structural change in non-stationary ar(1)...

Structural Change in Non-stationary AR(1)

Models∗

May 15, 2016

Tianxiao Pang1,†, Terence Tai-Leung Chong2,3, Danna Zhang4 and Yanling Liang1,

1School of Mathematical Sciences, Yuquan Campus, Zhejiang University, Hangzhou 310027, P.R.China

2Department of Economics and Lau Chor Tak Institute of Global Economics and Finance,

The Chinese University of Hong Kong, Hong Kong

3Department of International Economics and Trade, Nanjing University, Nanjing 210093, P.R.China

4Department of Statistics, University of Chicago, 5734 S. University Ave., Chicago, Illinois 60637, USA

Abstract. This paper revisits the asymptotic inference for non-stationary AR(1) models

of Phillips and Magdalinos (2007a) by incorporating a possible structural change in the AR

parameter at an unknown time k0. Consider the model yt = β1yt−1I{t ≤ k0}+β2yt−1I{t >

k0} + εt, t = 1, 2, · · · , T, where I{·} denotes the indicator function and one of β1 and β2

depends on the sample size T and the other is equal to one. We examine four cases: Case

(I): β1 = β1T = 1 − c/kT , β2 = 1; (II): β1 = 1, β2 = β2T = 1 − c/kT , (III): β1 = 1,

β2 = β2T = 1+ c/kT , and case (IV): β1 = β1T = 1+ c/kT , β2 = 1, where c is a fixed positive

constant and kT is a sequence of positive constants increasing to ∞ such that kT = o(T ).

In addition, we assume that {εt, t ≥ 1} is a sequence of i.i.d. random variables which are

in the domain of attraction of the normal law (DAN) with zero means and possibly infinite

variances. We derive the limiting distributions of the least squares estimators of β1 and

β2 and that of the change-point estimator for the aforementioned cases under some mild

conditions. It is shown that: (1) the asymptotic properties of the least squares estimator of

the second AR parameter in mildly explosive AR(1) model with an unknown change point

∗Tianxiao Pang and Yanling Liang’s research was supported by the Department of Education of Zhejiang

Province in China (N20140202).

E-mail addresses: [email protected], [email protected], [email protected] and

[email protected].†Corresponding author.

1

are very different from those in mildly explosive AR(1) model or unit root model without a

change point; (2) when a unit root model switches to a mildly integrated or mildly explo-

sive AR(1) model at time k0, the asymptotic properties of the least squares estimator of k0

remain the same, and the phase transition for the estimation error of k0 occurs when kT has

the same order of magnitude as√T . Monte Carlo simulations are conducted to examine

the finite-sample properties of the estimators.

Keywords: AR(1) model, Least squares estimator, Limiting distribution, Mildly ex-

plosive, Mildly integrated, Structural change, Unit root.

JEL Classification: C22

2

1 Introduction and Main Results

The change-point problem in time series regression models has received considerable atten-

tion in the literature over the past decades. Many economic time series data are characterized

by single or multiple structural changes (Stock and Watson (1996, 1999), Hansen (2001)).

Bai and Perron (1998) provided the estimation and test procedure for the linear models

with multiple structural changes. Leybourne et al. (2003), Harvey et al. (2006), Halunga

and Osborn (2012) and Kejriwal et al. (2013) investigated structural changes in persistence.

Chong (2003), Pitarakis (2004) and Bai et al. (2008) studied the estimations and tests of

the change point under model misspecification. Qu (2008) tested for the structural change

in regression quantiles. Recent development in this area include that of Lee et al. (2016),

who investigated the change-point problem in high-dimension regression models.

