � M

for �1 2 B(�1�n; "n) and (�1�n; Fn) 2 FAR=AR: This veri�es Assumption (viii).Assumption (x) requires EFn

b�1n = O(1); where b�1n := maxs;u�p1 sup�12B(�1�1;")

jj(@2=@�1s�1u)bgn(�1)jj: By the de�nition of �1i in (8.2), b�1n � n�1Pni=1 �1i for " > 0 su¢ ciently

small that B(�1�1; ") � �1: Hence, Assumption (x) holds by the moment condition EF �21i � M

8(�1; F ) 2 FAR=AR:Assumption (xi) requires sup�12B(�1�n;K=n1=2) jjbn(�1) � n(�1)jj !p 0 for some nonrandom

functions fn(�) : n � 1g: We verify Assumption (xi) with n(�1) = EFngi(�1)gi(�1)0 � EFngi(�1)

65

� EFngi(�1)0: We have

sup�12B(�1�n;K=n1=2)

jjbn(�1)� n(�1)jj (19.20)

� sup�12B(�1�n;K=n1=2)

jjbn(�1)� bnjj+ sup�12B(�1�n;K=n1=2)

jjn(�1)� njj+ jjbn � njjusing the triangle inequality. Next, we have

jjbn � njj = n�1

nXi=1

gig0i � bgnbg0n � (EFngig0i � EFnbgnEFnbg0n)

(19.21)

� n�1

nXi=1

gig0i � EFngig0i

+ (jjbgnjj+ jjgnjj) n�1

nXi=1

gi � EFngi

= op(1);

where the �rst equality holds by the triangle inequality and standard manipulations and the second

equality holds by the WLLN for independent L1+ =2-bounded random variables for > 0 as in

FAR=AR using the moment conditions in FAR=AR and jjbgnjj+jjgnjj = Op(1) using Markov�s inequality

and the moment conditions in FAR=AR:To verify Assumption (xi), it remains to show that the �rst and second summands on the

rhs of (19.20) are op(1): Assumption (xii) requires sup�12B(�1�n;K=n1=2) jjn(�1)� njj ! 0: Hence,

verifying Assumption (xii) shows that the second summand on the rhs of (19.20) is op(1):

Now we verify Assumption (xii). For notational simplicity, we assume k = p1 = 1 when

verifying Assumption (xii). (The results for k; p1 � 1 hold by analogous arguments.) For �1 2B(�1�n;K=n1=2); element-by-element two-term Taylor expansions of gi(�1) about �1�n give

gi(�1) = gi +G1i � (�1 � �1�n) + (�1 � �1�n)@

@�1G1i(e�1i)(�1 � �1�n); (19.22)

where e�1i lies between �1 and �1�n: Taking expectations in (19.22) givesgn(�1) = gn +G1n � (�1 � �1�n) + (�1 � �1�n)EFn

@

@�1G1i(e�1i)(�1 � �1�n) and

sup�12B(�1�n;K=n1=2)

jgn(�1)� gnj � jG1nj � j�1 � �1�nj+ EFn�1ij�1 � �1�nj2 = o(1);

(19.23)

where gn := EFngi; G1n := EFnG1i; and the inequality uses the conditions in FAR=AR that EFn�21i �M; EFn jjvec(G1i)jj2+ := EFn jjvec(G1i(�1�n))jj2+ �M for (�1�n; Fn) 2 FAR=AR:

66

Using (19.22) and taking expectations, we have: uniformly over �1 2 B(�1�n;K=n1=2);

EFngi(�1)gi(�1) = EFngigi + EFnG21i � (�1 � �1�n)2 + EFn

�@

@�1G1i(e�1i)�2 (�1 � �1�n)4

+2EFngiG1i � (�1 � �1�n) + 2EFngi@

@�1G1i(e�1i)(�1 � �1�n)2

+2EFnG1i@

@�1G1i(e�1i)(�1 � �1�n)3

= EFngigi + o(1); (19.24)

where the second equality holds using j�1 � �1�nj � K=n1=2; the moment conditions in FAR=ARreferred to above, the inequality j(@=@�1j)G1i(e�1i)j � �1i; and the Cauchy-Bunyakovsky-Schwarz

inequality. Equations (19.23) and (19.24) yield

sup�12B(�1�n;K=n1=2)

jn(�1)� nj

= sup�12B(�1�n;K=n1=2)

jEFngi(�1)gi(�1)� EFngigi � EFngi(�1)EFngi(�1) + EFngiEFngij

= o(1); (19.25)

where the second equality uses the triangle inequality and EFngi(�1) = O(1); which holds by the

moment conditions in FAR=AR: This completes the veri�cation of Assumption (xii).Next, we show that the �rst summand on the rhs of (19.20) is op(1); which is needed to complete

the veri�cation of Assumption (xi). For notational simplicity, we assume k = p1 = 1 in this

paragraph. Using (19.22), we have: uniformly over �1 2 B(�1�n;K=n1=2);

n�1nXi=1

gi(�1)gi(�1)

= n�1nXi=1

gigi + n�1

nXi=1

G21i � (�1 � �1�n)2 + n�1nXi=1

�@

@�1G1i(e�1i)�2 (�1 � �1�n)4

+2n�1nXi=1

giG1i � (�1 � �1�n) + 2n�1nXi=1

gi@

@�1G1i(e�1i)(�1 � �1�n)2

+2n�1nXi=1

G1i@

@�1G1i(e�1i)(�1 � �1�n)3

= n�1nXi=1

gigi + op(1); (19.26)

where the second equality holds using jj�1 � �1�njj � K=n1=2; the inequality jj(@=@�1j)G1i(e�1i)jj ��1i; the WLLN for independent L1+ =2-bounded random variables for > 0 as in FAR=AR; the

67

moment conditions in FAR=AR; and the Cauchy-Bunyakovsky-Schwarz inequality. By similar, butsimpler calculations, uniformly over �1 2 B(�1�n;K=n1=2);

n�1nXi=1

gi(�1) = n�1nXi=1

gi + op(1): (19.27)

Equations (19.26) and (19.27) imply that the �rst summand on the rhs of (19.20) is op(1): This

completes the veri�cation of Assumption (xi).

Assumption (xiii) holds by the condition in FAR=AR that �min(n) = �min(EFngi(�1�n)gi(�1�n)0)

� � 8(�1�n; Fn) 2 FAR=AR:Assumption (xv) requires sup�12B(�1�n;K=n1=2) jjb�1n(�1)��1n(�1)jj = op(1) for some nonrandom

functions f�1n(�) : n � 1g; where b�1n(�) is de�ned in (14.2). We verify Assumption (xv) with�1n(�1) de�ned as in (14.2), i.e., �1n(�1) := EFnvec(G1i(�1) � EFnG1i(�1))gi(�1)

0 using the iden-

tical distribution assumption in FAR=AR: The veri�cation is quite similar to that of Assumption(xi) with vec(G1i(�1))gi(�1)0 and vec(G1i(�1)) in place of gi(�1)gi(�1)0 and gi(�1); respectively. In

consequence, we do not provide all of the details. Analogues of (19.20), (19.21), (19.26), and (19.27)

hold by analogous arguments, so it su¢ ces to show that an analogue of the second summand on the

rhs of (19.20) holds. Assumption (xvi) requires sup�12B(�1�n;K=n1=2) jj�1n(�1) � �1njj ! 0: Hence,

verifying Assumption (xvi) shows that the analogue of the second summand on the rhs of (19.20)

is op(1): We verify Assumption (xvi) below. This completes the veri�cation of Assumption (xv).

Assumption (xvi) requires sup�12B(�1�n;K=n1=2) jj�1n(�1) � �1njj ! 0; where �1n(�1) is de�ned

in the previous paragraph. For notational simplicity, we assume k = p1 = 1 in the veri�cation of

Assumption (xvi). Using (19.18) with "n = K=n1=2 and (19.22) and taking expectations, we obtain

EFnG1i(�1)gi(�1)� EFnG1igi

= EFnG21i(�1 � �1�n) + EFnG1i

@

@�1G1i(e�1i)(�1 � �1�n)2 + EFn @

@�1G1i(�1i)gi(�1 � �1�n)

+EFn@

@�1G1i(�1i)G1i(�1 � �1�n)2 + EFn

@

@�1G1i(�1i)

@

@�1G1i(e�1i)(�1 � �1�n)3

= o(1); (19.28)

where the second equality holds uniformly over �1 2 B(�1�n;K=n1=2) using sup�12B(�1�n;K=n1=2)

j�1 � �1�nj = o(1); j(@=@�1)G1i(�1i)j � �1i; j(@=@�1)G1i(e�1i)j � �1i; EFng2i + EFnG

21i + EFn�

21i

= O(1) by the moment conditions in FAR=AR; and the Cauchy-Bunyakovsky-Schwarz inequality.

