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Panels with nonstationary multifactor error structuresG. Kapetanios, M. Hashem Pesaran, T. Yamagata

To cite this version:G. Kapetanios, M. Hashem Pesaran, T. Yamagata. Panels with nonstationary multifactor error struc-tures. Econometrics, MDPI, 2010, 160 (2), pp.326. �10.1016/j.jeconom.2010.10.001�. �hal-00768190�

Accepted Manuscript

Panels with nonstationary multifactor error structures

G. Kapetanios, M. Hashem Pesaran, T. Yamagata

PII: S0304-4076(10)00202-2DOI: 10.1016/j.jeconom.2010.10.001Reference: ECONOM 3409

To appear in: Journal of Econometrics

Received date: 15 June 2009Revised date: 14 April 2010Accepted date: 5 October 2010

Please cite this article as: Kapetanios, G., Hashem Pesaran, M., Yamagata, T., Panels withnonstationary multifactor error structures. Journal of Econometrics (2010),doi:10.1016/j.jeconom.2010.10.001

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Panels with Nonstationary Multifactor Error Structures�

G. KapetaniosQueen Mary, University of London

M. Hashem PesaranCambridge University and USC

T. YamagataUniversity of York

September 27, 2010

Abstract

The presence of cross-sectionally correlated error terms invalidates much inferentialtheory of panel data models. Recently, work by Pesaran (2006) has suggested a methodwhich makes use of cross-sectional averages to provide valid inference in the case of sta-tionary panel regressions with a multifactor error structure. This paper extends this workand examines the important case where the unobservable common factors follow unit rootprocesses. The extension to I(1) processes is remarkable on two counts. Firstly, it is ofgreat interest to note that while intermediate results needed for deriving the asymptoticdistribution of the panel estimators di¤er between the I(1) and I(0) cases, the �nal resultsare surprisingly similar. This is in direct contrast to the standard distributional resultsfor I(1) processes that radically di¤er from those for I(0) processes. Secondly, it is worthnoting the signi�cant extra technical demands required to prove the new results. Thetheoretical �ndings are further supported for small samples via an extensive Monte Carlostudy. In particular, the results of the Monte Carlo study suggest that the cross-sectionalaverage based method is robust to a wide variety of data generation processes and haslower biases than the alternative estimation methods considered in the paper.Keywords: Cross Section Dependence, Large Panels, Unit Roots, Principal Components, Com-

mon Correlated E¤ects.JEL-Classi�cation: C12, C13, C33.

�The authors thank Vanessa Smith and Elisa Tosetti, and seminar participants at Kyoto University, Univer-sity of Amsterdam, University of Nottingham and the annual conference at the Granger Centre for most helpfulcomments on a previous version of this paper. This version has also bene�ted greatly from constructive com-ments and suggestions by the Editor (Cheng Hsiao), an associate editor and three anonymous referees. HashemPesaran and Takashi Yamagata acknowledge �nancial support from the ESRC (Grant No. RES-000-23-0135).

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1 Introduction

Panel data sets have been increasingly used in economics to analyze complex economic phenom-

ena. One of their attractions is the ability to use an extended data set to obtain information

about parameters of interest which are assumed to have common values across panel units.

Most of the work carried out on panel data has usually assumed some form of cross sectional

independence to derive the theoretical properties of various inferential procedures. However,

such assumptions are often suspect and as a result recent advances in the literature have focused

on estimation of panel data models subject to error cross sectional dependence.

A number of di¤erent approaches have been advanced for this purpose. In the case of

spatial data sets where a natural immutable distance measure is available the dependence is

often captured through �spatial lags�using techniques familiar from the time series literature.

In economic applications, spatial techniques are often adapted using alternative measures of

�economic distance�. This approach is exempli�ed in work by Lee and Pesaran (1993), Conley

and Dupor (2003), Conley and Topa (2002) and Pesaran, Schuermann, and Weiner (2004), as

well as the literature on spatial econometrics recently surveyed by Anselin (2001). In the case

of panel data models where the cross section dimension (N) is small (typically N < 10) and

the time series dimension (T ) is large the standard approach is to treat the equations from the

di¤erent cross section units as a system of seemingly unrelated regression equations (SURE)

and then estimate the system by the Generalized Least Squares (GLS) techniques.

The SURE approach is not applicable if the errors are correlated with the regressors and/or

if the panels under consideration have a large cross sectional dimension. This has led a number

of investigators to consider unobserved factor models, where the cross section error correlations

are de�ned in terms of the factor loadings. The use of unobserved factors also allows for certain

degree of correlation between the idiosyncratic errors and the unobserved factors. Use of factor

models is not new in economics and dates back to the pioneering work of Stone (1947) who

applied the principal components (PC) analysis of Hotelling to US macroeconomic time series

over the period 1922-1938 and was able to demonstrate that three factors (namely total income,

its rate of change and a time trend) explained over 97 per cent of the total variations of all the

17 macro variables that he had considered. Until recently, subsequent applications of the PC

approach to economic times series has been primarily in �nance. See, for example, Chamberlain

and Rothschild (1983), Connor and Korajzcyk (1986) and Connor and Korajzcyk (1988). But

more recently the unobserved factor models have gained popularity for forecasting with a large

number of variables as advocated by Stock and Watson (2002). The factor model is used very

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much in the spirit of the original work by Stone, in order to summarize the empirical content

of a large number of macroeconomics variables by a small set of factors which, when estimated

using principal components, is then used for further modelling and/or forecasting. A related

literature on dynamic factor models has also been put forward by Forni and Reichlin (1998)

and Forni, Hallin, Lippi, and Reichlin (2000).

Recent uses of factor models in forecasting focus on consistent estimation of unobserved

factors and their loadings. Related theoretical advances by Bai and Ng (2002) and Bai (2003)

are also concerned with estimation and selection of unobserved factors and do not consider the

estimation and inference problems in standard panel data models where the objects of interest

are slope coe¢ cients of the conditioning variables (regressors). In such panels the unobserved

factors are viewed as nuisance variables, introduced primarily to model the cross section de-

pendencies of the error terms in a parsimonious manner relative to the SURE formulation.

Despite these di¤erences knowledge of factor models could still be useful for the analysis of

panel data models if it is believed that the errors might be cross sectionally correlated. Disre-

garding the possible factor structure of the errors in panel data models can lead to inconsistent

parameter estimates and incorrect inference. Coakley, Fuertes, and Smith (2002) suggest a pos-

sible solution to the problem using the method of Stock and Watson (2002). But, as Pesaran

(2006) shows, the PC approach proposed by Coakley, Fuertes, and Smith (2002) can still yield

inconsistent estimates. Pesaran (2006) suggests a new approach by noting that linear combi-

nations of the unobserved factors can be well approximated by cross section averages of the

dependent variable and the observed regressors. This leads to a new set of estimators, referred

to as the Common Correlated E¤ects (CCE) estimators, that can be computed by running

standard panel regressions augmented with the cross section averages of the dependent and

independent variables. The CCE procedure is applicable to panels with a single or multiple

unobserved factors and does not necessarily require the number of unobserved factors to be

smaller than the number of observed cross section averages.

In this paper we extend the analysis of Pesaran (2006) to the case where the unobserved

common factors are integrated of order 1, or I(1). Our analysis does not require an a priori

knowledge of the number of unobserved factors. It is only required that the number of un-

observed factors remains �xed as the sample size is increased. The extension of the results of

Pesaran (2006) to the I(1) case is far from straightforward and involves the development of new

intermediate results that could be of relevance to the analysis of panels with unit roots. It is

also remarkable in the sense that whilst the intermediate results needed for deriving the asymp-

totic distribution of the panel estimators di¤er between the I(1) and I(0) cases, the �nal results

are surprisingly similar. This is in direct contrast to the usual phenomenon whereby distribu-

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tional results for I(1) processes are radically di¤erent to those for I(0) processes and involve

functionals of Brownian motion whose use requires separate tabulations of critical values.

It is very important to appreciate that our primary focus is on estimating the coe¢ cients

of the panel regression model. We do not wish to investigate the (co-)integration properties

of the unobserved factors. Rather, our focus is robustness to the properties of the unobserved

factors, for the estimation of the coe¢ cients of the observed regressors that vary over time as

well as over the cross section units. In this sense the extension provided by our work is of

great importance in empirical applications where the integration properties of the unobserved

common factors are typically unknown. In the CCE approach the nature of the factors does

not matter for inferential analysis of the coe¢ cients of the observed variables. The theoretical

�ndings of the paper are further supported for small samples via an extensive Monte Carlo

study. In particular, the results of the Monte Carlo study clearly show that the CCE estimator

is robust to a wide variety of data generation processes and has lower biases than all of the

alternative estimation methods considered in the paper.

The structure of the paper is as follows: Section 2 provides an overview of the method

suggested by Pesaran (2006) in the case of stationary factor processes. Section 3 provides the

theoretical framework of the analysis of nonstationarity. In this section the theoretical prop-

erties of the various estimators are presented. Section 4 presents an extensive Monte Carlo

study, and Section 5 concludes. The Appendix contains proofs of the theoretical results. Some

more technical results and proofs of Lemmas are relegated to a supplementary appendix that

is provided separately from the main paper.

Notations: K stands for a �nite positive constant, kAk = [Tr(AA0)]1=2 is the Frobenius

norm of the m� n matrix A, and A+ denotes the Moore-Penrose inverse of A. rk(A) denotes

the rank of A. supiWi is the supremum of Wi over i. an = O(bn) states the deterministic

sequence fang is at most of order bn, xn = Op(yn) states the vector of random variables, xn; is

at most of order yn in probability, and xn = op(yn) is of smaller order in probability than yn,q:m:!

denotes convergence in quadratic mean (or mean square error),p! convergence in probability,

d! convergence in distribution, and ds asymptotic equivalence of probability distributions. Allasymptotics are carried out under N !1, either with a �xed T , or jointly with T !1. Jointconvergence of N and T will be denoted by (N; T )

j!1. Restrictions (if any) on the relativerates of convergence of N and T will be speci�ed separately.

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2 Panel Data Models with Observed and UnobservedCommon E¤ects

In this section we review the methodology introduced in Pesaran (2006). Let yit be the obser-

vation on the ith cross section unit at time t for i = 1; 2; :::; N ; t = 1; 2; :::; T; and suppose that

it is generated according to the following linear heterogeneous panel data model

yit = �0idt + �

0ixit +

0if t + "it; (1)

where dt is a n � 1 vector of observed common e¤ects, which is partitioned as dt = (d01t;d02t)0

where d1t is a n1�1 vector of deterministic components such as intercepts or seasonal dummiesand d2t is a n2 � 1 vector of unit root stochastic observed common e¤ects, with n = n1 + n2,

xit is a k � 1 vector of observed individual-speci�c regressors on the ith cross section unit attime t, f t is the m� 1 vector of unobserved common e¤ects, and "it are the individual-speci�c(idiosyncratic) errors assumed to be independently distributed of (dt;xit). The unobserved

factors, f t, could be correlated with (dt;xit), and to allow for such a possibility the following

speci�cation for the individual speci�c regressors will be considered

xit = A0idt + �

0if t + vit; (2)

whereAi and �i are n�k andm�k factor loading matrices with �xed and bounded components,vit = (vi1t; :::; vikt)

0 are the speci�c components of xit distributed independently of the common

e¤ects and across i; but assumed to follow general covariance stationary processes. In our setup,

"it is assumed to be stationary, which implies that in the case where ft and/or dt contain unit

root processes, then yit, xit;dt and ft must be cointegrated.1 Some of the implications of this

property are explored further in Remark 6.

Combining (1) and (2) we now have

zit(k+1)�1

=

�yitxit

�= B0

i(k+1)�n

dtn�1

+ C 0i

(k+1)�mf tm�1

+ uit(k+1)�1

; (3)

where

uit =

�"it + �

0ivit

vit

�=

�1 �0i0 Ik

��"itvit

�; (4)

Bi =��i Ai

�� 1 0�i Ik

�, Ci =

� i �i

�� 1 0�i Ik

�; (5)

1However, as will be shown later, our results on the estimators of � hold even if the factor loadings i and/or�i are zero (or weak in the sense of Chudik, Pesaran, and Tosetti (2010)), and it is not necessary that xit andft are cointegrated. What is required for our results is that conditional on dt and ft, the idiosyncratic errors "itand vit are stationary.

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Ik is an identity matrix of order k, and the rank of Ci is determined by the rank of the

m� (k + 1) matrix of the unobserved factor loadings

~�i =� i �i

�: (6)

As discussed in Pesaran (2006), the above set up is su¢ ciently general and renders a variety

of panel data models as special cases. In the panel literature with T small and N large, the

primary parameters of interest are the means of the individual speci�c slope coe¢ cients, �i,

i = 1; 2; :::; N . The common factor loadings, �i and i, are generally treated as nuisance

parameters. In cases where both N and T are large, it is also possible to consider consistent

estimation of the factor loadings, but this topic will not be pursued here. The presence of

unobserved factors in (1) implies that estimation of �i and its cross sectional mean cannot be

undertaken using standard methods. Pesaran (2006) has suggested using cross section averages

of yit and xit to deal with the e¤ects of proxies for the unobserved factors in (1). To see why

such an approach could work, consider simple cross section averages of the equations in (3)2

�zt = �B0dt + �C

0f t + �ut; (7)

where

�zt =1

N

NXi=1

zit, �ut =1

N

NXi=1

uit;

and

�B =1

N

NXi=1

Bi, �C =1

N

NXi=1

Ci. (8)

We distinguish between two important cases: when the rank condition

rk( �C) = m � k + 1, for all N; and as N !1; (9)

holds, and when it does not. Under the former, the analysis simpli�es considerably since it is

possible to proxy the unobserved factors by linear combinations of cross section averages, �ztand the observed common components, dt. But if the rank condition is not satis�ed this is not

possible, although as we shall see it is still possible to consistently estimate the mean of the

regression coe¢ cients, �, by the CCE procedure.