An important strand of literature on structural change focuses on the autoregressive

models. Structural changes in autoregressive model are of interest as the time series prop-

erties of the model, such as stationarity, may be different before and after the change. As

a result, the rate of convergence and the asymptotic distributions of the estimators are dif-

ficult to derive. Chong (2001) investigated the statistical inference for the change point in

various AR(1) models. Berkes et al. (2011) studied the structural change from an AR(1)

model to a threshold AR(1) model. An important application of the AR(1) change-point

model was given by Mankiw and Miron (1986) and Mankiw et al. (1987), who found that

the short term interest rate has changed from a stationary process to a near random walk

since the Federal Reserve System was founded at the end of 1914. Other applications can

be found in Barsky (1987) and Burdekin and Siklos (1999) for the inflation rate series, in

Hakkio and Rush (1991) for the government budget deficits, in Phillips et al. (2011) for

1990s NASDAQ and in Phillips and Yu (2011) and Phillips et al. (2015a, 2015b, 2015c) for

financial bubbles.

This paper revisits and generalizes the model studied in Chong (2001). Consider the

following AR(1) model with a change in the AR parameter at an unknown time k0,

yt = β1yt−1I{t ≤ k0}+ β2yt−1I{t > k0}+ εt, t = 1, 2, · · · , T, (1.1)

where I{·} denotes the indicator function and {εt, t ≥ 1} is a sequence of i.i.d. random

variables. Under the regularity conditions that E(εt) = 0, V ar(εt) < ∞ and E(y20) < ∞,

the consistency and limiting distributions of the least squares estimators of fixed β1 and

β2 and the change-point estimator of τ0(= k0/T ) were developed in Chong (2001) for the

3

following three cases: (1) |β1| < 1 and |β2| < 1; (2) |β1| < 1 and β2 = 1; (3) β1 = 1 and

|β2| < 1.

Since heavy-tailed distributions, such as the Student’s t distribution with degrees of

freedom 2, Pareto distribution with index 2 and stable random variables are often used

to model asset returns in empirical studies (Mandelbrot, 1963), Pang and Zhang (2015)

employed the truncation technique to weaken the moment conditions of y0 and εt’s in case

(1) of Chong (2001). For any constant c, Pang et al. (2014) examined the asymptotics for

the case where |β1| < 1 and β2 = β2T = 1− c/T and the case where β1 = β1T = 1− c/T and

|β2| < 1. In these two cases, one of the β1 and β2 is fixed and smaller than one in absolute

value while the other is local to unity. The limiting distributions obtained in Pang et al.

(2014) are complicated. In the special case of c = 0, the results in Pang et al. (2014) are

reduced to the two main theorems in Chong (2001). Both Pang et al. (2014) and Pang and

Zhang (2015) only require εt’s to be in the domain of attraction of the normal law (DAN)

with zero mean and possibly infinite variance, and y0 to be any random variable of order

smaller than√T in probability.

The asymptotic theory for the near unit root model was first studied by Phillips (1987)

and Chan and Wei (1987) independently. Their studies bridge the gap between stationary

AR(1) models and the unit root model. This paper is related to that of Phillips and

Magdalinos (2007a), who attempted to bridge the gap between the asymptotic theories of

the stationary case, explosive case and the local to unity case for AR(1) model. In particular,

they investigated the limiting distribution of the least squares estimator of the AR parameter

for the following AR(1) model: yt = βyt−1 + εt, t = 1, 2, · · · , T, with β = βT = 1 − c/kT

and β = βT = 1 + c/kT , where c is a positive constant and kT is a sequence of positive

constants increasing to ∞ such that kT = o(T ). They proved the asymptotic normality for

the least squares estimator of βT with the convergence rate√TkT for the mildly integrated

AR(1) model when β = βT = 1 − c/kT and showed a Cauchy limiting distribution for the

least squares estimator of βT with convergence rate kTβTT for the mildly explosive AR(1)

model when β = βT = 1 + c/kT . The mildly explosive AR(1) model has been widely used

to model financial bubbles. The reader is referred to Phillips and Yu (2011), Phillips et al.

(2011) and Phillips et al. (2015a, 2015b, 2015c) for more details.