68

Next, we have

sup�12B(�1�n;K=n1=2)

j�1n(�1)� �1nj

= sup�12B(�1�n;K=n1=2)

jEFnG1i(�1)gi(�1)� EFnG1igi � EFnG1i(�1)EFngi(�1) + EFnG1iEFngij

= o(1); (19.29)

where the second equality holds by (19.19) with "n = K=n1=2; (19.23), (19.28), and the moment

conditions in FAR=AR; which imply that sup�12B(�1�n;K=n1=2)(jEFnvec(G1i(�1))j + jEFngi(�1)j) =O(1) (when k = p1 = 1): This completes the veri�cation of Assumption (xvi).

Assumption (xxiii) requires sup�12B(�1�n;K=n1=2) jb�21sn(�1)��21sn(�1)j !p 0 for some nonrandom

functions f�21sn(�) : n � 1g 8s = 1; :::; p1; where b�21sn(�1) is de�ned in (7.4). We verify Assumption(xxiii) with �21sn(�1) := EFn jjG1si(�1)jj2 � (EFn jjG1si(�1)jj)2: Provided Assumption (xxiv) holds,Assumption (xxiii) holds by the same argument as for Assumption (xv) with jjG1si(�1)jj2 andjjG1si(�1)jj in place of vec(G1i(�1))gi(�1)0 and vec(G1i(�1)); respectively (which in turn relies onthe veri�cation of Assumption (xi)).

Now we verify Assumption (xxiv), which requires that sup�12B(�1�n;K=n1=2) j�21sn(�1)��21snj ! 0

8s = 1; :::; p1: The latter is implied by sup�12B(�1�n;K=n1=2) jEFn jjG1si(�1)jj2 � EFn jjG1sijj2j ! 0

and sup�12B(�1�n;K=n1=2) jEFn jjG1si(�1)jj � (EFn jjG1sijj)j ! 0: For notational simplicity, we suppose

k = p1 = 1: Using (19.18) with "n = K=n1=2 and taking expectations, we have

sup�1

jEFn jG1i(�1)j2 � EFn jG1ij2j

= sup�1

��EFn��G1i + @

@�1G1i(�1i)(�1 � �1�n)

��2 � EFn jG1ij2��

� sup�1

EFn

�� @@�1G1i(�1i)(�1 � �1�n)��2

+2 sup�1

EFn

�� @@�1G1i(�1i)(�1 � �1�n)G1i��

� EFn�21iK

2=n+ 2EFn(�1ijG1ij)K=n1=2

= op(1); (19.30)

where sup�1 denotes sup�12B(�1�n;K=n1=2); the �rst equality uses (19.18), the �rst inequality holds

by the triangle inequality, the second inequality uses jj(@=@�1)G1i(�1i)jj � �1i; and the last equality

uses EFn�21i � M and EF jG1ij2+ � M (when k = p1 = 1) for (�1�n; Fn) 2 FAR=AR and the

Cauchy-Bunyakovsky-Schwarz inequality. Establishing the analogous result to that in (19.30) with

69

jG1si(�1)j2 and jG1sij2 replaced by jG1si(�1)j and jG1sij is quite similar.Assumption (xxv) requires that lim infn!1 �21sn > 0 8s = 1; :::; p1; where �21sn := EFn jjG1sijj2�

(EFn jjG1sijj)2: This holds by the condition in FAR=AR that V arF (jjG1sijj) � � > 0 8s = 1; :::; p1:By (19.6), f�n;h : n � 1g denotes a sequence f�n 2 �AR=AR : n � 1g for which hn(�n)! h 2 H:

By (19.7), we have

n1=2�1;�1�n;Fn : = n1=2(�11Fn(�1�n); :::; �1p1Fn(�1�n))0 ! h1;

�2;�1�n;Fn : = B1Fn(�1�n)! h2 =: B11;

�3;�1�n;Fn : = C1Fn(�1�n)! h3 =: C11;

�4;�1�n;Fn : = EFnG1i(�1�n)! h4 =: G11;

�5;�1�n;Fn : = EFn

0@ gi(�1�n)


1A0@ gi(�1�n)


1A0

! h5 =: V11 :=

0@ 1 �011

�11 h5;G1G1

1A ; (19.31)

�6;�1�n;Fn : = �1�n ! h6 =: �1�1; and

�7;�1�n;Fn : = (�1p1Fn(�1�n); ��1p1Fn(�1�n))

0 := (�1n; ��1n)

0 ! h7 =: (lim �1n; lim ��1n)

0;

where 1 2 Rk�k; �11 2 R(p1k)�k; h5;G1G1 2 R(p1k)�(p1k); and the second equality in the secondlast line holds by the notation introduced in (10.1) and (10.2). The convergence results in (19.31)

verify Assumptions (iii), (ix), (xiv), (xvii), (xviii), (xix), and (xx). Note that h6 =: �1�1 lies in �1

as required by Assumption (iii) because �1� is bounded and B(�1�; ") � �1 for some " > 0 by theassumptions of the present lemma. In addition, the last convergence result in (19.31) guarantees

that lim �1n and lim ��1n; which appear in Theorem 18.1(b), exist.

Assumption (v) holds by the univariate CLT for triangular arrays of rowwise independent L2+ -

bounded random variables (where L2+ -boundedness holds by the moment conditions in FAR=AR);the convergence condition V arFn(n

1=2b0(bg0n; vec( bG1n � EFn bG1n))0) ! b0V11b 8b 2 R(p1+1)k with

jjbjj > 0 (which holds by the convergence results for �5;�1�n;Fn in (19.31)), and the Cramér-Wold

device.

Assumptions (xxi) and (xxii) hold by the assumptions of the lemma on fcn : n � 1g; which arethe same as in Theorem 8.1, that cn ! 0 and ncn !1:

This completes the veri�cation of Assumption HL1AR=AR and of the existence of lim �1n and

lim ��1n: �

The proof of Lemma 19.3 uses the following lemma when verifying Assumption HL2AR=AR(ii).

70

Lemma 19.4 Let B be a pseudometric space with pseudometric �: Let f�n(�) : n � 1g be a se-quence of real-valued stochastic processes on B: Suppose (i) B is totally bounded, (ii) f�n(�) :n � 1g is stochastically equicontinuous under �; i.e., 8"; � > 0 9� > 0 such that lim supn!1

P (sup�1;�22B:�(�1;�2)<� j�n(�1) � �n(�2)j > �) < "; and (iii) �n(�) = Op(1) 8� 2 B. Then,

sup�2B j�n(�)j = Op(1):

Comments: (i). The result of Lemma 19.4 also holds if the stochastic equicontinuity condition

is weaken by replacing �8� > 0�to �for some � > 0:�(ii). The result of Lemma 19.4 could be obtained by establishing the weak convergence of

f�n(�) : n � 1g to some limit process and applying the continuous mapping theorem. But, condi-tion (iii) of Lemma 19.4 is noticeably weaker than the weak convergence of all �nite-dimensional

distributions of �n(�); which would be needed to establish weak convergence. Condition (iii) can beveri�ed straightforwardly using Markov�s inequality when �n(�) is a sample average for � 2 B.

Proof of Lemma 19.4. Let B(�; �) denote a closed ball in B centered at � with radius � underthe pseudometric �: Because B is totally bounded, there exists a �nite number of balls in B; say J�balls, that cover B: Let the centers of these balls be f�j� 2 B : j = 1; :::; J�g: We have

sup�2B

j�n(�)j = maxj�J�

sup�2B(�j�;�)

j�n(�)j � maxj�J�

j�n(�j�)j+ ��n �J�Xj=1

j�n(�j�)j+ ��n; where

��n := maxj�J�

sup�2B(�j�;�)

j�n(�)� �n(�j�)j (19.32)

and the �rst inequality holds by the triangle inequality.

Given any " > 0 and some � > 0; e.g., � = 1 su¢ ces, take � > 0 such that lim supn!1 P (��n >

�) < "=2: Such a value � exists by the stochastic equicontinuity condition (ii). For 0 < K <1; wehave

lim supn!1

P

sup�2B

j�n(�)j > K

!� lim sup

n!1P

0@ J�Xj=1

j�n(�j�)j+ ��n > K

1A� lim sup

n!1

0@P0@ J�Xj=1

j�n(�j�)j+ ��n > K; ��n � �

1A+ P (��n > �)

1A�

J�Xj=1

lim supn!1

P

�j�n(�j�)j >

K � �J�

�+ "=2

< "; (19.33)

where the �rst inequality holds by (19.32), the second and third inequalities hold by standard

71

manipulations, and the last inequality holds for K su¢ ciently large using the assumption that

�n(�j�) = Op(1) 8j � J� by condition (iii). �

Proof of Lemma 19.3. We have: Assumption SI(ii) ) Assumption HL2AR=AR(i), SI(i) )HL2AR=AR(iv), SI(iii) ) HL2AR=AR(vi), and SI(vi) ) HL2AR=AR(vii).