In the case where the rank condition is met we have

f t =��C �C

0��1

�C��zt � �B

0dt � �ut

�: (10)

2Pesaran (2006) considers cross section weighted averages that are more general. But to simplify the expo-sition we con�ne our discussion to simple averages throughout.

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But since

�utq:m:! 0, as N !1, for each t; (11)

and�C

p! C = ~�

�1 0� Ik

�; as N !1; (12)

where~� = (E ( i) ; E (�i)) = ( ;�), (13)

it follows, assuming that Rank(~�) = m, that

f t � (CC 0)�1C��zt � �B

0dt

�q:m:! 0, as N !1:

This suggests that for su¢ ciently large N , it is valid to use �ht = (d0t; �z

0t)0 as observable proxies

for f t. This result holds irrespective of whether the unobserved factor loadings, i and �i, are

�xed or random.

When the rank condition is not satis�ed the use of cross section averages alone do not

allow consistent estimation of all of the unobserved factors and as a result the estimation of

the individual coe¢ cients �i by means of the cross section averages alone will not be possible.

But interestingly enough consistent estimates of the mean of the slope coe¢ cients, �, and their

asymptotic distribution can be obtained if it is further assumed that the factor loadings are

distributed independently of the factors and the individual-speci�c error processes.

2.1 The CCE Estimators

We now discuss the two estimators for the means of the individual speci�c slope coe¢ cients

proposed by Pesaran (2006). One is the Mean Group (MG) estimator proposed in Pesaran and

Smith (1995) and the other is a generalization of the �xed e¤ects estimator that allows for the

possibility of cross section dependence. The former is referred to as the �Common Correlated

E¤ects Mean Group�(CCEMG) estimator, and the latter as the �Common Correlated E¤ects

Pooled�(CCEP) estimator.

The CCEMG estimator is a simple average of the individual CCE estimators, bi of �i,

bMG = N�1NXi=1

bi; (14)

where

bi = (X0i�MX i)

�1X 0i�Myi; (15)

X i = (xi1;xi2; :::;xiT )0, yi = (yi1; yi2; :::; yiT )

0, �M is de�ned by

�M = IT � �H��H0 �H��1

�H0; (16)

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�H = (D; �Z), D and �Z being, respectively, the T �n and T � (k+1) matrices of observationson dt and �zt. We also de�ne for later use

M g = IT �G (G0G)�1G0; (17)

and

M q = IT �Q (Q0Q)+Q0, with Q = G �P , (18)

where G = (D;F ), D = (d1;d2; :::;dT )0, F = (f 1;f 2; :::;fT )

0 are T � n and T � m data

matrices on observed and unobserved common factors, respectively, (A)+ denotes the Moore-

Penrose inverse of A, and

�P(n+m)�(n+k+1)

=

�In �B0 �C

�; �U

�= (0; �U ); (19)

where �U � has the same dimension as �H and �U = (�u1; �u2; :::; �uT )0 is a T � (k + 1) matrix of

observations on �ut. E¢ ciency gains from pooling of observations over the cross section units can

be achieved when the individual slope coe¢ cients, �i, are the same. Such a pooled estimator

of �, denoted by CCEP, is given by

bP =

NXi=1

X 0i�MX i

!�1 NXi=1

X 0i�Myi; (20)

which can also be viewed as a generalized �xed e¤ects (GFE) estimator, and reduces to the

standard FE estimator if �H = � T with � T being a T � 1 vector of ones.

3 Theoretical Properties of CCE Estimators in Nonsta-tionary Panel Data Models

The following assumptions will be used in the derivation of the asymptotic properties of the

CCE estimators.

Assumption 1 (non-stationary common e¤ects): The (n2+m)�1 vector of stochastic commone¤ects, gt = (d

02t;f

0t)0, follows the multivariate unit root process

gt = gt�1 + �gt

where �gt is a (n2 + m) � 1 vector of L2+�, � > 0, stationary near epoque dependent (NED)

processes of size 1/2, on some �-mixing process of size �(2+ �)=�, distributed independently ofthe individual-speci�c errors, "it0 and vit0 for all i, t and t0.

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Assumption 2 (individual-speci�c errors): (i) The individual speci�c errors "it and vjt are

distributed independently of each other, for all i; j and t. "it have uniformly bounded positive

variance, supi �2i < K; for some constant K, and uniformly bounded fourth-order cumulants.

vit have covariance matrices, �vi, which are nonsingular and satisfy supi k�vik < K < 1,autocovariance matrices, �iv (s), such that supi

P1s=�1 k�iv (s)k < K <1, and have uniformly

bounded fourth-order cumulants. (ii) For each i, ("it;v0it)0 is an (k+1)�1 vector of L2+�, � > 0,

stationary near epoque dependent (NED) processes of size 2�2��4 on some �-mixing process it

of size �(2 + �)=� which is partitioned conformably to ("it;v0it)0 as ( "it;

0vit)

0 where "it and

vjt are independent for all i and j.

Assumption 3 The coe¢ cient matrices, Bi and Ci are independently and identically distrib-

uted across i, and of the individual speci�c errors, "jt and vjt, the common factors, �gt, for

all i; j and t with �xed means B and C, and uniformly bounded second-order moments. In

particular,

vec(Bi) = vec(B) + �B;i, �B;i v IID (0;B�); for i = 1; 2; :::; N; (21)

and

vec(Ci) = vec(C) + �C;i, �C;i v IID (0;C�); for i = 1; 2; :::; N; (22)

where B� and C� are (k+ 1)n� (k+ 1)n and (k+ 1)m� (k+ 1)m symmetric non-negative

de�nite matrices, kBk < K, kCk < K, kB�k < K and kC�k < K; for some constant K.

Assumption 4 (random slope coe¢ cients): The slope coe¢ cients, �i, follow the random co-

e¢ cient model

�i = � + {i, {i v IID (0;{); for i = 1; 2; :::; N; (23)

where k�k < K, k{k < K, for some constant K, { is a k � k symmetric non-negative

de�nite matrix, and the random deviations, {i, are distributed independently of j;�j,"jt, vjt,and �gt for all i, j and t. {i has �nite fourth moments uniformly over i.

Assumption 5 (identi�cation of �i and �):�X0i�MXi

T

��1exists for all i and T , and limN!1

1N

PNi=1�vi is nonsingular.

Assumption 6�X0iMgXi

T

��1exists for all i and T , and supiE

X0i�MXi

T

2 < K <1.

Assumption 7 When the rank condition (9) is not satis�ed, (i) 1N

PNi=1

X0iMqXi

T 2and � =

limN;T!1

�1N

PNi=1�iT

�, where �iT = E (T�2X 0

iM qX i), are nonsingular. (ii) If m � 2k+1,

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then�X0iMqXi

T 2

��1exists for all i and T and supiE

�X0iMqXi

T 2

��1 �X0iMqF

T 2

� 2 < 1. (iii) If

m < 2k + 1, then E F 0FT 2

2 <1 and E

�F 0FT 2 ��1 2 <1.Remark 1 Assumption 1 departs from the standard practice in the analysis of large panels

with common factors and speci�es that the factors are non-stationary. Assumption 2 concerns

the individual speci�c errors and relaxes the assumption that "it are serially uncorrelated, often

adopted in the literature (see, e.g., Pesaran (2006)). Assumptions 2-6 are standard in large

panels with random coe¢ cients. But some comments on Assumption 7 seems to be in order.

This Assumption is only used when the rank condition (9) is not satis�ed. It is made up of

three regularity conditions.3 The last two are of greater signi�cance and only relate to the

Mean Group estimator presented in the next Section. In e¤ect, these assumptions ensure that

the individual slope coe¢ cient estimators possess second-order moments asymptotically, which

seems plausible in most economic applications.

Remark 2 Note that Assumption 3 implies that i are independently and identically distributed

across i, and

i = + �i, �i v IID (0;�); for i = 1; 2; :::; N; (24)

where � is a m�m symmetric non-negative de�nite matrix, and k k < K, and k�k < K;

for some constant K.

For each i and t = 1; 2; :::; T , writing the model in matrix notation we have

yi =D�i +X i�i + F i + "i; (25)

where "i = ("i1; "i2; :::; "iT )0. Using (25) in (15) we have

bi � �i =�X 0

i�MX i

T

��1�X 0

i�MF

T

� i +

�X 0

i�MX i

T

��1�X 0

i�M"iT

�; (26)

which shows the direct dependence of bi on the unobserved factors through T�1X 0i�MF . To

examine the properties of this component, we �rst note that (2) and (7) can be written in

matrix notations as

X i = G�i + V i; (27)

and�H = (D;�Z) = (D;D�B+ F�C+ �U) = G �P + �U

�; (28)

3E T�2F 0F

2 <1, which is part of Assumption 7(iii), can be established under mild regularity conditions(see Lemma 4 of Phillips and Moon (1999)).

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where �i = (A0i;�

0i)0, V i = (vi1;vi2; :::;viT )

0, G = (D;F), and �P and �U � are de�ned by (19).

Using Lemmas 3 and 4 in Appendix A and assuming that the rank condition (9) is satis�ed,

it follows thatX 0

i�MF

T= Op

�1pNT

�+Op

�1

N

�; uniformly over i; (29)

X 0i�MX i

T� X

0iM gX i

T= Op

�1pN

�; uniformly over i; (30)

andX 0

i�M"iT

� X0iM g"iT

= Op

�1pNT

�+Op

�1

N

�; uniformly over i: (31)

If the rank condition does not hold then by Lemma 6 in Appendix A it follows that

X 0i�MF

T� X

0iM qF

T= Op

�1pN

�; uniformly over i; (32)

X 0i�MX i

T� X

0iM qX i

T= Op

�1pN

�; uniformly over i; (33)

andX 0

i�M"iT

� X0iM q"iT

= Op

�1pNT

�+Op

�1

N

�; uniformly over i: (34)

In the next subsections we discuss our main theoretical results.

3.1 Results for Pooled Estimators

We now examine the asymptotic properties of the pooled estimators. Focusing �rst on the MG

estimator, and using (26) we have

pN�bMG � �

�=

1pN

NXi=1

{i +1

N

NXi=1

�1iT

pNX 0

i�MF

T

! i+

1

N

NXi=1

�1iT

pNX 0

i�M"i

T

!; (35)

where iT = T�1X 0i�MX i. In the case where the rank condition (9) is satis�ed, by (29) we

have pN�X 0

i�MF

�T

= Op

�1pT

�+Op

�1pN

�: (36)

Using this, we can formally show that

pN�bMG � �

�=

1pN

NXi=1

{i +Op

�1pT

�+Op

�1pN

�:

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Hence pN�bMG � �

�d! N(0;�MG); as (N; T )

j!1: (37)

The variance estimator for �MG suggested by Pesaran (2006) is given by

�MG =1

N � 1

NXi=1

�bi � bMG

��bi � bMG

�0, (38)

which can be used here as well. The following theorem summarises the results for the mean

group estimator. The result is proven in Appendix B.

Theorem 1 Consider the panel data model (1) and (2). Let assumptions 1-6 and 7(ii),(iii)

hold. Then, for the Common Correlated E¤ects Mean Group estimator, bMG, de�ned by (14),

we have, as (N; T )j!1, that

pN�bMG � �

�d! N(0;�MG);

where

�MG = { +�; (39)

� = limN;T!1

"1

N

NXi=1

�iqT

#: (40)

and �iqT is de�ned in (A28). �MG can be consistently estimated by (38).

Note that this theorem does not require that the rank condition, (9), holds for any number,

m, of unobserved factors so long as m is �xed. Also, it does not impose any restrictions on the

relative rates of expansion of N and T . The following Theorem summarizes the results for the

second pooled estimator, bP . The proof is provided in Appendix B.

Theorem 2 Consider the panel data model (1) and (2), and suppose that Assumptions 1-6 and

7(i) hold. Then, for the Common Correlated E¤ects Pooled estimator, bP , de�ned by (20), as

(N; T )j!1, we have that p

N�bP � �

�d! N(0;��

P );

where ��P is given by

��P = ��1 (�+�)��1 (41)

where

� = limN;T!1

1

N

NXi=1

�Ti

!; � = lim

N;T!1

1

N

NXi=1

�Ti

!; � = lim

N;T!1

1

N

NXi=1

�Ti

!

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�Ti = V ar [T�2X 0iM qX i{i], and �Ti and �Ti are given by (A48) and (A45), respectively.

��P can be estimated consistently by

��P =

��1R���1

; (42)

where

�= N�1

NXi=1

X 0i�MX i

T; (43)

R�=

1

(N � 1)

NXi=1

�X 0

i�MX i

T

��bi � bMG

��bi � bMG

�0�X 0i�MX i

T

�: (44)

Overall we see that despite a number of di¤erences in the above analysis, especially in terms

of the results given in (29)-(34), compared to the results in Pesaran (2006), the conclusions are

remarkably similar when the factors are assumed to follow unit root processes.

Remark 3 The formal analysis in the Appendices focuses on the case where the factor is an

I(1) process and no cointegration is present among the factors. But, as shown by Johansen

(1995, pp. 40), when the factor process is cointegrated and there are l < m cointegrating

vectors, we have that F i = F 1�1i+F 2�2i where F 1 is an m� l-dimensional I(1) process withno cointegration whereas F 2 is an l-dimensional I(0) process. This implies that the cointegration

case is equivalent to a case where the model contains a mix of non-cointegrated I(1) and I(0)

factor processes. Since we know that the results of the paper hold for both non-cointegrated I(1)

and, by Pesaran (2006), I(0) factor processes, we conjecture that they hold for the cointegrated

case, as well. However, we feel that a formal proof of this statement is beyond the scope of the

present paper. We consider a case of cointegrated factors in the Monte Carlo study. The results

clearly support the above claim.