Motivated by the works of Chong (2001) and Phillips and Magdalinos (2007a) and the

importance of the mildly explosive AR(1) model in modelling the bubble process, in this

article, we aim to study the structural change in mildly integrated and mildly explosive

4

AR(1) models from the theoretical perspective. In particular, we are interested in the

following four cases: (I) β1 = β1T = 1−c/kT , β2 = 1; (II) β1 = 1, β2 = β2T = 1−c/kT ; (III)

β1 = 1, β2 = β2T = 1+c/kT and (IV) β1 = β1T = 1+c/kT , β2 = 1, where c > 0 and kT shares

the same assumption in Phillips and Magdalinos (2007a). Cases (III) and (IV) are related to

the works of Phillips and Yu (2011), Phillips et al. (2011) and Phillips et al. (2015a, 2015b,

2015c). In the aforementioned papers, the authors proposed a structural change model with

a bubble process occurring in a unit root process and dated the origination of the explosive

episode based on a test procedure and then estimated the explosive AR parameter by using

a demeaning procedure and the least squares method. Since the models before and after the

time k0 are both non-stationary and the difference between β1 and β2 converges to zero as

the sample size tends to infinity, obtaining the closed form limiting distributions of the least

squares estimators of β1 and β2 and the estimator of the change point will be a challenging

task. The main contribution of this paper is to derive the closed form solution of the limiting

distributions of these estimators for the aforementioned cases when the distribution of the

error term belongs to the DAN.

A sequence of i.i.d. random variables {Xi, i ≥ 1} belongs to the DAN if there exist two

constant sequences {An, n ≥ 1} and {Bn, n ≥ 1} such that Zn := B−1n (X1 + · · ·+Xn)−An

converges to a standard normal random variable in distribution as n tends to infinity (Feller,

1971), where Bn takes the form√nh(n) and h(n) is a slowly varying function at infinity.

For the models studied in this article, we make the following assumptions:

• C1: {εt, t ≥ 1} is a sequence of i.i.d. random variables which are in the domain of

attraction of the normal law with zero means and possibly infinite variances.

• C2: {kT , t ≥ 1} is a sequence of positive constants increasing to infinity slowly such

that kT = o(T ).

• C3: y0 is an arbitrary random variable with y0 = op(√kT ).

• C4: τ0 ∈ [τ , τ ] ⊂ (0, 1).

Remark 1.1 The assumption of i.i.d. in C1 is only for the convenience of exposition in the

proofs. If the DAN condition is replaced by some appropriate moment conditions, one can

extend our results to some cases that allow for dependence. One may refer to Phillips and

Magdalinos (2007b) and Magdalinos (2012) for details. Assumption C2 is the same as that

in Phillips and Magdalinos (2007a). Assumption C3 means y0 will not affect the asymptotic

5

properties of the estimators of β1, β2 and the change point. Assumption C4 is standard in

the change-point literature (Bai (1997), Chong (2001) and Pitarakis (2004)), which ensures

the identifiability of the AR parameters and the change point. In empirical studies, one can

set [τ , τ ] = [0.05, 0.95].

Let [a] denote the integer part of a. For any given 0 < τ < 1, the least squares estimators

of the AR parameters β1 and β2 are given by

β1(τ) =

∑[τT ]t=1 ytyt−1∑[τT ]t=1 y2t−1

, β2(τ) =

∑Tt=[τT ]+1 ytyt−1∑Tt=[τT ]+1 y

2t−1

,

The change-point estimator is defined as

τT = argminτ∈(0,1)

RSST (τ),

where

RSST (τ) =

[τT ]∑t=1

(yt − β1(τ)yt−1

)2+

T∑t=[τT ]+1

(yt − β2(τ)yt−1

)2.

Once we obtain the change-point estimator τT , the final least squares estimators of β1 and

β2 are defined by β1(τT ) and β2(τT ) respectively.