Next, we verify Assumption HL2AR=AR(iii). It su¢ ces to show Assumption HL2AR=AR(iii)

holds for sequences Kn ! 1 such that Kn=n1=2 ! 0 because inf�1 =2B(�1�n;K=n1=2) jjn

1=2gn(�1)jj isnonincreasing in K: Hence, we assume that Kn=n

1=2 ! 0:

For n � 1; let �y1n 2 B(�1�n;Kn=n1=2) satisfy

n1=2jjgn(�y1n)jj = inf�1 =2B(�1�n;Kn=n1=2)

n1=2jjgn(�1)jj+ "n; (19.34)

where "n > 0 and "n ! 0: Such values f�y1n : n � 1g always exist. Assumption HL2AR=AR(iii) holdsi¤ n1=2jjgn(�y1n)jj ! 1:

De�ne

dyn := jj�y1n � �1�njj and syn := n1=2jjgn(�y1n)jj: (19.35)

We want to show that syn ! 1: This holds if every subsequence fmn : n � 1g of fng has asubsubsequence fvn : n � 1g such that syvn ! 1: Given an arbitrary subsubsequence fvng eitherlim infn!1 dyvn > 0 or lim infn!1 dyvn = 0:

First, suppose lim infn!1 dyvn > 0: Let " 2 (0; lim infn!1 dyvn): We have

syvn = v1=2n jjgvn(�y1vn)jj � inf

�1 =2B(�1�vn ;")v1=2n jjgvn(�

y1vn)jj ! 1; (19.36)

as desired, where the inequality holds using the de�nitions of dyn and " and the convergence holds

by Assumption SI(i) and v1=2n !1:Second, suppose lim infn!1 dyvn = 0: Then, there exists a subsequence frng of fvng for which

limn!1 dyrn = 0 and, in this case, we that show syrn !1; which completes the proof of Assumption

HL2AR=AR(iii). For notational simplicity, we replace rn by n and assume lim dyn = 0: By de�nition,

�y1n =2 B(�1�n;Kn=n1=2): Hence, Kn=n

1=2 � jj�y1n��1�njj = dyn ! 0: Element-by-element mean-value

expansions about �1�n give

gn(�y1n) = G1n(e�1n)(�y1n � �1�n) = (G1n + o(1))(�y1n � �1�n); (19.37)

where e�1n lies between �y1n and �1�n and may di¤er across the rows ofG1n(e�1n); the �rst equality usesthe assumption that gi(�) satis�es the di¤erentiability condition in Theorem 8.1 and the condition in

72

FAR=AR that EFngi(�1�n) = 0k; and the second equality uses jje�1n��1�njj � jj�y1n��1�njj = dyn ! 0

and sup�12B(�1�n;"n) jjG1n(�1) = G1njj = op(1) for all sequences of positive constants "n ! 0; which

is Assumption HL1AR=AR(viii), and is veri�ed in (19.18) and (19.19).

We have

njjgn(�y1n)jj2 = njj(G1n + o(1))(�y1n � �1�n)jj2

� n inf�2Rp1 :jj�jj=1

jj(G1n + o(1))�jj2 � jj�y1n � �1�njj2

� n

�inf

�2Rp1 :jj�jj=1jjG1n�jj2 + o(1)

�(Kn=n

1=2)2

��

inf�2Rp1 :jj�jj=1

jj�1=2n G1n�jj2=�2max(�1=2n ) + o(1)

�K2n

=��21n�min(n) + o(1)

�K2n

! 1; (19.38)

where the �rst equality uses (19.37), the second inequality holds because G1n = O(1) (by the

moment condition EF jjvec(G1i(�1))jj2+ � M in FAR=AR) and because �y1n =2 B(�1�n;Kn=n1=2);

the last equality uses the de�nition of �1n; and the convergence to 1 uses the assumption that

Kn ! 1; lim infn!1 �min(n) > 0 (which holds by the condition �min(F (�1)) � � in FAR=AR),and the fact that lim infn!1 �1n > 0; which we now show. We obtain lim infn!1 �1n > 0 using

Assumption SI(ii) (i.e., lim infn!1 ��1n > KU ); the de�nition of ��1n in (10.3), the de�nition of �1n

in (14.3), and the conditions V arF (jjG1si(�1)jj) � � for all s = 1; :::; p1 in FAR=AR. This completesthe veri�cation of Assumption HL2AR=AR(iii).

Now, we show that Assumptions SI(iv) and (v) imply Assumption HL2AR=AR(ii). Because

Assumption HL2AR=AR(ii) can be veri�ed element-by-element, we assume without loss of generality

that k = 1: We use Lemma 19.4 with � = �1; B = �1; and

�n(�1) := n1=2(bgn(�1)� EFnbgn(�1)): (19.39)

Condition (iii) of Lemma 19.4, i.e., �n(�1) = Op(1) 8�1 2 �1; holds because for any " > 0;

PFn(�n(�1) > K) � EFn�2n(�1)=K

2 = EFn(gi(�1)� EFn

gi(�1))2=K2 < " (19.40)

for all n � 1; where the �rst inequality holds by Markov�s inequality, the equality holds because

fgi(�1) : i � ng are i.i.d. under Fn for each n � 1; and the last inequality holds for K su¢ ciently

large using Assumption SI(iii).

73

We verify conditions (i) and (ii) of Lemma 19.4, i.e., �1 is totally bounded under � and f�n(�) :n � 1g is stochastically equicontinuous under �; using Theorem 4 in Andrews (1994, p. 2277).

Here, � is de�ned by �(�a; �b) := lim supn!1(EFn(gi(�a) � gi(�b))2)1=2 for �a; �b 2 �1: Theorem

4 requires Assumptions B-D of Andrews (1994) to hold. Assumption C on p. 2269 of Andrews

(1994) holds by the independence assumption in FAR=AR: Assumption B on p. 2268 (an envelopecondition) holds by Assumption SI(iii), which states that lim supn!1EFn sup�12�1 jjgi(�1)jjr <1for some r > 2:

We verify Assumption D (Ossiander�s Lp entropy condition) in Andrews (1994) using Theorem

5 in Andrews (1994, p. 2281) with p = 2: To apply Theorem 5 it su¢ ces that the functions

fgi(�1) : �1 2 �1g are a type II class of functions (i.e., Lipschitz functions, see p. 2270 of Andrews(1994)) with �1 bounded and lim supn!1EFnB

ri < 1 for some r > 2; where Bi is a random

Lipschitz �constant.�This holds because, by the mean-value theorem, using the di¤erentiability of

gi(�1); the convexity of �1 imposed in Assumption SI(v), and the moment condition in Assumption

SI(iv), we have

jjgi(�a)� gi(�b))jj � B1ijj�a � �bjj 8�a; �b 2 �1 for B1i := sup�12�1

jj @@�1

gi(�1)jj; (19.41)

where lim supn!1EFnBr1i < 1: And, �1 is bounded by Assumption SI(v). This completes the

veri�cation of Assumption HL2AR=AR(ii).

Next, we verify Assumption HL2AR=AR(v), i.e., sup�12�1 jjbn(�1) � n(�1)jj = op(1); wherebn(�1) is de�ned in (3.6), and n(�1) = EFngi(�1)gi(�1)0 � EFngi(�1)EFngi(�1)

0: We do so by

obtaining uniform WLLN�s over �1 for averages over i � n of gi(�1) and gi(�1)gi(�1)0: For the

average over i � n of gi(�1); a uniform WLLN�s holds (i.e., sup�12�1 jjbgn(�1) � EFnbgn(�1)jj !p 0)

by Assumption HL2AR=AR(ii), which is veri�ed above.

To obtain a uniform WLLN�s for the average over i � n of gi(�1)gi(�1)0; we use the following

generic uniform WLLN�s. Let fsi(�1) : i � n; n � 1g be some vector-valued random functions on

�1; where si(�1) := s(Wi; �1): Let bsn(�1) := n�1Pni=1 si(�1): Su¢ cient conditions for a uniform

WLLN�s for these random functions under fFn : n � 1g (i.e., sup�12�1 jjbsn(�1)�EFnbsn(�1)jj !p 0)

are

(a) bsn(�1)� EFnbsn(�1)!p 0 8�1 2 �1;

(b) jjsi(�a)� si(�b)jj � Bsijj�a � �bjj; 8�a; �b 2 �1; where lim supn!1

n�1nXi=1

EFnBsi <1; and

(c) �1 is bounded, (19.42)

74

e.g., see Theorem 1(a) and Lemma 2(a) of Andrews (1991a).

Now, consider si(�1) = gi(�1)gi(�1)0: A pointwise WLLN�s for n�1

Pni=1 gi(�1)gi(�1)

0 holds for

each �xed �1 2 �1 under the i.i.d. condition in FAR=AR for each �xed F and Assumption SI(iii).

Hence, condition (a) of (19.42) holds. Condition (c) of (19.42) holds immediately by Assumption

SI(v).