Remark 4 In the case of standard linear panel data models with strictly exogenous regressors

and homogeneous slopes, and without unobserved common factors, Pesaran, Smith, and Im

(1996) show that in general the �xed e¤ect estimator is asymptotically at least as e¢ cient as

the mean group estimator. It is reasonable to expect that this result also applies to the CCE type

estimators, namely that under �i = � for all i, the CCEP estimator would be at least as e¢ cient

as the CCEMG estimator. Although, a formal proof is beyond the scope of the present paper,

the Monte Carlo results reported below provide some evidence in favour of this conjecture.

As we noted above, the whole analysis does not depend on whether the rank condition holds

or not. But in the case where the rank condition is satis�ed, a number of simpli�cations arise.

In particular, the technical Assumption 7 is not needed, and Assumption 3 can be relaxed.

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Namely the factor loadings, i, need not follow the random coe¢ cient model. It would be

su¢ cient that they are bounded. Also the expressions for the theoretical covariance matrices

of the estimators change, although crucially the estimators of these covariance matrices do not.

For completeness, we present Corollaries on the theoretical properties of the pooled estimators

when the rank condition holds, below. Proofs are provided in Supplementary Appendix B.

Corollary 1 Consider the panel data model (1) and (2). Assume the rank condition, (9), is

met and suppose that Assumptions 1-6 hold. Then, for the Common Correlated E¤ects Mean

Group estimator, bMG, de�ned by (14), we have, as (N; T )j!1, that

pN�bMG � �

�d! N(0;�MG);

where �MG is given by {. �MG can be consistently estimated by (38).

Corollary 2 Consider the panel data model (1) and (2), and suppose that the rank condi-

tion, (9), is met and Assumptions 1-6 hold. Then, for the Common Correlated E¤ects Pooled

estimator, bP , de�ned by (20), as (N; T )j!1, we have that

pN�bP � �

�d! N(0;��

P );

where

��P = ��1R���1; (45)

R� = limN;T!1

"N�1

NXi=1

�viT

#; (46)

� = limN!1

N�1

NXi=1

�vi

!; (47)

and �viT denotes the variance ofX0iMgXi

T{i. ��

P can be estimated consistently by (42).

3.2 Estimation of Individual Slope Coe¢ cients

In panel data models where N is large the estimation of the individual slope coe¢ cients is likely

to be of secondary importance as compared to establishing the properties of pooled estimators.

However, it might still be of interest to consider conditions under which they can be consistently

estimated. In the case of our set up the following further assumption is needed.

Assumption 8 For each i, "it is a martingale di¤erence sequence. For each i, vit is an k � 1vector of L2+�, � > 0, stationary near epoque dependent (NED) process of size 1/2, on some

�-mixing process of size �(2 + �)=�.

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Then, we have the following result. The proof is provided in Supplementary Appendix C.

Theorem 3 Consider the panel data model (1) and (2) and suppose that Assumptions 1, 2(i)

and 3-8 hold. LetpT=N ! 0, as (N; T )

j! 1, and assume that the rank condition (9) issatis�ed. As (N; T )

j!1, bi, de�ned by (15), is a consistent estimator of �i. FurtherpT�bi � �i

�d! N(0;�bi): (48)

A consistent estimator of �bi is given by

�bi =��2i

�X 0

i�MX i

T

��1; (49)

where

��2i =

�yi �X ibi

�0�M�yi �X ibi

�T � (n+ 2k + 1) : (50)

Remark 5 Parts of the above result hold under weaker versions of Assumption 8. In partic-

ular we note that the central limit theorem in (B72), in Supplementary Appendix C, holds if

Assumption 2(ii) holds. However, in this case the asymptotic variance has a di¤erent form as

autocovariances of "itvit enter the asymptotic variance expression. If, then, a consistent esti-

mate of the asymptotic variance is required a Newey and West (1987) type correction needs to

be used. Consistency of this variance estimator requires more stringent assumptions than the

NED assumption 2(ii). It is su¢ cient to assume that ("it;v0it)0 is a strongly mixing process for

this consistency to hold.

Remark 6 It is worth noting that despite the fact that under our Assumptions f t, yit and xitare I(1) and cointegrated, implying that "it is an I(0) process, in the results of Theorem 3, the

rate of convergence of bi to �i as (N; T )j!1 is

pT and not T . It is helpful to develop some

intuition behind this result. Since for N su¢ ciently large f t can be well approximated by the

cross section averages, for pedagogic purposes we might as well consider the case where f t is

observed. Without loss of generality we also abstract from dt, and substitute (2) in (1) to obtain

yit = �0i (�

0if t + vit) +

0if t + "it = #

0if t + �it; (51)

where #i = �i�i+ i and �it = "it+�0ivit. First, it is clear that under our assumptions and for

all values of �i, �it is I(0) irrespective of whether f t is I(0) or I(1). But if f t is I(1), since

�it v I(0), then yit will also be I(1) and cointegrated with f t. Hence, it follows that #i can be

estimated superconsistently. However, the OLS estimator of �i need not be superconsistent. To

see this note that �i can be estimated equivalently by regressing the residuals from the regressions

of yit on f t on the residuals from the regressions of xit on f t. Both these sets of residuals are

stationary processes and the resulting estimator of �i will be at mostpT -consistent.

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Remark 7 An issue related to the above remark concerns the probability limit of the OLS

estimator of the coe¢ cients of xit in a regression of yit on xit alone. In general, such a

regression will be subject to the omitted variable problem and hence misspeci�ed. Also the

asymptotic properties of such OLS estimators can not be derived without further assumptions.

However, there is a special case which illustrates the utility of our method. Abstracting from dt,

assuming that k = m and that �i is invertible, and similarly to (51) write the model for yit as

yit = �0ixit +

0i�0�1i (xit � vit) + "it = %

0ixit + &it (52)

where %0i = �0i +

0i�0�1i and &it = "it � 0i�0�1i vit. Note that &it is, by construction, correlated

with vit. The question is whether estimating a regression of the form (52) provides a consistent

estimate of %i. For stationary processes this would not be case due to the correlation between

&it and vit. However, in the case of nonstationary data this is not clear and consistency would

depend on the exact speci�cation of the model. Under the assumptions we have made in this

remark, the estimator of %i would be consistent. However, even in this case it is clear that the

application of the least squares method to (52) can only lead to a consistent estimator of %i and

not of �i. To consistently estimate the latter we need to augment the regressions of yit on xitwith their cross-section averages.

4 Monte Carlo Design and Evidence

In this section we provide Monte Carlo evidence on the small sample properties of the CCEMG

and the CCEP estimators, which are de�ned by (14) and (20), respectively. We consider nine

alternative estimators. The �rst one is the CupBC estimator proposed by Bai, Kao, and Ng

(2009), which is a bias-corrected version of a continuously-updated estimator that estimates

both the slope parameters and the unobserved factors iteratively. The CupBC estimator, as

analysed by Bai, Kao, and Ng (2009), assumes the number of unobserved factors is known

and only considers the case where the slopes are homogeneous.4 In addition, we consider two

alternative principal component augmentation approaches discussed in Kapetanios and Pesaran

(2007). The �rst PC approach applies the Bai and Ng (2002) procedure to zit = (yit;x0it)0 to

obtain consistent estimates of the unobserved factors, and then uses the estimated factors

to augment the regression (1), and thus produces consistent estimates of �. We consider

both pooled and mean group versions of this estimator which we refer to as PC1POOL and

PC1MG. The second PC approach begins with extracting the principal component estimates

of the unobserved factors from yit and xit separately. In the second step yit and xit are regressed

4See Bai, Kao, and Ng (2009), for more details.

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on their respective factor estimates, and in the third step the residuals from these regressions

are used to compute the standard pooled and mean group estimators, with no cross-sectional

dependence adjustments. We refer to the estimators based on this approach as PC2POOL and

PC2MG, respectively. On top of these principal component estimators, we consider two sets

of benchmark estimators. The �rst set consists of infeasible mean group and pooled estimators,

which are obtained assuming the factors are observable (i.e. �zt for the CCE estimators is

replaced by true factor f t). The other set consists of naive mean group and pooled estimators,

which ignore the factor structure. The naive estimators are expected to illustrate the extent of

bias and size distortions that can occur if the error cross section dependence that induced by

the factor structure is ignored.

We report summaries of the performance of the estimators in the Monte Carlo experiments

in terms of average biases, root mean square errors, and rejection probabilities of the t-test for

slope parameters under both the null hypothesis and an alternative hypothesis. For computing

the t-statistics, the standard errors of mean group and pooled CCE estimators are estimated

using (38) and (42), respectively. The standard errors of PC1, PC2, infeasible and naive

estimators are estimated similarly to those of the CCE estimators. The standard errors of the

CupBC estimator is computed following Bai, Kao, and Ng (2009).

4.1 Baseline Design

The experimental design of the Monte Carlo study closely follows the one used in Pesaran

(2006). Consider the data generating process (DGP):

yit = �i1d1t + �i1x1it + �i2x2it + i1f1t + i2f2t + "it, (53)

and

xijt = aij1d1t + aij2d2t + ij1f1t + ij3f3t + vijt, j = 1; 2; (54)

for i = 1; 2; :::; N , and t = 1; 2; :::; T . This DGP is a restricted version of the general linear

model considered in Pesaran (2006), and sets n = k = 2, and m = 3, with �0i = (�i1; 0);

�0i = (�i1; �i2), and 0i = ( i1; i2; 0); and

A0i =

�ai11 ai12ai21 ai22

�; �0i =

� i11 0 i13 i21 0 i23

�:

The observed common factors and the individual-speci�c errors of xit are generated as inde-

pendent stationary AR(1) processes with zero means and unit variances:

d1t = 1; d2t = �dd2;t�1 + vdt; t = �49; :::1; :::; T ,

vdt � IIDN(0; 1� �2d), �d = 0:5; d2;�50 = 0;

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vijt = �vijvijt�1 + {ijt; t = �49; :::1; :::; T;

{ijt � IIDN�0; 1� �2vij

�, vji;�50 = 0;

and

�vij � IIDU [0:05; 0:95] ; for j = 1; 2:

But the unobserved common factors are generated as non-stationary processes:

fjt = fjt�1 + vfj;t, for j = 1; 2; 3, t = �49; ::; 0; ::; T; (55)

vfj;t � IIDN(0; 1); fj;�50 = 0, for j = 1; 2; 3:

The �rst 50 observations are discarded.

To illustrate the robustness of the CCE estimators and others to the dynamics of the

individual-speci�c errors of yit, these are generated as the (cross sectional) mixture of stationary

heterogeneous AR(1) and MA(1) errors. Namely,

"it = �i""i;t�1 + �i

q1� �2i"!it, i = 1; 2; :::; N1, t = �49; ::; 0; ::; T;

and

"it =�ip1 + �2i"

(!it + �i"!i;t�1) , i = N1 + 1; :::; N , t = �49; ::; 0; ::; T;

where N1 is the nearest integer to N=2,

!it � IIDN (0; 1) , �2i � IIDU [0:5; 1:5] , �i" � IIDU [0:05; 0:95] , �i" � IIDU [0; 1] .

�vij, �i", �i" and �i are not changed across replications. The �rst 49 observations are discarded.

The factor loadings of the observed common e¤ects, �i1 and vec(Ai) = (ai11; ai21; ai12; ai22)0 are

generated as IIDN(1; 1) and IIDN(0:5� 4; 0:5 I4) with � 4 = (1; 1; 1; 1)0, respectively, which

are not changed across replications. The parameters of the unobserved common e¤ects in the

xit equation are generated independently across replications as

�0i =

� i11 0 i13 i21 0 i23

�� IID

�N (0:5; 0:50) 0 N (0; 0:50)N (0; 0:50) 0 N (0:5; 0:50)

�.

For the parameters of the unobserved common e¤ects in the yit equation, i, we considered

two di¤erent sets that we denote by A and B. Under set A, i are drawn such that the rankcondition is satis�ed, namely

i1 � IIDN (1; 0:2) ; i2A � IIDN (1; 0:2) ; i3 = 0;

and

E�~�iA

�= (E ( iA) ; E (�i)) =

0@ 1 0:5 01 0 00 0 0:5

1A :

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Under set B i1 � IIDN (1; 0:2) ; i2B � IIDN (0; 1) ; i3 = 0;

so that

E�~�iB

�= (E ( iB) ; E (�i)) =

0@ 1 0:5 00 0 00 0 0:5

1A ;

and the rank condition is not satis�ed. For each set we conducted two di¤erent experiments:

� Experiment 1 examines the case of heterogeneous slopes with �ij = 1 + �ij; j = 1; 2,

and �ij � IIDN(0; 0:04), across replications.

� Experiment 2 considers the case of homogeneous slopes with �i = � = (1; 1)0.

The two versions of experiment 1 will be denoted by 1A and 1B, and those of experiment2 by 2A and 2B.Concerning the infeasible pooled estimator, it is important to note that although this esti-

mator is unbiased under all the four sets of experiments, it need not be e¢ cient since in these

experiments the slope coe¢ cients, �i, and/or error variances, �2i , di¤er across i. As a result

the CCE or PC augmented estimators may in fact dominate the infeasible estimator in terms

of RMSE, particularly in the case of experiments 1A and 1B where the slopes as well as theerror variances are allowed to vary across i.