We define some notations before proceeding to our main results. Let W (·), W (·) and

W (·) be three independent Brownian motions defined on [0, 1], [0, 1] and R+ respectively,

and W1(·) and W2(·) be two independent Brownian motions defined on R+. “⇒” denotes

the weak convergence of the associated probability measures. “p→” denotes convergence in

probability and “d=” denotes identical in distribution. The limits in this paper are all taken

as T → ∞ unless specified otherwise. In addition, we denote k = [T τT ] and the notation

aT ≍ bT means there exist two positive constants c1 and c2 such that c1 ≤ aT /bT ≤ c2 for

all large T , where aT and bT are two positive functions of T .

Under assumptions C1-C4, we have the following results.

Theorem 1.1 In Model (1.1), if β1 = β1T = 1−c/kT and β2 = 1, where c is a fixed positive

constant, under assumptions C1-C4, we have:

(a) τT is consistent and its limiting distribution is given by

cT

kT(τT − τ0) ⇒ argmax

ν∈R

{C∗(ν)

Bc(12)

− |ν|2

}, (1.2)

where Bc(12) is generated by

√c∫∞0 exp (−cs)dW1(s) and C∗(ν) is defined to be C∗(ν) =

W1(−ν) for ν ≤ 0 and

C∗(ν) = −W2(ν)−∫ ν

0

W2(s)

Bc(12)

dW2(s)−∫ ν

0

( W2(s)

2Bc(12)

+ 1)W2(s)ds

6

for ν > 0.

(b) β1(τT ) is consistent and its limiting distribution is given by√τ0T

1− β21T

(β1(τT )− β1T ) ⇒ N(0, 1). (1.3)

(c) β2(τT ) is also consistent and its limiting distribution is given by

(1− τ0)T (β2(τT )− 1) ⇒ (W (1)−W (τ0))2 − (1− τ0)

2∫ 1τ0(W (s)−W (τ0))2ds/(1− τ0)

d=

W2(1)− 1

2∫ 10 W

2(s)ds

. (1.4)

Theorem 1.2 In Model (1.1), if β1 = 1 and β2 = β2T = 1−c/kT , where c is a fixed positive


(a) (i) When kT = o(√T ), then τT is exactly T -consistent, i.e.,

P (k = k0) → 0.

(ii) When kT ≍√T , then τT is T -consistent, i.e.,

|k − k0| = Op(1).

(iii) When√T = o(kT ), then τT is not T -consistent, but τT is consistent and its limiting

distribution is given by

c2T 2

k2T(τT − τ0) ⇒ argmax

ν∈R

{W ∗(ν)

W1(τ0)− |ν|

2

}, (1.5)

where W ∗(ν) is a two-sided Brownian motion on R defined to be W ∗(ν) = W1(−ν) for ν ≤ 0

and W ∗(ν) = W2(ν) for ν > 0.

(b) β1(τT ) is consistent and its limiting distribution is given by

τ0T (β1(τT )− 1) ⇒ W 2(τ0)− τ02∫ τ00 W 2(s)ds/τ0

d=

W 2(1)− 1

2∫ 10 W 2(s)ds

. (1.6)

(c) β2(τT ) is also consistent and its limiting distribution is given by√T

1− β22T

(β2(τT )− β2T ) ⇒W (1)

W 2(τ0) + (1− τ0). (1.7)

7

Theorem 1.3 In Model (1.1), if β1 = 1 and β2 = β2T = 1+c/kT , where c is a fixed positive


(a) (i) When kT = o(√T ), then τT is exactly T -consistent, i.e.,

P (k = k0) → 0.

(ii) When kT ≍√T , then τT is T -consistent, i.e.,

|k − k0| = Op(1).

(iii) When√T = o(kT ), then τT is not T -consistent, but τT is consistent and its limiting

distribution is given by (1.5).