Using (19.41), we obtain

jjgi(�a)gi(�a)0 � gi(�b)gi(�b)0jj

� 2 sup�12�1

jjgi(�1)jj � jjgi(�a)� gi(�b)jj

� 2 sup�12�1

jjgi(�1)jjBijj�a � �bjj; 8�a; �b 2 �1; (19.43)

where for matrix arguments jj � jj denotes the Frobenious norm and the �rst inequality uses

the triangle inequality. Combining (19.43) with lim supn!1EFn sup�12�1 jjgi(�1)jj2 < 1 and

lim supn!1EFnB2i < 1; which hold by Assumptions SI(iii) and (iv), and using the Cauchy-

Bunyakovsky-Schwarz inequality veri�es condition (b) of (19.42). This completes the veri�cation

of a uniform WLLN�s for the average over i � n of gi(�1)gi(�1)0; which completes the veri�cation

of Assumption HL2AR=AR(v). �


The proof of Theorem 8.2 is similar to that of Theorem 8.1 given in Section 19. But, it uses an

adjusted de�nition of ��1;F in (19.4) and a parameter space �AR=LM;QLR1 (de�ned below) in place

of �AR=AR:

As above, F (�) := V arF (gi(�)): We write a SVD of �1=2F (�1)EFG2i(�1) as

�1=2F (�1)EFG2i(�1) = C2F (�1)�2F (�1)B2F (�1)

0; (20.1)




We write a SVD of �1=2F (�1)EFG2i(�1)�2F (�1) as

�1=2F (�1)EFG2i(�1)�2F (�1) = C�2F (�1)�

�2F (�1)B

�2F (�1)

0; (20.2)

where C�2F (�1) 2 Rk�k and B�2F (�1) 2 Rp2�p2 are orthogonal matrices and ��2F (�1) 2 Rk�p2 has

75

the singular values ��21F (�1); :::; ��2p2F

(�1) of �1=2F (�1)EFG2i(�1)�2F (�1) in nonincreasing order on

its diagonal and zeros elsewhere.

Let �pF (�1) denote the smallest singular value of �1=2F (�1)EFGi(�1) and let ��pF (�1) denote

the smallest singular value of �1=2F (�1)EFGi(�1)�F (�1); where �F (�1) is de�ned in (8.4).

The adjusted de�nition of ��1;F is as follows. First, (�21F (�1); :::; �2p2F (�1))0 2 Rp2 and

(��21F (�1); :::; ��2p2F

(�1))0 2 Rp2 are added onto �1;�1;F so that �1;�1;F 2 Rp1+2p2 : Second, Gi(�1)

appears in place of G1i(�1) in �4;�1;F and �5;�1;F : Third, the following six elements are added onto

��1;F : �9;�1;F := B2F (�1) 2 Rp2�p2 ; �10;�1;F := C2F (�1) 2 Rk�k; �11;�1;F := (�211F (�1); :::; �21p1F (�1);�221F (�1); :::; �

22p2F

(�1))0 2 Rp; where �2jsF (�1) := V arF (jjGjsi(�1)jj) 8s = 1; :::; pj ; 8j = 1; 2;

�12;�1;F := B�2F (�1) 2 Rp2�p2 ; �13;�1;F := C�2F (�1) 2 Rk�k; and �14;�1;F := (�pF (�1); ��pF (�1))

0:

Fourth, h5 in (19.31) is de�ned by

h5 := V1 :=

0@ 1 �01

�1 h5;GG

1A ; (20.3)

where 1 2 Rk�k; �1 2 R(pk)�k; h5;GG 2 R(pk)�(pk): Fifth, hn;j(�) in (19.5) is de�ned to equal �jfor j = 9; :::; 14:

The parameter space �AR=LM;QLR1 (for �) that we use here is de�ned analogously to �AR=AR

in (19.5), but is based on the adjusted de�nition of � and the parameter space FAR=LM;QLR1;rather than FAR=AR:We use the same function hn(�) here as de�ned as in (19.5), but based on theadjusted de�nition of �:

The proof of Theorem 8.2 uses the following lemma. This lemma veri�es Assumptions

HL1AR=LM and HL1AR=QLR1; which are employed in Lemmas 18.3 and 18.4, for all subsequences

f�wn;h : n � 1g: The subsequences f�wn;h : n � 1g considered in this lemma are based on theadjusted de�nitions of ��1;F and hn;j(�) given immediately above.

Lemma 20.1 Suppose bgn(�1) are the moment functions de�ned in (3.3), gi(�) satis�es the di¤er-entiability conditions in Theorem 8.2, cn ! 0; ncn !1; a > 0; and p2 � 1: Let the null parameterspace be FAR=LM;QLR1: Then, for all subsequences f�wn;h : n � 1g; Assumptions HL1AR=LM and

HL1AR=QLR1 hold and lim �mn and lim ��mn

exist.

Comment: When LM2n(�) is the pure C(�)-LM statistic (i.e., WIn(�) := 0); the parameter space

FAR=LM;QLR1 is restricted as in (8.12), and the de�nition of ��1;F is augmented by �15;�1;F :=

(r1n; r2n)0; Lemma 20.1 holds and, in addition, conditions (vi) and (vii) in Comment (v) to Lemma

16.1 hold. The same is true when QLR12n(�) is the pure C(�)-QLR1 statistic and WIn(�) := 0

in the QLR1 critical value function. (These results are proved following the proof of Lemma 20.1

76

below.) These results imply that the results of Lemma 18.3(a) and (b) hold (using the Comment

to Lemma 18.3).

Proof of Theorem 8.2. The proof of Theorem 8.2 is the same as that of Theorem 8.1 given in

Section 19, but (i) with the de�nitions of ��1;F and hn;j(�) adjusted as in the paragraph containing

(20.3), (ii) using Lemmas 18.3 and Lemma 18.4 in place of Theorem 18.1 and Lemma 18.2, (iii)

with FAR=LM;QLR1 and F�;AR=LM;QLR1 in place of FAR=AR and F�;AR=AR; and (iv) using Lemma20.1 in addition to Lemma 19.2. Lemma 20.1 shows that lim �mn and lim �

�mn

exist, which is used

when Lemma 18.3 is employed and in showing that a subsequence Sm satis�es Assumption B when

lim ��mn< KL and Assumption C when lim infn!1 ��mn

> K�U ; as in the proof of Theorem 8.1

(with lim infn!1 ��mn> K�

U in place of lim infn!1 �1mn > 0):

The proof of part (a) uses Lemma 18.3, which employs Assumptions HL1AR=AR and

HL1AR=LM ; and for the second-step C(�)-QLR1 test, Assumption HL1QLR1 as well. These assump-

tions are veri�ed for the parameter space FAR=LM;QLR1 by Lemmas 19.2 (using FAR=LM;QLR1 �FAR=AR) and 20.1.

The proof of part (b) uses Lemma 18.4, which employs Assumptions HL1AR=AR; HL1AR=LM ;

HL2AR=LM;QLR1; and W. Assumptions HL1AR=AR and HL1AR=LM are veri�ed in the proof of

part (a) of the theorem. Assumption HL2AR=LM;QLR1 is the same as Assumption HL2AR=AR ex-

cept for part (i). Hence, using FAR=LM;QLR1 � FAR=AR; Lemma 19.3 veri�es all of AssumptionHL2AR=LM;QLR1 except part (i). Part (i) of Assumption HL2AR=LM;QLR1 is implied by Assumption

SI2, which is imposed in part (b) of the theorem. Assumption W is imposed in the theorem, so it

holds by assumption.

Parts (c)�(f) of the theorem hold by the same arguments as given in the proof of these parts of

Theorem 8.1. �

Proof of Lemma 20.1. For notational simplicity, we consider a sequence f�n;h : n � 1g; ratherthan a subsequence f�wn;h : n � 1g:

Assumption HL1AR=LM (i) holds by the di¤erentiability conditions that are imposed in Theorem

8.2, but not in Theorem 8.1.

Assumption HL1AR=LM (ii) holds by the CLT using the moment conditions in FAR=LM;QLR1(including the condition EF jjvec(G2i(�1))0jj2+ � M; which does not appear in FAR=AR) by thesame argument as in the veri�cation of Assumption HL1AR=AR(v) given in the paragraph following

(19.31) in the proof of Lemma 19.2. Note that the variance matrix V11 in Assumption HL1AR=AR(v)

is the upper left (p1+1)k� (p1+1)k sub-matrix of V1; where V1 is the limit of �5;�1�n;Fn in (20.3).

The veri�cation of Assumptions HL1AR=LM (iii)�(v) is the same as for Assumptions HL1AR=AR

77

(vi), (viii), and (ix) in the proof of Lemma 19.2 (see (19.17), the two paragraphs that follow (19.17),

and (19.31)) with subscripts 2 in place of 1 and using EF �22i �M in place of EF �21i �M:

Assumption HL1AR=LM (vi) holds by the condition EF �12i �M in FAR=LM;QLR1 and Markov�sinequality.

The veri�cation of Assumptions HL1AR=LM (vii)�(ix) is the same as for Assumptions

HL1AR=AR(xv)�(xvii) in the proof of Lemma 19.2 (see (19.28), the paragraph preceding (19.28),

(19.29), and (19.31)) with subscripts 2 in place of 1; using the condition EF �22i � M in place of

E�21i �M; and using Gi(�1) in place of G1i(�1) in the de�nitions of �4;�1;F and �5;�1;F as speci�ed

in the paragraph that contains (20.3).

The veri�cation of Assumptions HL1AR=LM (x)�(xii) is the same as for Assumptions

HL1AR=AR(xviii)�(xx) in the proof of Lemma 19.2 (see (19.31)) using the adjusted de�nition of

�1;�1;F to include (�21F (�1); :::; �2p2F (�1))0 and the addition of �9;�1;F and �10;�1;F to ��1;F ; as

speci�ed in the paragraph that contains (20.3).