Another important consideration worth bearing in mind when comparing the CCE and the

principal component type estimators is the fact that the computation of the CupBC, PC1 and

PC2 estimators assumes thatm = 3; namely that the number of unobserved factors is known.

In practice, m might be di¢ cult to estimate accurately particularly when N or T happen to

be smaller than 50. By contrast the CCE type estimators are valid for any �xed m and do not

require an a prior estimate for m.

Each experiment was replicated 2000 times for the (N; T ) pairs withN; T = 20; 30; 50; 100; 200.

In what follows we shall focus on �1 (the cross section mean of �i1) and the results for �2, which

are very similar to those for �1, will not be reported. The results for all the estimators consid-

ered are reported in Tables 1. Since the performance of CCE and CupBC estimators dominates

other feasible estimators in most of the designs considered, to save space we do not report the

results of these estimators for the remaining experiments.

4.2 Designs for Robustness Checks

In this subsection we consider a number of Monte Carlo experiment designs that aim to check

the robustness of the estimators to a variety of empirical settings.

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4.2.1 The Number of Factors Exceeds k + 1

In order to show the e¤ect of a di¤erent type of violation of the rank condition from experiment

B, we consider the DGP 1A but an extra factor term i4f4t is added to the right hand side

of the y equation (53), where i4 � IIDN(0:5; 0:2), f4t = f4t�1 + vf4;t, vf4;t � IIDN(0; 1),

f4;�50 = 0. In this case, observe that

E( i;�i)0 =

0@ 1 1 0 0:50:5 0 0 00 0 0:5 0

1Awhose rank is k+1 = 3, which is less than the number of unobserved factors, m = 4. Under this

experiment the number of factors is treated as unknown and estimated, using the information

criterion �PCp2�which is proposed by (Bai and Ng, 2002, pp. 201).5 The information criterion

is applied to the �rst di¤erenced variables with the maximum number of factors set to six. The

results are reported in Table 5. However, recall that the CCE type estimators does not make

use of the number of the factors and is valid irrespective of whether k + 1 is more or less than

m.

4.2.2 Cointegrating Factors

In this design the unobserved common factors are generated as cointegrated non-stationary

processes. There are two underlying stochastic trends given by

f tjt = f tjt�1 + vtfj;t, for j = 1; 2, t = �49; ::; 0; ::; T; (56)

vtfj;t � IIDN(0; 1); f tj;�50 = 0, for j = 1; 2:

Then, this experiment uses the same design as 1A, but the I(1) factors in (53) and (54) arereplaced by

f1t = f t1t + 0:5ft2t + vf1;t, t = �49; ::; 0; ::; T;

f2t = 0:5ft1t + f t2t + vf2;t, t = �49; ::; 0; ::; T;

f3t = 0:75ft1t + 0:25f

t2t + vf3;t, t = �49; ::; 0; ::; T;

vfj;t � IIDN(0; 1); fj;�50 = 0, for j = 1; 2; 3:

The �rst 50 observations are discarded. The results are reported in Table 6.

5PCp2 is one of the information criteria which performed well in the �nite sample investigations reported inBai and Ng (2002).

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4.2.3 Semi-Strong Factor Structure

Chudik, Pesaran, and Tosetti (2010) introduce the notions of weak, semi-strong and strong

factor structures and prove that these di¤erent factor structures do not a¤ect the consistency

of the CCE type estimators with I(0) factors. Here we consider the e¤ect of having semi-strong

factor structure when the factors are I(1). For this purpose, the same DGP of the experiment

1A is used, but all factor loadings in (53) and (54) are multiplied by N�1=2. The results are

reported in Table 7. It is easily seen that when the factors are weak or semi-strong they can

not be consistently estimated by the principal components and this could adversely impact the

estimators of � that rely on the PC�s as estimators of the unobserved factors.

4.2.4 A Structural Break in the Means of the Unobserved Factors

Finally, the results of recent research by Stock and Watson (2008) suggest that the possible

structural breaks in the means of the unobserved factors will not a¤ect the consistency of the

CCE type estimators, as well as the principal component type estimators. In view of this, we

considered another set of experiments, corresponding to the DGPs speci�ed as 1A, but now theunobserved factors are generated subject to mean shifts. Speci�cally, under these experiments

the unobserved factors are generated as fjt = 'jt for t < [2T=3] and fjt = 1+'jt for t � [2T=3]with [A] being the greatest integer less than or equal to A, where 'jt = 'j;t�1 + �jt, and

�jt � IIDN(0; 1), for j = 1; 2; 3: Results are reported in Table 8.

4.3 Results

Results of experiments 1A, 2A, 1B, 2B are summarized in Tables 1 to 4, respectively. We alsoprovide results for the naive estimator (that excludes the unobserved factors or their estimates)

and the infeasible estimator (that includes the unobserved factors as additional regressors) for

comparison purposes. But for the sake of brevity we include the simulation results for these

estimators only for experiment 1A.As can be seen from Table 1 the naive estimator is substantially biased, performs very poorly

and is subject to large size distortions; an outcome that continues to apply in the case of other

experiments (not reported here). In contrast, the feasible CCE estimators perform well, have

bias that are close to the bias of the infeasible estimators, show little size distortions even for

relatively small values of N and T , and their RMSE falls steadily with increases in N and/or

T . These results are quite similar to the results presented in Pesaran (2006), and illustrate

the robustness of the CCE estimators to the presence of unit roots in the unobserved common

factors. This is important since it obviates the need for pre-testing of unobserved common

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factors for the possibility of non-stationary components.

The CCE estimators perform well, in both heterogeneous and homogeneous slope cases, and

irrespective of whether the rank condition is satis�ed, although the CCE estimators with rank

de�ciency have sightly higher RMSEs than those under the full rank condition. The RMSEs of

the CCE estimators of Tables 1 and 3 (heterogeneous case) are higher than those reported in

Tables 2 and 4 for the homogeneous case. The sizes of the t-test based on the CCE estimators

are very close to the nominal 5% level. In the case of full rank, the power of the tests for the

CCE estimators are much higher than in the rank de�cient case. Finally, not surprisingly the

power of the tests for the CCE estimators in the homogeneous case is higher than that in the

heterogeneous case.

It is also important to note that the small sample properties of the CCE estimator does not

seem to be much a¤ected by the residual serial correlation of the idiosyncratic errors, "it. The

robustness of the CCE estimator to the short run dynamics is particularly helpful in practice

where typically little is known about such dynamics. In fact a comparison of the results for

the CCEP estimator with the infeasible counterpart given in Table 1 shows that the former

can even be more e¢ cient (in the RMSE sense). For example the RMSE of the CCEP for

N = T = 50 is 3.97 whilst the RMSE of the infeasible pooled estimator is 4.31. This might

seem counter intuitive at �rst, but as indicated above the infeasible estimator does not take

account of the residual serial correlation of the idiosyncratic errors, but the CCE estimator does

allow for such possibilities indirectly through the use of the cross section averages that partly

embody the serial correlation properties of f t and "it�s.

Consider now the PC augmented estimators and recall that they are computed assuming

the true number of common factors is known. The results in Table 1 bear some resemblance to

those presented in Kapetanios and Pesaran (2007). The bias and RMSEs of the PC1POOL and

PC1MG estimators improve as both N and T increase, but the t-tests based on these estimators

substantially over-reject the null hypothesis. The PC2POOL and PC2MG estimators perform

even worse. The biases of the PC estimators are always larger in absolute value than the

respective biases of the CCE estimators. The size distortion of the PC augmented estimators is

particularly pronounced. Finally, it is worth noting that the performance of the PC estimators

actually gets worse when N is small and kept small but T rises. This may be related to the

fact that the accuracy of the factor estimates depends on the minimum of N and T .

Now consider the CupBC estimator and again recall that it is computed assuming the true

number of common factors is known. Let us begin with discussing results in the case in which

the rank condition is satis�ed, the results of which are reported in Tables 1 and 2. As is evident,

the average bias and RMSEs of CupBC estimator are comparable to those of CCE estimators.

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Because of this, the results of CCEMG, CCEP and CupBC estimators only are reported in

Table 2 onwards. In the case of heterogeneous slopes with the rank condition satis�ed, the

RMSEs of the CCE estimator are uniformly smaller than those of the CupBC estimator (as can

be seen from Table 1). This might be expected since the CupBC estimator is designed for the

model with homogeneous slopes. In the case of homogeneous slopes with the rank condition

satis�ed, as can be seen from Table 2, the RMSEs of CCEP estimator are smaller than those

of CupBC estimator when T is relatively small (T = 20 and 30). Turning our attention to the

performance of the t-test, it is apparent that the size of the test based on CupBC estimator is

far from the nominal level across all experiments. This is especially so for experiments where

the slopes are heterogeneous. In these cases, increases in N and T do not seem to help to

improve test performance. Even for homogeneous slope cases, the best rejection probability

result is 14.90% for T = N = 200 in Table 2. In contrast, the size of the t-test based on the

CCE estimators is close to 5% nominal level across all experiments. Tables 3 and 4 provide

the summary of experimental results in the rank de�cient case. For this design, even though

the size of the t-test based on the CupBC estimator is grossly oversized, the RMSEs of the

estimator are smaller than those of CCE estimators. However, note that in these experiments

the number of factors are treated as known, which is rarely expected in practical situation. We

return to this issue below.

Tables 5-8 report the results of the experiments carried out as robustness checks. 6 Table 5

reports the results of the experiments where the number of unobserved factors is four (m = 4)

which exceeds k + 1 = 3, in the case of heterogeneous slopes. In this experiment, CupBC

estimates are obtained supposing that m is unknown but estimated using the information

criterion PCP2, which is proposed by Bai and Ng (2002), applied to the �rst-di¤erences of

(yit; x1it; x2it). We set the maximum number of factors to six.7 Firstly, despite the number of

unobserved factors, m = 4; exceeding the number of regressors and regressand (k+ 1 = 3), the

RMSEs of CCE estimators decrease as N and T are increased, which con�rms the consistency

of the estimators in the rank de�cient case. Furthermore, the RMSEs of CCE estimators

dominate those of the CupBC estimator, except only when T is very large (� 100). We notethat, although not reported for brevity, the size of the t-test based on CCE estimators is very

close to the nominal 5% level, whilst the size distortion of the CupBC estimators is acute for

all cases considered. Tables 6, 7 and 8 report the results of experiments with the same DGP as

6For brevity the size and power of t-tests are not reported in Tables 5-8, since they are qualitatively similarto those in Tables 1-4. For similar reasons, the results for homogeneous slopes and/or rank de�cient cases (forTables 6-8) are not reported. A full set of results is available upon request from the authors.

7For small N and T the information criterion tends to over-estimate the number of the factors in the �rst-di¤erenced data (yit; x1it; x2it), and the estimates tend to four as N and T get larger.

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in Table 1 but where the unobserved factors are cointegrated, factor structures are semi-strong,

and the unobserved factors are subject to mean shifts, respectively. In all of these designs the

CCE estimators uniformly dominate the CupBC estimator in terms of both RMSEs (and the

size of the t-test, which is not reported in the tables). These are consistent with the �ndings

of Chudik, Pesaran, and Tosetti (2010) and Stock and Watson (2008).

5 Conclusions

Recently, there has been increased interest in analysis of panel data models where the standard

assumption that the errors of the panel regressions are cross-sectionally uncorrelated is violated.

When the errors of a panel regression are cross-sectionally correlated then standard estimation

methods do not necessarily produce consistent estimates of the parameters of interest. An in-

�uential strand of the relevant literature provides a convenient parameterisation of the problem

in terms of a factor model for the error terms.

Pesaran (2006) adopts an error multifactor structure and suggests new estimators that

take into account cross-sectional dependence, making use of cross-sectional averages of the

dependent and explanatory variables. However, he focusses on the case of weakly stationary

factors that could be restrictive in some applications. This paper provides a formal extension

of the results of Pesaran (2006) to the case where the unobserved factors are allowed to follow

unit root processes. It is shown that the main results of Pesaran continue to hold in this

more general case. This is certainly of interest given the fact that usually there are major

di¤erences between results obtained for unit root and stationary processes. When we consider

the small sample properties of the new estimators, we observe that again the results accord

with the conclusions reached in the stationary case, lending further support to the use of the

CCE estimators irrespective of the order of integration of the data observed. The Monte Carlo

experiments also show that the CCE type estimators are robust to a number of important

departures from the theory developed in this paper, and in general have better small sample

properties than alternatives that are available in the literature. Most importantly the tests

based on CCE estimators have the correct size whilst the factor-based estimators (including

the one recently proposed by Bai, Kao, and Ng (2009)) show substantial size distortions even

in the case of relatively large samples.

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Kapetanios, G., and M. H. Pesaran (2007): �Alternative Approaches to Estimation and In-ference in Large Multifactor Panels: Small Sample Results with an Application to Modelling ofAsset Returns,�in The Re�nement of Econometric Estimation and Test Procedures: Finite Sampleand Asymptotic Analysis, ed. by G. D. A. Phillips, and E. Tzavalis. Cambridge University Press,Cambridge.

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Appendix A

LemmasProofs of Lemmas are provided in Supplementary Appendix A.