(b) β1(τT ) is consistent and its limiting distribution is given by

τ0T (β1(τT )− 1) ⇒ W 2(τ0)− τ02∫ τ00 W 2(s)ds/τ0

d=

W 2(1)− 1

2∫ 10 W 2(s)ds

. (1.8)

(c) β2(τT ) is also consistent and its limiting distribution is given by√τ0T

β22T − 1

βT−[τ0T ]2T (β2(τT )− β2T ) ⇒

√2cX

W (1)

d= ζ, (1.9)

where X is a random variable obeying N(0, 12c) and is independent of W (·), and ζ stands

for a standard Cauchy variate.

Theorem 1.4 In Model (1.1), if β1 = β1T = 1+c/kT and β2 = 1, where c is a fixed positive

constant, under assumptions C1-C4, the estimators k, β1(τT ) and β2(τT ) are all consistent

and

P (k = k0) → 0, (1.10)

β[τ0T ]1T

β21T − 1

(β1(τT )− β1T ) ⇒ ζ, (1.11)√(1− τ0)T

β21T − 1

β[τ0T ]1T (β2(τT )− 1) ⇒ W (1)−W (τ0)√

2c(1− τ0)Y

d= ζ, (1.12)

where ζ is a standard Cauchy variate, Y is a random variable obeying N(0, 12c) and inde-

pendent of W (·).

Based on these results, it follows from assumption C4 that τ0 cannot be equal to zero

or one, but it can be close to zero or one if we let the interval [τ , τ ] be close to (0, 1)

enough. If we let τ0 be close to one in Theorem 1.1 and close to zero in Theorem 1.2,

8

Model (1.1) becomes an approximately mildly integrated AR(1) model. Then, since 1−(1−

c/kT )2 = 2c/kT (1+ o(1)), the limiting distributions of the statistics

√TkT (β1(τT )−β1T ) in

Theorem 1.1 and√TkT (β2(τT ) − β2T ) in Theorem 1.2 will be approximately N(0, 2c) and

W (2c), respectively, with the convergence rate√TkT . This agrees with the result in Phillips

and Magdalinos (2007a). Analogously, letting τ0 be close to zero in Theorem 1.1 and be

close to one in Theorem 1.2, Model (1.1) converges to a unit root model, and the limiting

distributions of the statistics T (β2(τT )− 1) in Theorem 1.1 and T (β1(τT )− 1) in Theorem

1.2 will be approximately W2(1)−1

2∫ 10 W

2(s)ds

and W 2(1)−1

2∫ 10 W 2(s)ds

respectively with a convergence rate

T , which agrees with the asymptotic theory for the unit root model. Therefore, when the

sample {y1, · · · , yT } is divided into two sub-samples {y1, · · · , y[τ0T ]} and {y[τ0T ]+1, · · · , yT }

if [τ0T ] is the change point in mildly integrated AR(1) model, then the first sub-sample

{y1, · · · , y[τ0T ]} will not affect the asymptotic properties of the least squares estimator of

β2. However, it is interesting to find that the results above are not applicable to Theorem

1.3 and Theorem 1.4. Even if τ0 is very close to zero, the influence of the first sub-sample

{y1, · · · , y[τ0T ]} is not negligible. In fact, it can be seen from the proofs of Theorem 1.3 and

Theorem 1.4 that the influence of the first sub-sample {y1, · · · , y[τ0T ]} dominates that of

the second sub-sample {y[τ0T ]+1, · · · , yT } even when τ0 is very close to zero, leading to the

difference in asymptotic properties of the least squares estimator of β2 from those in mildly

explosive model or unit root model. For example, in Theorem 1.3, affected by the first sub-

sample, the limiting distribution of the statistic√

τ0TkT2c β

T−[τ0T ]2T (β2(τT )− β2T ) is standard

Cauchy. This result is a little different from that in mildly explosive AR(1) model studied

in Phillips and Magdalinos (2007a). Moreover, the convergence rate for the least squares

estimator of β2 in Theorem 1.3 is slower than that in Phillips and Magdalinos (2007a).