The veri�cation of Assumptions HL1AR=LM (xiii) and (xiv) is the same as for Assumptions

HL1AR=AR(xxiii) and (xxiv) in the proof of Lemma 19.2 with subscripts 2 in place of 1 (see (19.30))

using the EF jjvec(G2i(�1))jj2+ �M and EF �22i �M conditions in the de�nition of FAR=LM;QLR1:By (19.6), f�n;h : n � 1g denotes a sequence f�n 2 �AR=LM;QLR1 : n � 1g for which hn(�n)!

h 2 H: For the sequence f�n;h : n � 1g; the convergence result of Assumption HL1AR=LM (xv) (i.e.,�2jsn ! �2js1 8s = 1; :::; pj ; 8j = 1; 2) holds by the addition of �11;�1;F to ��1;F ; as speci�ed in theparagraph that contains (20.3) because �2jsn := V arFn(jjGjsijj) 8s = 1; :::; pj ; 8j = 1; 2; see (17.1).The result of Assumption HL1AR=LM (xv) that �2js1 2 (0;1) 8s = 1; :::; pj ; 8j = 1; 2 holds by

the V arF (jjGjsijj) � � and EF jjvec(Gji(�1))jj2+ � M conditions for j = 1; 2 in the de�nitions of

FAR=AR and FAR=LM;QLR1: This completes the veri�cation of Assumption HL1AR=LM :By (19.5) and the adjusted de�nition of �1;�1�n;Fn ; for the sequence f�n;h : n � 1g; n1=2��2sn

converges to some value in [0;1] 8s � p2: Hence, Assumption HL1AR=QLR1(i) holds. The conver-

gence results of Assumptions HL1AR=QLR1(ii) and (iii) (i.e., C�2n ! C�21 and B�2n ! B�21) hold

by the addition of �12;�1;F and �13;�1;F to ��1;F ; as speci�ed in the paragraph that contains (20.3).

This completes the veri�cation of Assumption HL1AR=QLR1:

The limits lim �n and lim ��n exist by the addition of �14;�1;F to ��1;F : �

Now we prove the Comment to Lemma 20.1, which requires that we verify conditions (vi) and

(vii) stated in Comment (v) to Lemma 16.1. Condition (vi) (i.e., rjn = rj1 for n su¢ ciently large)

holds by the addition of �15;�1;F to ��1;F ; which implies that lim rjn exists, and the fact that rjn can

only take on a �nite number of values. Condition (vii) holds by (8.12) (i.e., �min(C�F (�1)0C�F (�1)) ��):

78

The Comments to Lemmas 18.3 and 20.1 imply that Lemma 18.3 holds for the pure C(�)-LM

and pure C(�)-QLR1 tests provided FAR=LM;QLR1 is restricted as in (8.12) and �15;�1;F is addedto ��1;F :

Lastly, the proof of Comment (iii) to Theorem 8.2 is the same as the proof of Theorem 8.2 given

above using the results of Lemma 18.3, which hold when FAR=LM;QLR1 is restricted as in (8.12)and the test is the pure C(�)-LM or pure C(�)-QLR1 test (for which WIn(�) := 0):


The proof of Theorem 11.1 is analogous to that of Theorems 8.1 and 8.2. In the time series

case, we de�ne �; f�n;h : n � 1g; and � as in (19.5) and (19.6) and the discussion around (19.8)for the AR/AR test and CS, respectively, and as in the discussion around (20.3) for the AR/LM

and AR/QLR1 tests and CS�s. But, we de�ne �5;�1;F and �5;�;F di¤erently from the i.i.d. case. We

de�ne

�5;�1;F := VF (�1) and �5;�;F := VF (�); (21.1)

where VF (�) is de�ned in (11.3), rather than as in (19.4). In consequence, �5;�1n;Fn ! h5 implies

that VFn(�1n) ! h5 and the condition VFn(�1�n) ! V1 in Assumption V holds with �1�n = �1n

and V1 = h5 (and analogously for CS�s and Assumption V-CS). We let �TS;AR=AR denote the time

series version of �AR=AR: It is de�ned as in (19.5), but with FTS;AR=AR in place of FAR=AR andwith the changes described above.

The proof of Theorem 11.1 uses the CLT given in the following lemma. This lemma employs

Corollary 1 in de Jong (1997) and is analogous to Lemma 20.1 in Section 20 in the SM to AG1.

Lemma 21.1 Let fi(�) := (gi(�)0; vec(Gi(�))0)0: We have: for tests, w

�1=2n

Pwni=1(fi(�1�n)�

EFnfi(�1�n))!d N(0(p+1)k; h5) under all subsequences fwng and all sequences f�wn;h 2 �TS;AR=AR :

n � 1g; and for CS�s the same result holds with ��n in place of �1�n:

We use the following stochastic equicontinuity result, which is a special case of Hansen (1996)

Theorem 3, in the proof of Theorem 11.1. The strong mixing numbers of a triangular array of

random vectors are de�ned in the usual way, e.g., see Hansen (1996).

Lemma 21.2 Suppose (i) fWni : i � n; n � 1g are row-wise identically distributed, strong mix-ing random vectors taking values in a set W, (ii) B is a bounded subset of Rd� ; (iii) FLip is aset of real-valued functions s(w; �) on W �B that satisfy js(w; �1) � s(w; �2)j � B(w)jj�1 � �2jjfor some Lipschitz function B(w) on W, (iv) lim supn!1Ejs(Wni; �)jr < 1 8� 2 B for some

79

r > 2; (v) lim supn!1EjB(Wni)jr < 1; and (vi) the strong mixing numbers of fWni : i �n; n � 1g satisfy

P1m=1 �

1=q�1=rm < 1 for some q > d� and 2 � q < r: Then, 8"; � > 0

9� > 0 such that lim supn!1 P (sup�1;�22B:jj�1��2jj<� j�n(�1) � �n(�2)j > �) < "; where �n(�) :=

n�1=2Pni=1(s(Wni; �)� Es(Wni; �)):

Comment: The same constant r appears in conditions (iv)�(vi).

Proof of Theorem 11.1. The proof is the same as the proofs of Theorems 8.1 and 8.2 and Lemmas

19.2, 19.3, and 20.1, given in Sections 19 and 20, with some modi�cations. The modi�cations a¤ect

the proofs of Lemmas 19.2, 19.3, and 20.1. No modi�cations are needed elsewhere. We describe

the modi�cations for tests. The modi�cations for CS�s are analogous with ��n in place of �1�n:

The �rst modi�cation is the change in the de�nition of �5;�1;F described in (21.1). Equation

(21.1) and the �min(�) condition in FTS;AR=AR imply that Assumptions HL1AR=AR(xiv) and (xvii)(stated in Section 18) hold.

The second modi�cation is the use of a WLLN for triangular arrays of strong mixing random

vectors, rather than i.i.d. random vectors, when verifying Assumptions HL1AR=AR(i) (in condi-

tion (a) in (19.11)), HL1AR=AR(vi) (in the paragraph following (19.17)), and HL1AR=AR(xxiii) (in

the paragraph following (19.29)), and when verifying Assumptions HL1AR=LM (iii) and (xiii) in

the proof of Lemma 20.1. For the WLLN, we use Example 4 of Andrews (1988), which shows

that for a strong mixing row-wise-stationary triangular array fWni : i � n; n � 1g we haven�1

Pni=1(�(Wni) � EFn�(Wni)) !p 0 for any real-valued function �(�) (that may depend on n)

for which supn�1EFn jj�(Wni)jj1+� <1 for some � > 0:

The third modi�cation is the use of a CLT for triangular arrays of strong mixing random vectors,

rather than i.i.d. random vectors, when verifying Assumption HL1AR=AR(v) at the end of the proof

of Lemma 19.2 and when verifying Assumption HL1AR=LM (ii) in the proof of Lemma 20.1. For the

CLT, we use Lemma 21.1.