Lemma 1 Under Assumptions 1-4,

�U0 �U

T= Op

�1

N

�(A1)

V 0i�U

T= Op

�1

N

�+Op

�1pNT

�,"0i�U

T= Op

�1

N

�+Op

�1pNT

�; uniformly over i (A2)

F 0 �U

T= Op

�1pN

�,D0 �U

T= Op

�1pN

�(A3)

X 0i�U

T= Op

�1pN

�; uniformly over i (A4)

Q0 �U

T= Op

�1pN

�(A5)

Q0Q

T 2= Op (1) (A6)

Q0Xi

T 2= Op (1) , uniformly over i (A7)

Q0G

T 2= Op (1) (A8)

�H0 �H

T 2= Op(1) (A9)

�H0G

T 2= Op(1) (A10)

�H0"iT

= Op(1), uniformly over i (A11)

�H0V i

T= Op(1), uniformly over i (A12)

�H0Xi

T 2= Op(1), uniformly over i (A13)

�H0 �U

T= Op

�1pN

�: (A14)

Lemma 2 Under assumptions 1-4,

V 0i�U

T= Op

�1pTN

�+Op

�1

N

�uniformly over i: (A15)

Lemma 3 Under Assumptions 1-4 and assuming that the rank condition (9) holds, then

X 0i�MXi

T� X

0iMgXi

T= Op

�1pN

�; uniformly over i; (A16)

X 0i�M"iT

� X0iMg"iT

= Op

�1pNT

�+Op

�1

N

�; uniformly over i: (A17)

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Lemma 4 Assume that the rank condition (9) holds. Then, under Assumptions 1-4

X 0i�MF

T= Op

�1

N

�+Op

�1pNT

�, uniformly over i: (A18)

Lemma 5 Under Assumptions 1-4,

X 0iMgXi

T��vi = Op

�1pT

�:

Lemma 6 Under Assumptions 1-4 and assuming that the rank condition (9) does not hold, then

X 0i�MXi

T� X

0iM qXi

T= Op

�1pN

�; uniformly over i; (A19)

X 0i�MF

T� X

0iM qF

T= Op

�1pN

�; uniformly over i; (A20)

X 0i�M"iT

� X0iM q"iT

= Op

�1pNT

�+Op

�1

N

�; uniformly over i: (A21)

Lemma 7 Under Assumptions 1-4 and assuming that the rank condition (9) does not hold,�X 0i�MF

T

��C = Op

�1

N

�+Op

�1pNT

�, uniformly over i: (A22)

Appendix B: Proofs of theorems for pooled estimators

Proof of Theorem 1We know that

�C =

� + ���+

1

N

NXi=1

�i{i; ��

!;

where �� = 1N

PNi=1 �i and � =

1N

PNi=1 i. Substituting this result in (A22) now yields�

X 0i�MF

T

� � + ���+

1

N

NXi=1

�i{i

!= Op

�1

N

�+Op

�1pNT

�, uniformly over i;�

X 0i�MF

T

��� = Op

�1

N

�+Op

�1pNT

�, uniformly over i;

which in turn yieldspNX 0

i�MF

T

� +

1

N

NXi=1

�i{i

!= Op

�1pN

�+Op

�1pT

�, uniformly over i:

But under Assumption 4, 1N

PNi=1 �i{i = Op

�N�1=2�, and therefore

pN�X 0i�MF

��

T= Op

�1pN

�+Op

�1pT

�, uniformly over i: (A23)

We next reconsider the second term on the RHS of (35), which is the only term a¤ected by the fact that rankcondition does not hold. The second term on the RHS in (35) can be written as

�NT �1

N

NXi=1

�X 0i�MXi

T 2

�+ pNX 0

i�MF

T 2

!(� + �i � ��) ; (A24)

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where �� = 1N

PNi=1 �i. By (A19) and (A20) it follows that

�NT �1

N

NXi=1

�X 0iM qXi

T 2

�+ pNX 0

iM qF

T 2

!(� + �i � ��) +Op

�1pN

�: (A25)

Note that for the above two expressions, we have changed the normalisation from T to T 2. This is because inthe case where the rank condition does not hold, the use of cross-sectional averages is not su¢ cient to removethe e¤ect of the I(1) unobserved factors and so X 0

i�MXi, X

0i�MF , X 0

iM qXi and X0iM qF would involve

nonstationary components. Then, since by (A23),pN(X0

i�MF )�

T 2 = Op

�1

TpN

�+ Op

�1

T 3=2

�, uniformly over i; it

is the case that for N and T large

pN�bMG � �

�d� 1p

N

NXi=1

{i +1pN

NXi=1

�X 0iM qXi

T 2

�+�X 0iM qF

T 2

�(�i � ��) : (A26)

We next focus on analysing the RHS of (A26). The �rst term on the RHS of (A26) tends to a Normaldensity with mean zero and �nite variance. The second term needs further analysis. Letting

Q1iT =

�X 0iM qXi

T 2

�+�X 0iM qF

T 2

�and �Q1T =

1N

PNi=1Q1iT , we have that

1pN

NXi=1

Q1iT (�i � ��) =1pN

NXi=1

�Q1iT � �Q1T

��i: (A27)

We note that �i is i.i.d. with zero mean and �nite variance and independent of all other stochastic quantitiesin the second term of the RHS on (A27). We de�ne

Q1iT;�i =

�X 0iM q;�iXi

T 2

�+�X 0iM q;�iF

T 2

�

and �Q1T;�i =1N

PNi=1Q1iT;�i, whereM q;�i = IT�Q�i

�Q0�iQ�i

�+Q0�i,Q�i = G �P�i, �P�i =

�In �B�i0 �C�i

��B�i =

1N

PNj=1;j 6=iBj and �C�i =

1N

PNj=1;j 6=iCj . Then, it is straightforward that�

Q1iT � �Q1T

���Q1iT;�i � �Q1T;�i

�= Op

�1

N

�, uniformly over i;

and1pN

NXi=1

�Q1iT � �Q1T

��i �

1pN

NXi=1

�Q1iT;�i � �Q1T;�i

��i = Op

�1

N1=2

�:

Then, it is easy to show that if zTi = xiyTi, xi is an i.i.d. sequence with zero mean and �nite variance and yTiis a triangular array of random variables with �nite variance then zTi is a martingale di¤erence triangular arrayfor which a central limit theorem holds (see, e.g., Theorem 24.3 of Davidson (1994)). But this is the case here,for any ordering over i, setting yTi =

�Q1iT;�i � �Q1T;�i

�and xi = �i. Using this result, it follows that the

second term on the RHS of (A26) tends to a Normal density if�Q1iT � �Q1T

��i has variance with �nite norm,

uniformly over i, denoted by �iqT , i.e.

�iqT = V ar��Q1iT � �Q1T

��i�: (A28)

In order to establish the existence of second moments, it is su¢ cient to prove that �Q1iT � �Q1T

� , or equiv-alently

�Q1iT;�i � �Q1T;�i� , has �nite second moments. We carry out the analysis for �Q1iT � �Q1T

� . Forthis, we need to provide further analysis of X

0iMqXi

T 2 and X0iMqFT 2 . First, note that Xi can be written as

Xi = QBi1 + SBi2 + V i; (A29)

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where S is the T �m� k � 1 dimensional complement of Q, i.e. Q and S are orthogonal and

F = QK1 + SK2; (A30)

where K1 and K2 are full row rank matrices of constants with bounded norm. Note that if m < 2k + 1, weassume, without loss of generality, that Bi2 has full row rank whereas if m � 2k+1, Bi2 has full column rank.Then,

X 0iM qXi =X

0iM q (QBi1 + SBi2 + V i) =X

0iM qSBi2 +X

0iM qV i =

B0i2S

0M qSBi2 + V0iM qV i +B

0i2S

0M qV i + V0iM qSBi2:

But, it easily follows thatV 0iM qV i

T 2= Op

�1

T

�, uniformly over i;

andB0i2S

0M qV i

T 2= Op

�1

T

�, uniformly over i:

Then,X 0iM qXi

T 2= B0

i2

S0S

T 2Bi2 +Op

�1

T

�, uniformly over i: (A31)

Similarly, using (A30),X 0iM qF

T 2= B0

i2

S0S

T 2K2 +Op

�1

T

�, uniformly over i:

Thus �X 0iM qXi

T 2

�+�X 0iM qF

T 2

�=

�B0i2

S0S

T 2Bi2

�+�B0i2

S0S

T 2K2

�+Op

�1

T

�, uniformly over i:

We need to distinguish between two cases. In the �rst case, m � 2k + 1. Then, it is easy to see that X0iMqXi

T 2

and B0i2S0ST 2 Bi2 have an inverse. Then, by Assumption 7(ii)

�Q1iT � �Q1T

� has �nite second moments. Thecase where m < 2k + 1 is more complicated. Denoting � = T�2S0S and ~Bi2 = �

1=2Bi2, we have

B0i2

S0S

T 2Bi2 = ~B

0i2~Bi2:

Then, noting that�~B0i2~Bi2

�+= ~B

+

i2~B0+

i2 and since in this case Bi2 has full row rank then

~B+

i2 = B0i2

�Bi2B

0i2

��1��1=2;

and we obtain �B0i2

S0S

T 2Bi2

�+= B0

i2

�Bi2B

0i2

��1�S0ST 2

��1 �Bi2B

0i2

��1Bi2: (A32)

Hence �X 0iM qXi

T 2

�+�X 0iM qF

T 2

�= B0

i2

�Bi2B

0i2

��1K2 +Op

�1

T

�, uniformly over i;

and the required result now follows by the boundedness assumption for Bi2 and K2. The assumption that Bi2

has full row rank if m < 2k+1 implies that the whole of S enters the equations for Xi. If that is not the casethen the argument above has to be modi�ed as follows: We have that

Xi = QBi1 + S1Bi2 + V i;

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where S1 is a subset of S. Then,

X 0iM qXi

T 2= B0

i2

S01S1T 2

Bi2 +Op

�1

T

�, uniformly over i:

and the analysis proceeds as above until�X 0iM qXi

T 2

�+�X 0iM qF

T 2

�= B0

i2

�Bi2B

0i2

��1�S01S1T 2

��1�S01S

T 2

�K2 +Op

�1

T

�, uniformly over i:

Then, the required result follows by Assumption 7(iii) which implies that E

�S01S1T 2

��1 <1 and E S01ST 2 <

1, and the boundedness assumption for Bi2 and K2.Thus, in general we have that

pN�bMG � �

�d! N(0;�MG); as (N;T )

j!1;

where�MG = { +�; (A33)

and

� = limN;T!1

"1

N

NXi=1

�iqT

#: (A34)

To complete the proof we have to consider two further issues. First we note that in (A26), we disregard a term

involving�X0

iMqXi

T 2

�+ �X0

iMq"iT

�. In particular we have to prove that

�1

T

�1pN

NXi=1

�X 0iM qXi

T 2

�+�X 0iM q"iT

�= Op

�1

T

�: (A35)

For this, it is enough to show that 1pN

PNi=1

�X0

iMqXi

T 2

�+ �X0

iMq"iT

�follows a CLT. This follows if (i) for any

ordering of the cross-sectional units, X0iMq"iT is a martingale di¤erence and (ii)

�X0

iMqXi

T 2

�+ �X0

iMq"iT

�has

�nite second moments. (ii) follows easily from the argument made in other parts of the appendix about the

existence of moments of�X0

iMqXi

T 2

�+ �X0

iMqFT

�. Then, one has to simply prove (i). We need to show that for

any orderingE(Q�i jQ�i�1) = 0; (A36)

where Q�i =�X0

iMqXi

T 2

�+ �X0

iMq"iT

�: Denote Q��i =

�X0

iMqXi

T 2

�+: Then Q�i = Q

��i

�X0

iMq"iT

�. Now X0

iMq"iT =

1T

PTt=1 st"it where st is a unit root process (cf. the de�nition of S in (A29) above). Then, for (A36) to hold it

is su¢ cient to note that for all t,l, E(Q��i st"itjQ��i sl"i�1l) = 0. This completes the proof of (A35).Finally, we need to show that the variance estimator given by

�MG =1

N � 1

NXi=1

�bi � bMG

��bi � bMG

�0, (A37)

which is consistent. To see this �rst note that

bi � � = {i + hiT +Op�1pN

�+Op

�1pT

�, uniformly over i; (A38)

where

hiT =

�X 0i�MXi

T 2

�+X 0i�M [F (�i � ��) + "i]

T 2; (A39)

31

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and so

bi � bMG = ({i � �{) +�hiT � �hT

�+Op

�1pN

�+Op

�1pT

�, uniformly over i; (A40)

where �hT = 1N

PNi=1 hiT . Since by assumption {i and hiT are independently distributed across i, then

1

N � 1

NXi=1

�bi � bMG

��bi � bMG

�0= �MG +Op

�1pN

�+Op

�1pT

�;

and the desired result follows.

Proof of Theorem 2As before the pooled estimator, bP , de�ned by (20), can be written as

pN�bP � �

�=

1

N

NXi=1

X 0i�MXi

T 2

!�1 "1pN

NXi=1

X 0i�M(Xi{i + "i)

T 2+ qNT

#; (A41)

where

qNT =1pN

NXi=1

�X 0i�MF

� i

T 2: (A42)

Assuming random coe¢ cients we note that i = � + �i � ��, where �� = 1N

PNi=1 �i. Hence

qNT =1

N

NXi=1

pNX 0

i�MF

T 2

!� +

1pN

NXi=1

�X 0i�MF

T 2

�(�i � ��) :

But by (A23), the �rst component of qNT is Op�

1TpN

�+ Op

�1

T 3=2

�. Substituting this result in (A41), and

making use of (33) and (34) we have

pN�bP � �

�=

1

N

NXi=1

X 0iM qXi

T 2

!�1 "1pN

NXi=1

X 0iM q(Xi{i + "i + F (�i � ��))

T 2

#+ (A43)

Op

�1

TpN

�+Op

�1

T 3=2

�:

Also by Assumption 7, when the rank condition is not satis�ed, 1N

PNi=1

X0iMqXi

T 2 is nonsingular. Further, by(A31),

1

N

NXi=1

X 0iM qXi

T 2=1

N

NXi=1

B0i2

S0S

T 2Bi2 +Op

�1

T

�:

We note that, by assumption 3, Bi2 is an i.i.d. sequence with �nite second moments. Further, by Assumption 7,

it follows that E S0ST 2 2 <1. Hence, T�2B0

i2S0SBi2 forms asymptotically a martingale di¤erence triangular

array with �nite mean and variance and, as a result, T�2B0i2S

0SBi2 obeys the martingale di¤erence triangulararray law of large numbers across i, (see, e.g., Theorem 19.7 of Davidson (1994)) and, therefore, its mean tendsto a nonstochastic limit which we denote by �, i.e.