Furthermore, the asymptotic properties of the least squares estimator of β2 are affected by

the first sub-sample more seriously in Theorem 1.4. It can be seen from Theorem 1.4 that

the limiting distribution of the least squares estimator of β2 is totally different from that

in unit root model, and the convergence rate is faster than that in unit root model. These

findings are useful in testing for a change point in mildly explosive AR(1) model.

Theorem 1.1 and Theorem 1.2 indicate that, when the two sub-samples do not dominate

each other, the convergence rate of the least squares estimator when the AR parameter

equals one is faster than that in the case when the AR parameter equals 1− c/kT , since the

signal from the regressor yt−1 in the former case is stronger than that from the regressor

yt−1 in the latter case. It should also be noted that the convergence rate of τT in Theorem

9

1.2 is faster than that in Theorem 1.1. This is because the signal from the regressor yt−1

when (β1, β2) = (1, 1− c/kT ) is stronger than that from the regressor yt−1 when (β1, β2) =

(1−c/kT , 1). However, these findings are not totally applicable to Theorem 1.3 and Theorem

1.4. Note that in Theorem 1.3, the convergence rate of the least squares estimator of β2

is faster than that of the least squares estimator of β1. Moreover, the convergence rate of

τT in Theorem 1.4 is faster than that in Theorem 1.3 since the signal from the regressor

yt−1 when (β1, β2) = (1 + c/kT , 1) is stronger than that from the regressor yt−1 when

(β1, β2) = (1, 1+ c/kT ). These are consistent with Theorem 1.1 and Theorem 1.2. However,

it is surprising that the convergence rate of the least squares estimator of β2 is faster than

that of the least squares estimator of β1 in Theorem 1.4. This is completely different from

the finding from Theorem 1.1 and Theorem 1.2. Compared with Theorem 1.3, the second

sub-sample is more affected by the first sub-sample in Theorem 1.4.

The precision of the estimation of k0 depends on both the strength of the signal from

the model and the difference between β1 and β2. Note that the difference between β1 and

β2 is c/kT and the strength of the signal from the model becomes increasingly stronger from

Theorem 1.1 to Theorem 1.4. In general, the signal from the model in Theorem 1.1 is too

weak for k0 to be located for any kT = o(T ), while the signal from the model in Theorem

1.4 is too strong such that k0 can be located for any kT = o(T ). For the models in Theorems

1.2 and 1.3, although the signal from the model in Theorem 1.3 is stronger than that from

the model in Theorem 1.2, it is surprising that k0 can be located only when kT = o(√T )

in both models. This is because the increment of signal from Theorem 1.2 to Theorem

1.3 is too small such that the difference between RSST (τ0) and RSST (τ0 ± mT ) for a fixed

integer m in Theorem 1.3 is the same as that in Theorem 1.2, which can be seen from the

proofs of Theorems 1.2 and 1.3. Although the increase in strength of signal from the model

in Theorem 1.2 to the model in Theorem 1.3 does not help to locate k0, it improves the

convergence rate for the least squares estimator of β2.

It is interesting to find that the conventional T -consistent for τT occurs only when

kT ≍√T in both Theorems 1.2 and 1.3. When kT is of order higher than

√T , β1 and

β2 will be very close, and the estimation error for k0 will be huge. When kT has smaller

magnitude than√T , it implies that β1 and β2 has enough difference, and the estimation

error for k0 reduces tremendously.

Suppose c0 is a fixed constant smaller than one in absolute value. It follows from Chong

(2001) and Pang et al. (2014) that the convergence rates of the least squares estimators

10

of β1 and β2 under the situations of (β1, β2) = (c0, 1 − c/T ) and (β1, β2) = (1 − c/T, c0)

with c an arbitrary fixed constant are symmetric. This is also the case in our paper when

(β1, β2) = (1−c/kT , 1) and (β1, β2) = (1, 1−c/kT ) with c a fixed positive constant. However,

it is surprising to see from Theorems 1.3 and 1.4 that the convergence rates of the least

squares estimators of β1 and β2 when (β1, β2) = (1, 1 + c/kT ) and (β1, β2) = (1 + c/kT , 1)

are asymmetric.