The fourth modi�cation is to use Assumption V to verify Assumptions HL1AR=AR(xi) and (xv)

in the proof of Lemma 19.2 and Assumption HL1AR=LM (vii) in the proof of Lemma 20.1 with

n(�); �1n(�); and �2n(�) de�ned by the submatrices of VFn(�); de�ned in (11.3) and partitioned

as in (11.4), e.g., �1n(�) := (�11Fn(�)0; :::;�1p1Fn(�)

0)0:

The �fth modi�cation is the veri�cation of Assumptions HL1AR=AR(xii) and (xvi) and

HL1AR=LM (viii). Assumption HL1AR=AR(xii) requires sup�12B(�1�n;K=n1=2) jjn(�1) � njj ! 0;

where n(�1) :=P1m=�1(EF gi(�)g

0i�m(�) � EF gi(�)EF gi(�)

0) is de�ned in (11.1). For notational

simplicity, we suppose that k = 1; so that n(�1) is a scalar. To verify Assumption HL1AR=AR(xii),

80

we use the two-term Taylor expansion in (19.22) and write

gi(�1) = gi + �ni(�1); where

�ni(�1) := G1i � (�1 � �1�n) + (�1 � �1�n)@

@�1G1i(e�1i)(�1 � �1�n) (21.2)

and e�1i lies between �1 and �1�n:By the standard strong mixing covariance inequality of Davydov (1968), for two function s1(�)

and s1(�) on W and some > 0 and C1 <1;

jjCovF (s1(Wi); s2(Wi�m))jj � C1jjs1(Wi)jjF;2+ jjs2(Wi)jjF;2+ � =(2+ )F (m)

� C1C =(2+ )jjs1(Wi)jjF;2+ jjs2(Wi)jjF;2+ m�d =(2+ ); where

jjs(Wi)jjF;2+ := (EF jjs(Wi)jj2+ )1=(2+ ); (21.3)

d =(2 + ) > 1; and the second inequality uses the condition on the strong mixing numbers in the

de�nition of FTS;AR=AR in (11.2).Using (21.2), we have

sup�12B(�1�n;K=n1=2)

jj�ni(�1)jjFn;2+ � jjG1ijjFn;2+ K=n1=2 + @

@�1G1i(e�1i)

Fn;2+

K2=n

= O(n�1=2); (21.4)

where the equality holds using the de�nition of �1i in (8.2) and the moment conditions EF �2+ 1i �M

and EF jjvec(G1i(�1))jj2+ �M 8(�1; F ) 2 FTS;AR=AR:Now we bound the mth term in the doubly in�nite sum over m = �1; :::;1 that de�nes

n(�1)� n: We have

Anm := sup�12B(�1�n;K=n1=2)

jjEFngi(�1)gi�m(�1)� (EFngi(�1))2 � EFngigi�m + (EFngi)2jj

= sup�12B(�1�n;K=n1=2)

jjEFngi�ni�m(�1)� EFngiEFn�ni(�1) + EFn�ni(�1)gi�m � EFn�ni(�1)EFngi

+EFn�ni(�1)�ni�m(�1)� (EFn�ni(�1))2jj

� C1C =(2+ )(2jjgijjFn;2+ jj�ni(�1)jjFn;2+ + jj�ni(�1)jj2Fn;2+ )m

�d =(2+ )

= O(n�1=2)m�d =(2+ ); (21.5)

where the O(n�1=2) term does not depend on m; the �rst equality holds by (21.2), the inequality

holds by (21.3) applied three times, and the last equality holds by (21.4) and jjgijjFn;2+ �M:

81

We have

sup�12B(�1�n;K=n1=2)

jjn(�1)� njj �1X

m=�1Anm = O(n�1=2)

1Xm=�1

m�d =(2+ ) = O(n�1=2); (21.6)

where the �rst inequality holds by the de�nition of n(�1) in (11.1), the �rst equality holds by (21.5),

and the last equality uses the condition on d in FTS;AR=AR that d > (2 + )= : This completes theveri�cation of Assumption HL1AR=AR(xii). The veri�cations of Assumptions HL1AR=AR(xvi) and

HL1AR=LM (viii) are similar and hence, for brevity, are not given.

The sixth modi�cation is in the veri�cation of Assumption HL2AR=AR(ii) in (19.39)�(19.41) in

the proof of Lemma 19.3. We verify condition (ii) of Lemma 19.4, i.e., f�n(�) : n � 1g is stochasti-cally equicontinuous for �n(�1) de�ned in (19.39), using Lemma 21.2 with s(w; �) = g(w; �1; �20);

� = �1; d� = p1; and B = �1; rather than using Theorem 4 in Andrews (1994, p. 2277). (We use

the latter result in the row-wise i.i.d. case because it yields weaker conditions than are obtained by

applying Lemma 21.2 in the i.i.d. case.) Conditions (i) and (ii) of Lemma 21.2 hold by the strong

mixing condition in FTS;AR=AR and Assumption SI(v), respectively. Condition (iii) of Lemma 21.2holds with B(Wni) = sup�12�1 jjG1i(�1)jj by a mean-value expansion using Assumption SI(v), asin (19.41). Conditions (iv), (v), and (vi) of Lemma 21.2 hold by Assumptions SI(iii), SI(iv), and

SI-TS(i) (because the conditions in Assumption SI-TS(i) imply that q > p1 and 2 � q < r);

respectively.

In addition, when verifying Assumption HL2AR=AR(ii), we verify condition (iii) of Lemma 19.4,

i.e., �n(�1) = Op(1) 8�1 2 �1 for �n(�1) de�ned in (19.39), using Markov�s inequality and thestrong mixing covariance inequality in (21.3), rather than Markov�s inequality combined with the

expression for the variance of an average of i.i.d. random variables, as in (19.40). It su¢ ces to

show that V arFn(n1=2bgn(�1)) = Op(1) 8�1 2 �1: By change of variables, we have

V arFn

n�1=2

nXi=1

gi(�1)

!=

n�1Xm=�n+1

�1� jmj

n

�CovFn(gi(�1); gi�m(�1))

� O(1)

1Xm=�1

�(r�2)=rFn

(m) = O(1); (21.7)

where the inequality holds using the �rst line of (21.3) with r in place of 2 + ; s1(Wi) =

s2(Wi) = gi(�1); and lim supn!1EFn sup�12�1 jjgi(�1)jjr < 1 (by Assumption SI(iii)), and the

second equality holds by the conditions on the strong mixing numbers in Assumption SI-TI(i) by

the following argument. Given the mixing number condition in Assumption SI-TI(i), it su¢ ces

to show that 1=q � 1=r � (r � 2)=r; or equivalently, 1=q � 1 � 1=r; or q � r=(r � 1): We have

82

q � (q + �1)=(1 + �1) � (q + �1)=(q + �1 � 1) = r=(r � 1); where the �rst inequality holds becauseq � 1 and �1 > 0 and the second inequality holds because q � 1 � 1:

The seventh modi�cation is to use Assumption SI-TS(ii) to verify Assumption HL2AR=AR(v).

This completes the proof of Theorem 11.1. �

Proof of Lemma 21.1. The proof is essentially the same as that of Lemma 20.1 in Section 20 in

the SM to AG1. For the CS case, it relies on the moment conditions EFn jjfi(��n)jj2+ � M < 1for some M < 1; 8(��n; Fn) 2 F�;TS;AR=AR (or, equivalently, 8�n 2 ��;TS;AR=AR); and on thestrong mixing numbers satisfying �Fn(m) � Cm�d for some d > (2 + )= and some C < 1;8(��n; Fn) 2 F�;TS;AR=AR; where for notational simplicity we consider the sequence fng; ratherthan a subsequence fwn : n � 1g: �

Proof of Lemma 21.2. The result of Lemma 21.2 is a special case of Theorem 3 in Hansen (1996)

where Hansen�s Lipschitz exponent � equals 1; the average over n in his equations (12) and (13)

disappear because of the assumption of row-wise identical distributions (and, hence, the square

and square root in his (12) and (13) cancel), his parameter dimension �a� is d� in our notation,

his metric �r is the Lr metric given our assumption of row-wise identical distributions, and the Lr

metric on B can be replaced by the Euclidean metric on B using the Lipschitz condition (iii) andthe moment condition (v) in the statement of the Lemma. �

22 Additional Simulation Results

This section provides additional simulation results to those given in Section 9. The details

concerning the models, tests, and simulation scenarios considered are given in Section 9.

22.1 Heteroskedastic Linear IV Model

Table SM-I provides NRP�s of the AR/QLR1 test in the heteroskedastic linear IV model of

Section 9.1 for sample sizes n = 50 and 500 for k = 4: Table SM-II does likewise for n = 100 and

250 for k = 8: The NRP results in Table SM-I are similar to those in Table I for n = 100 and

250: Even for n = 50; the maximum NRP is :050: On the other hand, the results in Table SM-II

for k = 8 show some over-rejection of the null with the maximum NRP probability being :064 for

n = 100 and :056 for n = 250:

83

TABLE SM-I. Null Rejection Probabilities of the Nominal :05 AR/QLR1 Test for n = 50 and

500 and k = 4; and Base Case Tuning Parameters in the Heteroskedastic Linear Instrumental

Variables Model

n = 50 n = 500

jj�2jj : 40 20 12 4 0 40 20 12 4 0

40 .043 .043 .043 .046 .050 .049 .049 .049 .047 .046

20 .040 .040 .040 .042 .048 .049 .049 .048 .046 .045

jj�1jj 12 .037 .037 .038 .040 .044 .048 .048 .047 .044 .043

4 .018 .018 .019 .022 .033 .037 .034 .030 .030 .037

0 .000 .000 .000 .000 .001 .000 .000 .001 .001 .001

TABLE SM-II. Null Rejection Probabilities of the Nominal :05 AR/QLR1 Test for n = 100

and 250 and k = 8; and Base Case Tuning Parameters in the Heteroskedastic Linear Instrumental

Variables Model

n = 100 n = 250

jj�2jj : 40 20 12 4 0 40 20 12 4 0

40 .043 .045 .049 .061 .064 .045 .045 .046 .054 .056

20 .040 .042 .046 .059 .063 .044 .044 .044 .053 .055

jj�1jj 12 .038 .039 .042 .057 .061 .042 .041 .040 .051 .054

4 .014 .014 .017 .036 .047 .019 .019 .020 .038 .044

0 .000 .000 .000 .001 .001 .000 .001 .001 .001 .001

22.2 Nonlinear IV Model: Inference on the Structural Function

Table SM-III shows little sensitivity of NRP�s of the AR/QLR1 test to �1; but some sensitivity

of power to �1 when jj�jj = 4 and �2 is positive. Table SM-III shows no sensitivity of the NRP�sand power of the AR/QLR1 test to KL; K

�L; and a: The table shows no sensitivity of NRP�s to

Krk; but some sensitivity of power to Krk: The smallest value of Krk; :25; yields noticeably lower

power than larger values.