� = limN;T!1

1

N

NXi=1

�iT

!; (A44)

where�iT = E

�T�2B0

i2S0SBi2

�: (A45)

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But, by similar arguments to those used for the mean group estimator in the case when the rank condition doesnot hold, we can show that

1pN

NXi=1

X 0iM qXi

T 2{i

d! N (0;�) ;

where

� = limN;T!1

1

N

NXi=1

�Ti

!; (A46)

and �Ti = V ar�T�2X 0

iM qXi{i�: Further, by independence of "i across i,

1pN

NXi=1

X 0iM q"iT 2

= Op

�1

T

�:

Further, letting Q2iT = T�2X 0

iM qF and �Q2T =1N

PNi=1Q2iT , we have

1pN

NXi=1

�X 0iM qF

T 2

�(�i � ��) =

1pN

NXi=1

�Q2iT � �Q2T

��i:

Then, similarly to the analysis used above for T�2X 0iM qXi, we have

1pN

NXi=1

�Q2iT � �Q2T

��i

d! N (0;�)

where

� = limN;T!1

1

N

NXi=1

�Ti

!(A47)

and�Ti = V ar

��Q2iT � �Q2T

��i�: (A48)

Thus, overall by the independence of {i and �i, it follows thatpN�bP � �

�d! N(0;��P ); as (N;T )

j!1; (A49)

where��P = �

�1 (�+�)��1 (A50)

proving the result for the pooled estimator. The result for the consistency of the variance estimator followsalong similar lines to that for the mean group estimator.

33

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Table

1:

Sm

all

Sam

ple

Propertie

sof

Com

mon

Correla

ted

E¤ects

Type

Estim

ato

rs

inth

eC

ase

of

Experim

ent

1A

(Hete

rogeneous

Slo

pes

+Full

Rank)

Bia

s(£

100)

Root

Mean

Square

Errors

(£

100)

Siz

e(5%

level,

0:

1=

10

0)

Pow

er

(5%

level,

1:

1=

09

5)

(N,T

)20

30

50

100

200

20

30

50

100

200

20

30

50

100

200

20

30

50

100

200

CC

ET

ype

Est

imato

rs

CC

EM

G20

0.0

5-0

.10

-0.0

30.0

6-0

.07

9.6

77.8

96.7

45.8

75.5

47.2

06.9

07.1

57.9

07.5

511.6

513.0

016.1

017.5

020.1

030

0.0

9-0

.01

-0.0

1-0

.13

0.1

07.6

96.0

95.1

14.5

44.2

26.9

55.3

05.9

06.2

56.3

511.4

014.2

518.0

522.0

526.8

550

-0.1

90.2

2-0

.11

0.1

4-0

.04

5.8

84.6

14.0

13.4

43.1

35.7

05.0

56.6

56.2

05.9

515.1

020.4

025.6

034.1

036.6

5100

0.0

00.0

40.0

40.0

30.0

44.2

53.4

62.8

92.3

32.2

75.7

55.8

55.2

54.9

06.2

023.3

534.3

044.4

056.0

063.2

5200

-0.0

5-0

.02

-0.0

30.0

50.0

03.0

72.4

92.0

11.7

21.5

14.4

05.1

54.9

05.6

05.1

035.5

552.6

568.7

083.6

590.5

0C

CE

P20

0.1

80.0

0-0

.05

-0.0

1-0

.13

8.7

57.6

76.8

56.3

26.2

17.7

08.1

07.3

08.0

57.1

512.7

513.5

016.0

516.8

018.3

030

-0.1

7-0

.12

0.0

9-0

.15

0.1

37.1

05.9

95.3

24.7

84.4

67.5

56.2

56.7

56.6

56.4

512.4

015.0

019.3

020.6

526.9

050

0.0

00.1

8-0

.07

0.1

2-0

.01

5.3

34.5

13.9

73.4

73.2

26.8

06.2

05.9

06.3

56.4

517.4

522.1

526.4

032.9

036.2

5100

0.0

00.0

90.0

30.0

00.0

23.7

83.2

52.8

52.3

42.2

85.7

05.6

55.6

05.1

56.2

528.1

537.4

044.8

055.2

061.7

5200

-0.0

7-0

.04

-0.0

50.0

50.0

02.7

12.2

91.9

51.7

01.5

35.1

04.3

55.0

54.7

04.7

544.7

556.8

070.3

083.5

589.7

5B

ai,

Kao

and

Ng

Prin

cip

al

Com

ponent

Estim

ato

rC

upB

C20

0.6

20.7

00.8

10.7

70.8

711.1

69.8

68.3

57.4

66.9

567.2

564.4

057.9

060.9

565.7

572.0

568.7

565.0

068.5

074.6

030

0.3

50.4

20.7

30.5

90.8

38.9

17.7

06.5

15.6

65.2

866.8

061.0

555.9

555.4

063.3

071.8

569.9

566.3

570.7

577.6

050

0.5

30.6

70.3

30.6

30.5

46.7

76.0

15.0

54.2

03.8

364.4

558.8

551.9

551.3

556.7

077.2

072.6

569.5

576.9

083.4

5100

0.2

10.3

40.3

50.2

80.3

34.8

34.1

53.3

92.7

62.5

564.6

056.4

047.8

543.3

552.6

580.7

080.3

582.5

087.9

092.5

0200

0.1

00.1

00.0

80.2

30.1

73.5

52.9

42.4

52.0

01.6

962.8

552.8

545.0

044.5

048.0

086.6

588.7

090.7

596.6

099.1

0In

feasib

leE

stim

ato

rs

(inclu

din

g1

and2 )

Infe

asib

leM

G20

0.0

1-0

.19

-0.0

80.1

5-0

.08

7.2

16.3

35.6

24.9

84.7

66.4

06.2

06.8

05.9

56.5

012.7

515.3

516.8

519.7

020.4

030

0.0

2-0

.14

0.0

1-0

.02

0.1

25.9

14.9

54.4

33.9

73.8

76.5

05.8

06.0

55.3

05.9

016.1

518.0

523.3

525.2

028.8

050

-0.1

00.0

7-0

.06

0.1

4-0

.04

4.4

83.7

53.3

93.0

92.9

46.4

55.2

55.9

05.2

55.2

021.7

027.3

531.4

538.4

540.2

5100

0.0

10.0

70.0

20.0

00.0

43.1

62.7

82.4

92.1

52.1

45.5

05.1

55.4

54.7

05.4

536.8

546.1

555.1

062.5

066.6

5200

-0.0

70.0

4-0

.07

0.0

60.0

12.2

21.9

31.6

91.5

71.4

44.8

55.0

05.0

05.6

04.7

059.1

572.8

582.2

590.4

092.7

5In

feasib

leP

oole

d20

0.1

5-0

.13

-0.1

5-0

.26

-0.2

17.3

06.9

66.9

27.1

17.4

06.4

06.8

06.6

07.0

05.1

013.7

013.7

514.5

514.1

012.6

530

-0.2

0-0

.15

0.2

2-0

.07

0.2

76.2

35.7

85.7

95.8

96.6

17.0

55.9

07.0

05.2

55.7

015.7

015.3

518.9

516.7

016.6

050

0.1

20.0

7-0

.08

0.2

10.0

24.6

14.4

04.3

14.7

15.0

25.7

05.8

05.5

06.2

55.0

022.2

022.5

523.6

525.5

021.0

0100

-0.0

50.0

70.0

90.0

60.0

03.3

03.2

63.1

23.3

03.5

25.2

55.6

05.2

05.2

05.3

033.4

538.2

038.8

536.7

532.3

0200

-0.0

80.0

6-0

.12

0.0

7-0

.02

2.3

52.2

22.2

02.4

52.4

94.9

54.7

04.5

05.8

54.7

056.1

562.1

059.5

059.0

552.2

0N

aïv

eE

stim

ators

(exclu

din

g1

and2 )

Naïv

eM

G20

22.1

823.1

326.8

229.9

632.6

231.7

632.9

737.3

741.4

947.0

432.0

532.9

534.8

535.4

531.5

041.0

042.6

543.5

041.9

538.0

530

22.2

325.0

628.3

631.3

334.0

130.5

133.3

137.8

741.4

645.3

240.4

544.1

046.6

543.8

539.4

551.0

053.9

557.4

552.2

047.1

550

22.2

123.9

125.6

529.6

133.6

429.7

531.1

232.7

537.7

342.6

655.8

059.3

058.0

059.2

554.7

568.3

070.8

570.3

069.2

065.0

5100

21.9

723.9

226.7

630.0

432.8

828.4

030.0

232.9

736.3

940.0

671.2

075.2

577.9

078.6

075.2

581.0

584.3

585.9

585.8

583.2

0200

22.1

524.0

927.4

930.0

933.2

327.8

729.4

432.8

035.7

139.3

481.8

586.0

087.8

588.0

587.9

588.7

591.9

592.3

092.9

092.0

5N

aïv

eP

oole

d20

25.2

526.6

031.2

733.5

934.8

435.3

037.0

142.6

645.4

247.6

742.1

543.6

547.7

545.2

044.5

052.5

052.6

555.9

553.4

051.9

530

25.7

629.3

932.4

535.3

735.4

635.4

839.1

342.7

045.9

746.8

151.5

556.7

057.6

559.5

556.2

061.0

566.6

066.5

567.7

564.5

550

26.5

428.7

530.3

934.0

135.8

835.6

137.3

939.0

544.0

445.9

364.7

567.1

569.2

570.3

569.3

573.5

576.2

578.2

578.6

577.4

5100

25.8

128.4

731.3

033.1

534.9

134.3

936.7

639.9

041.7

944.2

775.8

578.9

081.3

579.3

080.1

585.1

086.5

588.0

586.6

586.4

0200

25.9

528.3

231.8

933.6

534.1

134.2

036.2

139.6

342.3

942.6

883.4

586.2

587.7

087.4

087.2

089.9

591.9

093.5

592.2

092.2

0

34

ACC

EPTE

DM

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

(Table

1C

ontin

ued)

Bia

s(£

100)

Root

Mean

Square

Errors

(£

100)

Siz

e(5

%le

vel,

0:

1=

10

0)

Pow

er

(5%

level,

1:

1=

09

5)