The rest of the paper is organized as follows. Section 2 discusses some possible extensions

for the results obtained in this section. Section 3 demonstrates some finite sample Monte

Carlo results for our estimators in the presence of structural change, and for mildly explosive

AR(1) model of Phillips and Magdalinos (2007a) which does not have a structural change.

Section 4 concludes the paper. The proofs of Theorems 1 to 4 are relegated to Appendix A

to Appendix D respectively.

2 Discussions

2.1 Pivotal explore

The statistics in the theorems in last section depend on nuisance parameters when exploring

the limiting distributions of the least squares estimators of the AR parameters, but most of

them can be rewritten pivotally. Note that τT is always a consistent estimator of τ0 with

convergence rate at least kT /T or k2T /T2 in the four theorems in last section, implying that

T τTTτ0

p→ 1 andT − T τTT − Tτ0

p→ 1.

Note also that β1(τT ) and β2(τT ) are always a consistent estimator of β1T and β2T respec-

tively in our theorems,

(1± c/kT )k2T /T → 1

and the relationship (C.16) below. In Theorem 1.3, since

[τ0T ]− [τTT ] = Op(k2T /T ) = op(kT )

and

β2(τT ) = β2T +Op

( 1√TkTβ

T−[τ0T ]2T

),

we have

(β2(τT ))T−[τTT ]

βT−[τ0T ]2T

=

(β2(τT )

β2T

)T−[τ0T ]

· (β2(τT ))[τ0T ]−[τTT ] p→ 1.

11

Similarly, in Theorem 1.4, we have

(β1(τT ))[T τT ]

β[τ0T ]1T

p→ 1.

Then, the limiting distributions of the least squares estimators of AR parameters in the

theorems in last section can further be written as:

Theorem 2.1 In Theorem 1.1,

(a) β1(τT ) is consistent and its limiting distribution is given by√T τT

1− β21(τT )

(β1(τT )− β1T ) ⇒ N(0, 1),

(b) β2(τT ) is also consistent and its limiting distribution is given by

(T − T τT )(β2(τT )− 1) ⇒ W2(1)− 1

2∫ 10 W

2(s)ds

.


(a) β1(τT ) is consistent and its limiting distribution is given by

T τT (β1(τT )− 1) ⇒ W 2(1)− 1

2∫ 10 W 2(s)ds

,

(b) β2(τT ) is also consistent and its limiting distribution is given by√T

1− β22(τT )

(β2(τT )− β2T ) ⇒W (1)

W 2(τ0) + (1− τ0).



T τT (β1(τT )− 1) ⇒ W 2(1)− 1

2∫ 10 W 2(s)ds

,

(b) β2(τT ) is also consistent and its limiting distribution is given by√T τT

β22(τT )− 1

(β2(τT ))T−[T τT ](β2(τT )− β2T ) ⇒ ζ,

where ζ stands for a standard Cauchy variate.

12



(β1(τT ))[T τT ]

β21(τT )− 1

(β1(τT )− β1T ) ⇒ ζ,

(b) β2(τT ) is also consistent and its limiting distribution is given by√T − T τT

β21(τT )− 1

(β1(τT ))[T τT ](β2(τT )− 1) ⇒ ζ,

where ζ stands for a standard Cauchy variate.

2.2 The presence of an intercept in the model

In most applications, AR(1) model with an intercept is more realistic. Hence, it might be

useful to extend our analysis in Section 1 to non-stationary AR(1) models with an intercept.

Suppose the true structural change model is

yt = α+ β1yt−1I{t ≤ k0}+ β2yt−1I{t > k0}+ εt, t = 1, 2, · · · , T.