84

TABLE SM-III. Sensitivity of NRP and Power of the Nominal :05 AR/QLR1 Test to the Tuning


the Structural Function at y1 = 2 in the Nonlinear Instrumental Variables Model

Tuning jj�jj = 50 jj�jj = 4Parameter �2 : .00 -.130 -.094 .105 .155 .00 -1.15 -.88 2.7 8.8

.0010 .043 .796 .497 .507 .808 .038 .798 .496 .519 .808

.0025 .044 .799 .502 .507 .807 .038 .799 .496 .514 .805

�1 .0050 .044 .801 .504 .502 .803 .037 .799 .495 .504 .800

.0100 .042 .798 .502 .494 .794 .035 .797 .487 .486 .785

.0150 .041 .795 .498 .483 .784 .032 .794 .475 .465 .766

.01 .045 .804 .510 .513 .813 .039 .803 .503 .524 .800

KL .05 .044 .801 .504 .502 .803 .037 .799 .495 .504 .800

.10 .040 .791 .491 .493 .796 .035 .792 .486 .504 .800

.25 .046 .742 .445 .430 .735 .041 .769 .451 .465 .789

.50 .045 .776 .477 .468 .770 .037 .791 .474 .482 .792

Krk 1.0 .044 .801 .504 .502 .803 .037 .799 .495 .504 .800

2.0 .045 .811 .518 .524 .821 .037 .802 .500 .538 .810

4.0 .045 .812 .521 .530 .824 .038 .800 .500 .565 .826

.001 .044 .801 .504 .502 .803 .037 .799 .495 .504 .803

K�L .005 .044 .801 .504 .502 .803 .037 .799 .495 .504 .800

.010 .044 .801 .504 .502 .803 .037 .799 .495 .504 .707

.00 .044 .801 .504 .502 .803 .037 .799 .495 .504 .800

a 10�6 .044 .801 .504 .502 .803 .037 .799 .495 .504 .800

.01 .043 .801 .504 .502 .803 .036 .799 .495 .504 .798

85

TABLE SM-IV. Sensitivity of NRP and Power of the Nominal :05 AR/QLR1 Test to the Sample

Size, n; and Number of Instruments, k; for jj�jj = 50 and 4 and for Five Values of �2 for Inferenceon the Structural Function at y1 = 2 in the Nonlinear Instrumental Variables Model

jj�jj = 50 jj�jj = 4�2 : .00 -.130 -.094 .105 .155 .00 -1.15 -.88 2.7 8.8

50 .026 .271 .211 .191 .229 .018 .125 .071 .097 .256

100 .036 .596 .383 .348 .545 .025 .350 .186 .228 .536

n 250 .040 .790 .496 .479 .764 .034 .670 .381 .411 .763

500 .044 .801 .504 .502 .803 .037 .799 .495 .504 .800

1000 .048 .775 .481 .489 .787 .042 .868 .570 .572 .755

4 .044 .801 .504 .502 .803 .037 .799 .495 .504 .800

k 8 .044 .648 .362 .338 .620 .037 .742 .415 .348 .664

12 .049 .506 .268 .237 .462 .037 .658 .348 .278 .572

22.3 Nonlinear IV Model: Inference on the Derivative of the Structural

Function

Table SM-IV provides NRP�s for the nominal .05 AR/QLR1 test for hypotheses concerning the

derivative of the structural function at y1 = 2: The table shows that the NRP�s vary between :007

and :052 over these cases. The lowest NRP�s occur for jj�jj = 0: In the base case scenario, n = 500and k = 4; the NRP�s are in [:034; :047] for jj�jj � 4:

TABLE SM-V. Null Rejection Probabilities of the Nominal :05 AR/QLR1 Test for Base Case

Tuning Parameters for Inference on the Derivative of the Structural Function at y1 = 2 in the

Nonlinear Instrumental Variables

k n Errors jj�jj : 100 75 50 35 20 14 8 4 0

4 50 Homoskedastic .033 .031 .028 .024 .024 .026 .025 .019 .002

4 100 Homoskedastic .039 .039 .039 .039 .039 .037 .033 .026 .007

4 250 Homoskedastic .042 .043 .044 .043 .040 .037 .033 .031 .017

4 500 Homoskedastic .045 .046 .047 .043 .039 .038 .037 .034 .016

8 100 Homoskedastic .051 .052 .051 .050 .050 .051 .048 .036 .007

8 250 Homoskedastic .045 .046 .047 .047 .046 .044 .040 .033 .023

4 250 Heteroskedastic .032 .031 .030 .029 .026 .023 .017 .011 .008

86

The results of Table SM-VI are very similar to those of Table SM-III. The results of Table

SM-VII are broadly similar to those of Table SM-IV.

TABLE SM-VI. Sensitivity of NRP and Power of the Nominal :05 AR/QLR1 Test to the Tuning


the Derivative of the Structural Function in the Nonlinear Instrumental Variables Model

Tuning jj�jj = 50 jj�jj = 4Parameter �2 : .00 -.085 -.061 .070 .104 .00 -.80 -.60 1.6 4.5

.0010 .046 .796 .495 .497 .792 .035 .805 .505 .520 .811

.0025 .046 .796 .495 .503 .797 .035 .806 .505 .515 .809

�1 .0050 .046 .797 .495 .504 .800 .034 .805 .503 .505 .802

.0100 .046 .797 .495 .506 .804 .032 .799 .495 .485 .787

.0150 .046 .797 .495 .506 .804 .030 .792 .482 .463 .770

.01 .046 .797 .496 .505 .801 .036 .810 .508 .523 .813

KL .05 .046 .797 .495 .504 .800 .034 .805 .503 .505 .802

.10 .044 .789 .486 .489 .788 .032 .799 .494 .505 .802

.25 .048 .734 .425 .430 .736 .038 .778 .458 .470 .788

.50 .050 .770 .466 .469 .769 .035 .796 .482 .485 .793

Krk 1.0 .046 .797 .495 .504 .800 .034 .805 .503 .505 .802

2.0 .047 .809 .508 .521 .816 .036 .811 .509 .530 .816

4.0 .048 .810 .511 .526 .819 .039 .813 .516 .552 .839

.001 .046 .797 .495 .504 .800 .034 .805 .503 .505 .802

K�L .005 .046 .797 .495 .504 .800 .034 .805 .503 .505 .802

.010 .046 .797 .495 .504 .800 .034 .800 .498 .505 .796

.00 .046 .797 .495 .504 .800 .034 .805 .503 .505 .802

a 10�6 .046 .797 .495 .504 .800 .034 .805 .503 .505 .802

.01 .046 .797 .495 .504 .800 .034 .805 .503 .504 .802

87

TABLE SM-VII. Sensitivity of NRP and Power of the Nominal :05 AR/QLR1 Test to the

Sample Size, n; and Number of Instruments, k; for jj�jj = 50 and 4 and for Five Values of �2

for Inference on the Derivative of the Structural Function in the Nonlinear Instrumental Variables

Model

jj�jj = 50 jj�jj = 4�2 : .00 -.085 -.061 .070 .104 .00 -.80 -.60 1.6 4.5

50 .030 .371 .246 .213 .327 .018 .053 .035 .064 .126

100 .040 .697 .470 .433 .660 .024 .156 .090 .177 .404

n 250 .043 .834 .544 .520 .805 .031 .492 .278 .380 .723

500 .046 .797 .495 .504 .800 .034 .805 .503 .505 .802

1000 .048 .692 .403 .431 .726 .039 .959 .727 .590 .834

4 .046 .797 .495 .504 .800 .034 .805 .503 .505 .802

k 8 .046 .603 .338 .322 .591 .030 .776 .437 .356 .669

12 .053 .458 .244 .232 .439 .042 .693 .360 .284 .565

23 Additional Second-Step C(�) Tests


Here, we de�ne a C(�) version of the test in Andrews and Guggenberger (2015) (AG), which

we refer to as the C(�)-QLR2 test. The second-step C(�)-QLR2 test statistic is

QLR22n(�) := AR2n(�)� �min(n bQ2n(�)); wherebQ2n(�) := �egn(�); bD�2n(�)

�0 cM1n(�)�egn(�); bD�

2n(�)�2 R(p2+1)�(p2+1);bD�

2n(�) :=b�1=2n (�) bD2n(�)bL1=22n (�) 2 Rk�p2 ;bL2n(�) := (�2; Ip2)(

b�"2n(�))�1(�2; Ip2)0 2 Rp2�p2 ; (23.1)

egn(�) is de�ned in (7.10), and b�"2n(�) is de�ned below.19The C(�)-QLR2 test uses a conditional critical value that depends on the k � p2 matrix

n1=2 bD�2n(�1) and the k�k projection matrix cM1n(�1): For nonrandom D2 2 Rk�p2 and nonrandom

symmetric psd M 2 Rk�k; let

QLR2k;p2(D2;M) := Z 0MZ � �min((Z;D2)0M(Z;D2)); where Z � N(0k; Ik): (23.2)

19Unlike the random perturbation of b�1=2n (�) bD1n(�) by an�1=2�1 in (7.10) and the random perturbation ofb�1=2n (�) bD2n(�) by an�1=2�2 in (7.13), no random perturbation of bD�2n(�) is needed in the de�nition of QLR22n(�):

88

De�ne cQLR2k;p2(D2;M; 1 � �) to be the 1 � � quantile of the distribution of QLR2k;p2(D2;M): For

given D2 and M; cQLR2k;p2(D2;M; 1� �) can be computed by simulation very quickly and easily.