(N,T

)20

30

50

100

200

20

30

50

100

200

20

30

50

100

200

20

30

50

100

200

Prin

cip

al

Com

ponent

Estim

ato

rs,

Augm

ente

dP

C1M

G20

-12.2

7-1

1.1

5-1

0.3

0-8

.87

-8.9

017.0

914.8

113.2

411.5

111.5

522.5

525.3

530.0

533.4

037.4

012.1

512.9

513.3

012.7

013.7

530

-9.2

5-7

.86

-6.4

6-5

.72

-5.2

513.5

510.8

48.9

87.8

07.1

520.6

020.9

021.6

524.7

524.7

010.7

58.2

57.3

57.4

06.7

550

-6.8

4-5

.05

-3.8

9-3

.01

-3.1

210.1

07.7

95.8

64.6

74.4

719.9

517.6

516.2

514.9

517.9

08.7

08.2

07.6

511.4

09.7

5100

-4.7

8-3

.21

-2.0

3-1

.57

-1.4

57.4

45.3

43.6

82.8

72.7

220.1

016.8

011.4

59.7

511.1

09.5

512.1

520.2

528.8

536.7

5200

-4.3

1-2

.54

-1.3

9-0

.81

-0.7

86.3

94.1

92.6

01.9

31.7

125.2

017.9

510.9

58.1

57.6

513.8

521.9

542.8

567.6

577.1

5P

C1P

OO

L20

-11.9

7-1

1.0

4-1

0.3

5-9

.09

-9.2

315.8

814.3

813.0

711.5

912.0

725.5

028.3

532.0

534.4

538.9

512.0

514.1

014.9

014.5

514.9

030

-8.8

6-7

.66

-6.3

4-5

.73

-5.3

712.4

810.4

58.8

97.8

07.3

421.4

523.7

522.0

524.7

025.5

011.0

08.8

07.5

57.9

56.3

550

-6.2

0-4

.86

-3.8

1-3

.07

-3.1

99.0

67.5

25.7

24.7

34.5

421.4

018.7

516.0

016.0

518.9

08.5

59.5

58.1

010.9

09.6

5100

-4.3

6-3

.00

-2.0

1-1

.60

-1.4

96.6

15.0

13.6

12.8

82.7

421.0

516.8

511.2

59.3

510.8

011.2

514.5

520.8

527.9

036.3

0200

-3.6

2-2

.32

-1.3

6-0

.81

-0.7

95.3

93.8

12.5

11.9

11.7

325.1

517.6

010.5

07.8

07.8

016.3

526.7

545.4

568.0

076.1

5P

rin

cip

al

Com

ponent

Estim

ato

rs,

Orthogonalis

ed

PC

2M

G20

-31.2

6-2

7.0

6-2

4.0

1-2

2.6

7-2

3.1

132.8

328.3

425.0

023.4

423.8

386.5

088.4

591.2

595.2

097.4

074.1

073.9

575.8

082.0

588.2

030

-25.5

0-2

1.2

1-1

8.2

7-1

6.6

9-1

6.3

326.8

222.2

519.1

317.3

516.9

286.8

587.1

089.1

093.3

595.9

570.1

567.8

066.1

069.2

574.7

050

-20.6

5-1

6.2

3-1

3.3

2-1

1.4

1-1

0.8

921.6

817.0

613.9

811.9

511.3

790.1

588.3

588.8

089.0

591.7

070.8

060.2

552.2

045.8

046.1

0100

-16.1

7-1

2.4

4-9

.69

-7.6

1-6

.60

16.8

712.9

710.1

87.9

97.0

293.6

593.3

089.7

587.5

083.3

072.3

556.2

037.6

019.3

013.6

0200

-14.6

1-1

0.7

8-8

.12

-5.7

9-4

.59

15.1

111.1

98.4

56.0

84.8

598.9

597.8

595.4

590.7

583.7

579.6

560.2

033.3

010.0

06.7

5P

C2P

OO

L20

-31.9

7-2

7.4

7-2

4.2

7-2

3.1

8-2

4.1

933.3

928.6

925.2

323.9

924.9

991.0

090.7

093.2

095.5

598.5

080.6

578.6

078.8

083.3

590.4

530

-26.3

2-2

1.5

1-1

8.2

4-1

6.8

3-1

6.7

527.5

322.4

819.1

317.5

117.3

791.3

590.4

089.7

093.3

596.1

578.5

071.8

066.6

570.6

576.9

050

-21.2

2-1

6.3

5-1

3.1

7-1

1.3

5-1

0.9

922.1

017.1

513.8

211.9

111.4

895.0

590.9

088.9

588.2

091.7

079.6

563.8

052.9

546.2

048.2

5100

-16.7

7-1

2.5

2-9

.62

-7.5

5-6

.60

17.4

313.0

610.1

17.9

57.0

397.9

595.0

590.5

086.4

582.3

080.9

060.8

038.1

018.3

014.2

5200

-15.1

6-1

0.9

1-8

.00

-5.6

6-4

.53

15.6

711.3

38.3

45.9

64.7

999.7

598.4

595.9

589.3

582.5

088.6

565.8

533.3

58.4

06.3

0

Note

s:T

he

DG

Pis

=11

+11

+22

+11

+22

+

with

=

¡1

+ (1

¡2)12 ,

=

12

[2],

and

= (1

+

2)¡

12(

+

¡1),

=

[2]+

1

,

»

(01),

2

»

[0515

],

»

[00

509

5],

»

[01].

Regresso

rsare

gen

era

tedby

=11+22+11+33 +

v ,

=12,

for=

12

.1

=12

=052

¡1

+ ,

»

(01

¡05

2),2¡

50

=0;

=¡

1+ ,

»

(01),¡

50

=0,fo

r

=123;v

=v¡

1+ ,

»

(01

¡2),

v¡

50

=0

and

»

[00

509

5]fo

r

=12,fo

r=

¡49

with

the

…rst

50

observ

atio

ns

disca

rded

;1

»

(11);

»

(0505

)fo

r

=12,

=12;11

and23

»

(0505

0),13

and21

»

(005

0);

1

and2

»

(102

);

=1

+

with

»

(000

4)

for

=12.,,,

21,

for

=12,

=12

are

…xed

acro

ssre

plic

atio

ns.

CC

EM

G

and

CC

EP

are

de…ned

by

(14)

and

(20).

CupB

Cis

bia

s-corre

cted

iterate

dprin

cip

al

com

ponen

testim

ato

rof

Bai

etal.

(2009).

The

PC

1and

PC

2estim

ato

rsare

of

Kapeta

nio

sand

Pesa

ran

(2007).

The

varia

nce

estimato

rsof

all

mean

gro

up

and

pooled

estimato

rs(ex

cep

tth

at

of

CupB

C)

are

de…ned

by

(38)

and

(42),

respectiv

ely.

The

PC

type

estimato

rsare

com

puted

assu

min

gth

enum

ber

of

unobse

rved

facto

rs,

=3,

isknow

n.

All

experim

ents

are

base

don

2000

replic

atio

ns.

35

ACC

EPTE

DM

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

Table

2:

Sm

all

Sam

ple

Propertie

sof

Com

mon

Correla

ted

E¤ects

Type

Estim

ators

inth

eC

ase

of

Experim

ent

2A

(Hom

ogeneous

Slo

pes

+Full

Rank)

Bia

s(£

100)

Root

Mean

Square

Errors

(£

100)

Siz

e(5

%le

vel,

0:

1=

10

0)

Pow

er

(5%

level,

1:

1=

09

5)

(N,T

)20

30

50

100

200

20

30

50

100

200

20

30

50

100

200

20

30

50

100

200

CC

EM

G20

0.0

5-0

.15

0.0

2-0

.15

0.0

98.4

56.2

95.1

03.7

83.1

47.1

56.4

06.8

06.7

56.8

511.7

013.8

021.7

531.2

547.9

030

-0.1

40.1

20.0

40.0

30.0

06.4

45.1

13.8

02.6

72.0

76.0

56.7

57.2

56.4

06.4

512.7

020.4

530.7

050.9

071.6

050

0.0

8-0

.06

0.0

20.0

50.0

35.0

83.7

92.8

01.9

41.3

96.1

05.9

04.8

55.4

05.3

518.0

026.9

044.4

575.6

595.0

0100

-0.0

4-0

.08

0.0

6-0

.04

-0.0

13.5

92.7

62.0

21.3

50.9

84.5

55.5

06.0

55.1

06.1

028.3

043.0

072.3

595.2

099.9

0200

0.0

6-0

.02

0.0

30.0

10.0

02.8

32.0

51.5

21.0

00.6

85.6

04.4

56.3

55.2

05.7

044.2

067.9

591.9

099.9

0100.0

0C

CE

P20

0.1

80.0

00.0

3-0

.14

0.0

86.9

55.5

64.9

43.9

83.7

46.6

06.7

57.3

06.7

56.8

014.2

516.2

525.2

533.7

046.2

530

-0.1

40.1

40.0

70.0

10.0

15.2

04.5

03.5

52.6

72.2

65.1

05.9

07.2

56.2

56.4

015.2

524.5

534.9

052.9

570.7

050

0.0

50.0

7-0

.02

0.0

40.0

34.0

83.2

92.5

61.8

41.3

95.4

05.4

05.4

56.2

05.3

024.6

034.3

551.7

078.6

595.0

0100

-0.0

2-0

.04

0.0

6-0

.04

-0.0

12.8

72.3

71.7

81.2

40.9

35.6

06.2

06.4

05.2

55.9

541.6

558.3

581.8

597.8

0100.0

0200

0.0

7-0

.03

0.0

10.0

20.0

02.1

71.6

31.3

20.9

20.6

55.6

03.9

55.7

05.6

05.3

565.2

584.4

096.9

5100.0

0100.0

0C

upB

C20

0.1

20.1

00.0

8-0

.01

0.0

18.2

56.1

34.1

42.3

21.2

964.0

052.4

038.2

025.1

518.8

570.4

066.6

565.9

084.7

598.3

530

0.0

40.0

80.0

70.0

2-0

.01

6.4

04.7

33.0

81.7

20.9

661.8

550.0

035.4

023.2

519.1

571.3

071.3

579.3

095.0

099.9

050

-0.0

40.2

2-0

.06

0.0

40.0

34.8

93.5

62.3

11.2

70.7

059.9

049.2

534.4

521.9

015.4

077.2

081.6

088.3

598.8

5100.0

0100

0.0

30.0

10.0

2-0

.05

0.0

13.2

72.4

31.6

60.8

60.4

860.3

048.4

034.4

020.2

517.1

587.1

591.6

597.4

0100.0

0100.0

0200

0.0

70.0

10.0

30.0

30.0

02.4

31.7

31.1

60.6

30.3

359.9

546.6

032.6

020.7

014.9

094.7

097.7

099.8

0100.0

0100.0

0N

otes:

The

DG

Pis

the

sam

eas

that

of

Table

1,

exce

pt

=1

for

all

and,=

12

,

=12.

See

notes

toT

able

1.

Table

3:

Sm

all

Sam

ple

Propertie

sof

Com

mon

Correla

ted

E¤ects

Type

Est

imato

rs

inthe

Case

of

Experim

ent

1B

(H

eterogeneous

Slo

pes

+R

ank

De…cie

nt)

Bia

s(£

100)

Root

Mean

Square

Errors

(£

100)

Siz

e(5

%le

vel,

0:

1=

10

0)

Pow

er

(5%

level,

1:

1=

09

5)

(N,T

)20

30

50

100

200

20

30

50

100

200

20

30

50

100

200

20

30

50

100

200

CC

EM

G20

0.3

3-0

.19

0.2

00.1

40.2

315.0

213.9

012.6

113.3

513.7

86.8

06.9

06.7

56.6

07.2

09.4

08.9

510.1

510.1

510.1

530

0.3

00.1

40.0

9-0

.17

0.3

512.9

112.0

310.7

010.0

710.5

95.5

06.8

05.2

56.1

54.8

08.4

010.0

59.4

510.3

511.6

550

-0.1

50.6

3-0

.20

-0.1

70.0

29.8

28.4

67.8

77.4

27.3

45.8

05.1

06.1

05.7

55.9

09.7

512.9

013.4

014.0

015.2

0100

0.2

50.1

30.2

70.0

00.0

67.0

16.5

55.8

55.2

55.0

15.7

55.9

55.4

55.4

56.1

014.5

017.7

521.6

522.6

527.3

0200

0.0

5-0

.11

-0.1

7-0

.07

-0.0

55.3

54.6

54.1

53.6

13.3

14.8

05.0

54.7

55.1

54.5

519.4

523.7

029.7

537.2

543.4

5C

CE

P20

0.4

80.0

6-0

.04

0.1

60.1

013.1

312.8

112.2

113.5

715.3

06.7

57.4

07.0

06.6

56.7

59.9

010.2

010.4

010.3

510.2

530

-0.2

3-0

.06

0.1

8-0

.25

0.4

311.4

810.7

010.3

99.9

511.0

46.1

06.9

05.7

06.0

05.5

09.0

59.9

510.5

510.2

510.6

050

0.0

00.4

8-0

.18

-0.1

7-0

.02

8.4

27.5

77.2

37.2

27.2

25.2

55.9

06.2

55.3

05.5

011.4

014.0

514.1

514.3

515.2

0100

0.1

10.1

80.2

4-0

.06

0.0

55.8

75.7

25.2

74.8

74.9

85.1

06.0

05.4

04.9

56.0

017.2

519.6

023.5

023.5

527.0

0200

0.0

4-0

.10

-0.1

6-0

.04

-0.0

34.3

53.9

93.7

53.3

03.1

55.4

04.7

05.2

54.1

03.9

525.7

528.5

034.5

041.1

046.0

5C

upB

C20

1.3

40.8

31.0

71.1

21.3

511.2

49.5

28.2

47.5

97.2

467.3

560.2

056.7

060.8

566.8

070.4

566.0

566.0

571.2

576.8

530

0.5

10.8

51.1

40.8

61.2

38.9

77.5

26.4

75.7

85.6

067.4

059.8

055.3

556.9

565.3

572.3

568.9

569.1

572.9

580.2

050

0.5

70.7

00.6

20.9

10.8

16.7

75.8

54.9

84.3

24.0

564.6

557.3

552.4

052.0

059.7

074.9

072.2

570.4

078.6

584.5

0100

0.3

00.4

40.4

50.4

20.4

64.8

64.2

03.4

42.7

62.6

166.4

056.5

548.2

044.0

053.0

079.3

580.4

083.1

089.4

093.6

0200

0.1

40.1

40.1

30.2

70.2

63.5

32.9

92.4

52.0

01.6

964.8

053.3

545.1

543.9

546.9

586.9

088.4

591.2

096.9

099.3

5N

otes:

The

DG

Pis

the

sam

eas

that

of

Table

1,

exce

pt2

»

(01),

soth

at

the

rank

conditio

nis

not

satis…

ed.

See

notes

toT

able

1.

36

ACC

EPTE

DM

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

Table

4:

Sm

all

Sam

ple

Propertie

sof

Com

mon

Correla

ted

E¤ects

Type

Estim

ato

rs

inth

eC

ase

of

Experim

ent

2B

(Hom

ogeneous

Slo

pes

+R

ank

De…cie

nt)

Bia

s(£

100)

Root

Mean

Square

Errors

(£

100)

Siz

e(5

%le

vel,

0:

1=

10

0)

Pow

er

(5%

level,

1:

1=

09

5)

(N,T

)20

30

50

100

200

20

30

50

100

200

20

30

50

100

200

20

30

50

100

200

CC

EM

G20

-0.2

8-0

.26

0.4

1-0

.31

0.7

314.4

512.8

512.0

212.0

713.4

77.3

55.4

56.4

06.7

06.0

09.3

59.1

510.9

511.5

510.9

030

-0.1

10.0

70.0

90.4

5-0

.05

11.9

910.7

89.8

29.5

210.3

35.2

05.9

05.9

56.5

06.5

57.8

510.5

012.4

014.3

514.9

050

0.0

00.2

3-0

.07

-0.0

20.0

09.0

17.9

77.6

26.7

96.7

25.0

54.8

05.0

05.4

54.9

59.4

012.2

015.7

517.6

021.1

5100

0.1

4-0

.08

-0.1

2-0

.03

0.0

66.6

65.9

25.1

64.7

84.5

64.6

55.4

05.6

04.6

06.3

515.1

018.1

523.9

528.5

034.8

5200

0.1

40.1

10.0

1-0

.17

-0.0

75.1

34.4

53.8

83.2

73.3

45.4

55.1

05.4

54.6

55.1

522.3

528.8

036.6

044.7

556.7

0C

CE

P20

-0.1

2-0

.19

0.3

5-0

.26

0.6

612.6

611.5

311.5

612.1

215.0

77.4

57.0

07.5

56.3

56.5

09.8

510.0

012.6

012.6

511.5

030

-0.0

90.0

50.0

60.3

90.0

310.0

09.5

79.2

69.3

611.0

55.5

55.7

56.8

06.7

06.7

59.9

011.7

013.3

015.2

014.5

050

-0.1

40.3

9-0

.08

0.0

10.0

37.2

96.9

26.8

46.5

86.7

94.9

55.2

55.4

55.6

04.8

511.2

515.6

016.6

519.9

520.4

0100

0.2

0-0

.13

-0.1

1-0

.05

0.0

45.4

44.9

74.5

54.4

54.3

94.8

05.3

55.4

04.9

56.0

520.6

022.6

528.3

531.4

036.8

0200

0.1

90.1

1-0

.08

-0.1

3-0

.07

3.9

73.7

13.3

52.9

63.0

95.2

55.1

55.0

55.0

05.6

031.9

538.4

544.3

050.7

060.4

0C

upB

C20

0.4

40.3

30.2

90.2

60.2

08.1

16.0

24.1

12.4

11.3

359.6

548.2

534.4

026.1

519.0

069.4

565.9

068.1

585.8

599.3

030

0.1

80.2

20.2

30.1

40.0

96.3

34.6

43.0

41.7

21.0

060.0

548.7

533.8

521.6

020.0

071.4

572.1

579.2

095.1

0100.0

050

0.1

20.3

60.0

30.1

30.0

74.9

03.6

22.3

21.2

90.7

060.9

047.1

032.8

520.0

014.7

577.0

082.2

588.3

598.9

5100.0

0100

0.1

80.0

20.0

9-0

.01

0.0

43.2

32.4

81.6

50.8

60.4

859.6

548.7

033.2

019.8

016.9

087.8

591.1

097.8

5100.0

0100.0

0200

0.1

00.0

30.0

60.0

50.0

22.3

91.7

21.1

70.6

30.3

359.5

045.6

532.1

521.5

015.8

095.0

598.5

099.6

5100.0

0100.0

0N

otes:

The

DG

Pis

the

sam

eas

that

of

Table

1,

exce

pt2

»

(01),

soth

at

the

rank

conditio

nis

not

satis…

ed,

and

=1

for

all

and,=

12

,

=12.

See

note

sto

Table

1.

37

ACC

EPTE

DM

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

Table

5:

Sm

all

Sam

ple

Propertie

sof

Com

mon

Correla

ted

E¤ects

Type

Esti

mators,

The

Num

ber

of

Facto

rs

=4

Exceeds

+

1=

3,

Inth

ecase

of

Hete

rogeneous

Slo

pes

Bia

s(£

100)

Root

Mean

Square

Errors

(£

100)

(N,T

)20

30

50

100

200

20

30

50

100

200

CC

EM

G20

0.2

30.2

90.0

6-0

.23

-0.1

610.9

79.5

98.2

97.6

17.7

030

0.2

00.0

8-0

.07

0.1

4-0

.03

8.9

87.6

56.8

46.4

26.2

950

-0.0

40.0

0-0

.16

-0.1

90.1

46.8

16.0

35.1

24.7

14.6

7100

0.1

2-0

.06

0.0

1-0

.01

0.1

24.8

14.2

53.6

93.5

33.4

6200

0.0

1-0

.04

0.0

3-0

.04

-0.1

03.7

83.0

82.8

42.6

12.5

3C

CE

P20

0.0

90.5

0-0

.02

-0.2

2-0

.11

9.5

78.9

48.0

77.7

07.8

330

0.0

3-0

.05

-0.0

80.0

4-0

.09

7.9

67.2

16.6

06.3

66.2

550

-0.0

4-0

.05

-0.1

3-0

.14

0.1

36.0

65.5

94.8

54.5

44.4

9100

0.0

6-0

.07

-0.0

10.0

10.1

14.2

13.8

53.5

13.3

73.3

8200

-0.0

4-0

.05

0.0

0-0

.03

-0.1

03.1

32.7

42.6

22.4

22.3

7C

upB

C20

0.4

90.3

20.0

60.1

10.1

111.5

610.2

68.9

47.0

96.3

030

0.0

10.1

20.1

20.2

10.0

79.3

87.9

86.6

85.5

84.6

250

-0.1

10.2

5-0

.08

-0.0

20.2

17.0

76.2

95.0

34.0

43.5

4100

0.0

60.0

40.1

00.0

4-0

.04

4.8

14.3

23.5

82.8

22.5

4200

0.0

5-0

.11

0.0

00.0

0-0

.03

3.5

53.1

42.6

12.0

01.6

7N

otes:

Th

eD

GP

isth

esa

me

as

th

at

of

Ta

ble

1,

ex

cep

ta

nex

tra

term

44

isa

dd

ed

to

th

e

eq

ua

tio

n,

wh

ere4

»

(0502

),4

=4¡

1+4

,4

»

(01),4¡

50

=0

.

Fo

rC

up

BC

estim

ato

r,

th

en

um

ber

of

un

ob

serv

ed

facto

rs

istrea

ted

un

kn

ow

nb

ut

estim

ated

by

th

ein

form

atio

ncrit

erio

n

2,

wh

ich

isp

ro

po

sed

by

Ba

ia

nd

NG

(2

00

2).

We

set

th

em

ax

imu

mn

um

ber

of

facto

rs

to

six

.S

ee

als

oth

en

otes

to

Ta

ble

1.

Table

6:

Sm

all

Sam

ple

Properti

es

of

Com

mon

Correla

ted

E¤ects

Type

Esti

mato

rs,

Hete

rogeneous

Slo

pes

and

Full

Rank,

Coin

tegrated

Facto

rs,

inth

eC

ase

of

Experim

ent

1A

(Hete

rogeneous

Slo

pes

+Full

Rank)

Bia

s(£

100)

Root

Mean

Square

Errors

(£

100)

(N,T

)20

30

50

100

200

20

30

50

100

200

CC

EM

G20

0.0

5-0

.05

-0.2

20.0

80.0

09.2

67.8

76.5

85.6

95.2

930

-0.1

40.0

90.0

3-0

.02

0.0

27.3

56.0

25.1

84.5

44.1

650

-0.0

30.1

4-0

.05

0.1

10.1

15.8

54.7

04.0

63.4

93.1

4100

-0.0

5-0

.01

0.0

3-0

.05

0.0

04.1

53.4

02.8

72.4

92.1

9200

-0.0

50.1

40.0

30.0

4-0

.04

3.0

82.4

62.0

21.7

21.5

9C

CE

P20

-0.0

6-0

.01

-0.2

30.0

6-0

.01

8.5

27.5

46.6

55.9

55.6

830

-0.0

6-0

.07

-0.0

7-0

.02

0.0

16.7

85.9

05.2

54.7

04.2

950

-0.0

30.1

4-0

.09

0.1

20.1

35.3

54.5

44.0

53.5

53.1

9100

-0.0

20.0

30.0

6-0

.03

-0.0

23.7

73.1

82.8

42.5

02.2

2200

-0.0

40.1

0-0

.01

0.0

5-0

.04

2.7

02.3

31.9

91.7

21.6

0C

upB

C20

0.5

40.8

50.6

10.6

80.7

811.0

19.5

88.0

16.9

46.3

230

0.6

90.5

20.5

40.5

00.6

88.6

57.4

86.2

65.3

94.9

150

0.4

90.5

40.5

00.5

30.5

86.8

25.6

94.9

94.2

43.7

0100

0.3

30.3

70.3

80.2

20.2

64.6

13.8

63.4

32.8

42.5

2200

0.1

30.3

10.1

30.2

50.0

93.3

92.8

82.4

12.0

31.8

2N

otes:

Th

eD

GP

of

th

esa

me

as

th

at

of

Ta

ble

1,

ex

cep

tth

efa

cto

rs

are

gen

era

ted

as

co

integ

ra

ted

no

n-s

ta

tio

na

ry

pro

cesses:1

= 1

+05 2

+1

,

2

=05 1

+ 2

+2

,3

=07

5 1

+02

5 2

+3

,w

ith

»

(01)

¡

50

=0

,fo

r

=123

,w

here

= ¡

1+

wit

h

»

(01)

for

=12

,

=¡

490

See

als

oth

en

otes

to

Ta

ble

1.

Table

7:

Sm

all

Sam

ple

Propertie

sof

Com

mon

Correla

ted

E¤ects

Type

Esti

mato

rs,

Sem

i-Str

ong

Factors,

inthe

Case

of

Experim

ent

1A

(Hete

rogeneous

Slo

pes

+Full

Rank)

Bia

s(£

100)

Root

Mean

Square

Errors

(£

100)

(N,T

)20

30

50

100

200

20

30

50

100

200

CC

EM

G20

-0.0

9-0

.22

-0.0

70.0

9-0

.09

9.9

28.0

16.5

75.6

35.1

730

0.0

20.0

10.0

1-0

.11

0.1

07.7

46.2

15.1

44.4

34.1

050

-0.1

20.1

6-0

.11

0.1

4-0

.04

5.9

64.5

73.9

93.4

23.1

0100

0.0

10.0

30.0

50.0

20.0

44.2

33.5

12.8

72.3

32.2

6200

-0.0

60.0

1-0

.01

0.0

50.0

03.0

62.4

62.0

01.7

21.5

1C

CE

P20

0.0

9-0

.07

-0.0

60.0

4-0

.12

8.6

47.4

96.3

45.6

55.3

430

-0.1

9-0

.10

0.0

9-0

.08

0.1

37.1

25.9

05.1

24.4

94.2

150

0.0

10.1

3-0

.05

0.1

3-0

.02

5.2

74.4

63.9

33.4

33.1

6100

0.0

40.0

80.0

20.0

00.0

33.7

73.2

82.8

42.3

52.2

8200

-0.0

7-0

.03

-0.0

40.0

50.0

02.6

82.3

01.9

61.7

01.5

3C

upB

C20

0.2

30.4

60.1

70.4

30.4

512.2

910.5

58.0

96.7

55.8

030

-0.2

00.0

90.3

80.2

00.4

99.5

38.0

36.3

95.1

44.5

850

0.3

90.3

70.0

60.2

00.1

57.3

46.0

85.0

73.9

93.4

0100

0.1

80.1

80.0

60.0

50.0

94.9

94.4

03.6

12.6

92.4

5200

0.0

00.0

30.0

30.0

90.0

13.7

73.0

32.5

51.9

81.6

4N

otes:

Th

eD

GP

of

th

esa

me

as

th

at

of

Ta

ble

1,

ex

cep

tth

efa

cto

rlo

ad

ing

s

ma

trix

¡0

ism

ult

ipli

ed

by

¡12

for

all.

See

als

oth

en

otes

to

Ta

ble

1.

38

Table

8:

Sm

all

Sam

ple

Properti

es

of

Com

mon

Correla

ted

E¤ects

Type

Esti

mators,

One

Break

inthe

Means

of

Unobse

rved

Facto

rs,

inthe

Case

of

Experim

ent

1A

(H

eterogeneous

Slo

pes

+Full

Rank)

Bia

s(£

100)

Root

Mean

Square

Errors

(£

100)

(N,T

)20

30

50

100

200

20

30

50

100

200

CC

EM

G20

0.0

1-0

.10

-0.0

20.0

6-0

.07

9.6

67.8

26.7

45.8

75.5

430

0.1

4-0

.03

-0.0

2-0

.13

0.1

07.6

86.0

85.1

14.5

44.2

250

-0.2

10.2

0-0

.11

0.1

4-0

.04

5.9

14.6

44.0

13.4

33.1

3100

0.0

20.0

30.0

50.0

30.0

44.2

63.4

82.8

82.3

32.2

6200

-0.0

8-0

.02

-0.0

20.0

60.0

03.0

82.4

92.0

11.7

21.5

1C

CE

P20

0.1

70.0

0-0

.05

0.0

0-0

.13

8.7

37.6

16.8

66.3

06.2

130

-0.1

5-0

.13

0.0

7-0

.14

0.1

47.1

05.9

85.3

14.7

84.4

650

-0.0

30.1

8-0

.06

0.1

1-0

.01

5.3

04.5

33.9

73.4

73.2

1100

0.0

50.0

90.0

40.0

10.0

23.8

03.2

62.8

52.3

42.2

8200

-0.0

6-0

.04

-0.0

50.0

50.0

02.7

22.2

91.9

51.7

11.5

3C

upB

C20

0.5

20.7

70.7

90.8

00.8

911.1

89.8

78.3

97.5

26.9

730

0.3

20.5

80.7

70.5

80.8

48.9

17.8

06.5

55.6

85.2

750

0.5

80.7

50.3

80.6

10.5

46.7

86.0

15.0

34.1

83.8

2100

0.2

80.3

50.3

80.2

90.3

24.8

54.2

23.4

12.7

52.5

5200

0.1

00.0

80.0

80.2

30.1

73.5

72.9

32.4

42.0

11.6

9N

otes:

Th

eD

GP

isth

esa

me

as

th

at

of

Ta

ble

1,

ex

cep

tth

at

=

for

b23

ca

nd

=1

+

for

¸b23

cw

ith

b

cb

ein

gth

eg

rea

test

integ

er

pa

rt

of

,w

here

=

¡1

+

,

»

(01),

=

123

See

als

oth

en

otes

to

Ta

ble

1.