For given �1 2 �1; the nominal level � second-step C(�)-QLR2 test rejects H0 : �2 = �20 when

�QLR22n (�1; �) := QLR22n(�1; �20)� cQLR2k;p2(n1=2 bD�

2n(�1; �20);cM1n(�1; �20); 1� �) > 0: (23.3)

When p2 = 1; the �min(n bQ2n(�)) term that appears in (23.1) can be solved in closed form. In

this case, the QLR22n(�) statistic can be written as

QLR22n(�) :=1

2

�AR2n(�)� rk�2n(�) +

q(AR2n(�)� rk�2n(�))2 + 4LM�

2n(�) � rk�2n(�)�; where

LM�2n(�) := negn(�)0P bD�

2n(�)egn(�);

rk�2n(�) := n bD�2n(�)

0cM1n(�) bD�2n(�); (23.4)

and AR2n(�) is de�ned in (7.10). When p2 = 1; the C(�)-QLR2 critical value, for a nominal level

� test, is as in (7.17) with rk2n(�) = rk�2n(�) and WIyn(�) = 0: cQLR1(1� �; rk�2n(�); 0):Now, we de�ne b�"2n(�): De�nebR2n(�) := �B2(�)0 Ik� bV2n(�) (B2(�) Ik) 2 R(p2+1)k�(p2+1)k; wherebV2n(�) := n�1

nXi=1

�f2i(�)� bf2n(�)��f2i(�)� bf2n(�)�0 2 R(p2+1)k�(p2+1)k; (23.5)

f2i(�) :=

0@ gi(�)

vec(G2i(�))

1A ; bf2n(�) :=0@ bgn(�)

vec( bG2n(�))1A ; and B2(�) :=

0@ 1 00p2

��2 �Ip2

1A :

Let bR2j`n(�) denote the (j; `) k � k submatrix of bR2n(�) for j; ` � p2 + 1:20

We de�ne b�2n(�) 2 R(p2+1)�(p2+1) to be the symmetric pd matrix whose (j; `) element isb�2j`n(�) = tr( bR2j`n(�)0b�1n (�))=k (23.6)

for j; ` � p2 + 1: AG use an eigenvalue-adjusted version of b�2n(�); denoted b�"2n(�):The eigenvalue adjustment is de�ned as follows. LetH 2 RdH�dH be any non-zero positive semi-

de�nite (psd) matrix with spectral decomposition AH�HA0H ; where �H = Diagf�H1; :::; �HdHg isthe diagonal matrix of eigenvalues of H with nonnegative nonincreasing diagonal elements and

AH is a corresponding orthogonal matrix of eigenvectors of H: For " > 0; the eigenvalue-adjusted

20That is, bR2j`n(�) contains the elements of bR2n(�) indexed by rows (j � 1)k+1 to jk and columns (`� 1)k to `k:

89

matrix H" is

H" := AH�"HA

0H ; where �

"H := Diagfmaxf�H1; �max(H)"g; :::;maxf�HdH ; �max(H)"gg; (23.7)

where �max(H) denotes the maximum eigenvalue of H: Note that H" = H whenever the condition

number of H is less than or equal to 1=" (for " � 1):21

The matrix b�"2n(�) is de�ned in (23.7) with H = b�2n(�): Based on the �nite-sample simulations,AG recommend using " = :05:


In this section we introduce a C(�) version of the CQLR test in I. Andrews and Mikusheva

(2016) (AM), which we refer to as the C(�)-QLR3 test. For given �1 2 �1; the second-step C(�)-QLR3 test statistic is

AR2n(�1; �20)� inf�22�2

AR2n(�1; �2): (23.8)

Next, we de�ne AM�s data-dependent critical value. For � := (�1; �2) 2 � and �0 := (�01; �020)0 2 �;let

h2n(�) := n1=2bgn(�)� bn(�; �0)b�1n (�0; �0)n1=2bgn(�0) andg�2n(�) := h2n(�) + bn(�; �0)b�1n (�0; �0)��2; (23.9)

where ��2 � N(0; bn(�0; �0)) given bn(�; �) and bn(�) := bn(�; �): We view h2n(�1; �2) as a sto-

chastic process indexed by �2 with �1 �xed. It is designed to be asymptotically independent of

n1=2bgn(�1; �20) when �0 = (�01; �020)0 is the true value. De�neQLR3�2n(�1) := AR�2n(�1; �20)� inf

�22�2AR�2n(�1; �2); where

AR�2n(�) := g�2n(�)0b�1=2n (�)cM1n(�)b�1=2n (�)g�2n(�): (23.10)

Let cvQLR32n (h2n; �1; �) denote the 1�� quantile of the conditional distribution of QLR3�2n(�1) givenh2n := h2n(�) and bn(�; �):

For given �1 2 �1; the �1-orthogonalized nominal � second-step CQLR3 test rejects H0 : �2 =21AG shows that the eigenvalue-adjustment procedure possesses the following desirable properties: (i) H" is

uniquely de�ned, (ii) �min(H") � �max(H)"; (iii) �max(H")=�min(H") � maxf1="; 1g; (iv) for all c > 0; (cH)" = cH";

and (v) H"n ! H" for any sequence of psd matrices fHn : n � 1g with Hn ! H:

90

�20 when

�QLR32n (�1; �) := AR2n(�1; �20)� inf�22�2

AR2n(�1; �2)� cvQLR32n (h2n; �1; �) > 0: (23.11)

The second-step CQL3 test is applicable in moment condition models, but not minimum distance

models.

91

References

Aitchison, J. and S. D. Silvey (1959): �Maximum-Likelihood Estimation of Parameters Subject

to Restraints,�Annals of Mathematical Statistics, 29, 813�828.

Andrews, D. W. K. (1988): �Laws of Large Numbers for Dependent Non-identically Distributed

Random Variables,�Econometric Theory, 4, 458�467.

� � � (1991a): �Generic Uniform Convergence,�Econometric Theory, 8, 241�257.

� � � (1991b): �Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estima-

tion,�Econometrica, 59, 817�858.

� � � (1994): �Empirical Processes in Econometrics,�Ch. 37 in Handbook of Econometrics, Vol.

IV, ed. by R. F. Engle and D. L. McFadden. New York: Elselvier.

Andrews, D. W. K., X. Cheng, and P. Guggenberger (2011): �Generic Results for Establishing

the Asymptotic Size of Con�dence Sets and Tests,�Cowles Foundation Discussion Paper No.

1813, Yale University.

Andrews, D. W. K. and P. Guggenberger (2015): �Identi�cation- and Singularity-Robust Inference

for Moment Condition Models,� Cowles Foundation Discussion Paper No. 1978, revised

December 2015, Yale University.

� � � (2017): �Asymptotic Size of Kleibergen�s LM and Conditional LR Tests for Moment Con-

dition Models,�Econometric Theory, 33, forthcoming. Also available as Cowles Foundation

Discussion Paper No. 1977, revised September 2015, Yale University.

Andrews, I. and A. Mikusheva (2016): �Conditional Inference with a Functional Nuisance Para-

meter,�Econometrica, 84, 1571�1612.

Cramér, H. (1946): Mathematical Methods of Statistics. Princeton, NJ: Princeton University

Press.

Crowder, M. J. (1976): �Maximum Likelihood Estimation for Dependent Observations,�Journal

of the Royal Statistical Society, Ser. B, 38, 45�53.

Davydov, Y. A. (1968): �Convergence of Distributions Generated by Stationary Stochastic Processes,�

Theory of Probability and Its Applications, 13, 691�696.

92

de Jong, R. M. (1997): �Central Limit Theorems for Dependent Heterogeneous Random Vari-

ables,�Econometric Theory, 13, 353�367.

Newey, W. K., and K. West (1987): �A Simple, Positive Semi-de�nite, Heteroskedasticity and

Autocorrelation Consistent Covariance Matrix,�Econometrica, 55, 703�708.

van der Vaart, A. W. (1998): Asymptotic Statistics. Cambridge, UK: Cambridge University Press.

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supplemental material for identification-robust subvector

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