After demeaning the model using yt = yt − 1T

∑Tt=1 yt, we have

yt =

β1y1,t−1 − β21T

∑Tj=k0+1 yj−1 + εt, 1 ≤ t ≤ k0

β2y2,t−1 − β11T

∑k0j=1 yj−1 + εt, k0 < t ≤ T

with

y1,t−1 = yt−1 −1

T

k0∑t=1

yt−1, y2,t−1 = yt−1 −1

T

T∑t=k0+1

yt−1 and εt = εt −1

T

T∑t=1

εt.

However, the structural-change model becomes more complicated and the previous analysis

is not applicable here.

Another possible approach is to estimate the parameters directly without the demeaning

procedure. For any given 0 < τ < 1, we use

β1(τ) =

∑[τT ]t=1 y1,ty1,t−1∑[τT ]t=1 y21,t−1

, β2(τ) =

∑Tt=[τT ]+1 y2,ty2,t−1∑Tt=[τT ]+1 y

22,t−1

and

α(τ) =1

T

T∑t=1

yt − β1(τ)1

T

[τT ]∑t=1

yt−1 − β2(τ)1

T

T∑t=[τT ]+1

yt−1

to denote the least squares estimators of β1, β2 and α respectively, where

y1,t = yt −1

[τT ]

[τT ]∑t=1

yt, y1,t−1 = yt−1 −1

[τT ]

[τT ]∑t=1

yt−1

13

and

y2,t = yt −1

T − [τT ]

T∑t=[τT ]+1

yt, y2,t−1 = yt−1 −1

T − [τT ]

T∑t=[τT ]+1

yt−1.

The change-point estimator is then defined as

τT = argminτ∈(0,1)

RSST (τ),

where

RSST (τ) =

[τT ]∑t=1

(yt − α(τ)− β1(τ)yt−1

)2+

T∑t=[τT ]+1

(yt − α(τ)− β2(τ)yt−1

)2.

However, since our analysis in last section relies heavily on the existence of closed form

expression of RSST (τ)−RSST (τ0), which is not easy to be formulated here.

Therefore, the analysis of structural change in AR(1) models with an intercept is ex-

tremely complicated, and we leave it for future work.

3 Simulations

For empirical applications, we perform the following Experiments 1-4 to see how well the

finite sample properties of the estimators follow our asymptotic results. In all experiments,

the sample size is set at T = 600, the interval [τ , τ ] is taken as [0.05, 0.95] (hence the break

fraction is searched within this interval in our experiments), the true break fraction is set

at τ0 = 0.5 (hence k0 = 300) and the number of replications is set at N = 50, 000; {yt}Tt=1

is generated from Model (1.1), y0 is set at zero for simplicity and {εt}Tt=1 are generated

independently from N(0, 1). In addition, we take c = 3 and kT = Tα with α = 0.3, 0.5, 0.7

respectively. The case where α = 0.3 presents kT = o(√T ), the case where α = 0.5 presents

kT ≍√T and the case where α = 0.7 presents

√T = o(kT ). Note that the two AR

parameters have a large difference when α = 0.3 (= 3/6000.3 = 0.440), have a moderate

difference when α = 0.5 (= 3/6000.5 = 0.122) and have a very small difference when α = 0.7

(= 3/6000.7 = 0.034).

3.1 Experiment 1

First, we conduct experiments to verify Theorem 1.1. Figure 1 and Figure 2 are the his-

tograms of k and the distributions of β1(τT ) and β2(τT ) respectively. Our result predicts

that k is not a consistent estimator of k0 and the estimation error is of Op(kT ). This is

supported by Figure 1. Figure 1 also indicates that the smaller is the difference between β1

14

and β2, the larger is the estimation error, which agrees with the intuition. In addition, the

bottom graph in Figure 1 demonstrates that there exists a “boundary effect” when β1 and

β2 has no enough difference. Part (b) of Theorem 1.1 predicts that β1(τT ) should have a

normal distribution and part (c) of Theorem 1.1 predicts that β2(τT ) should have a Dickey-

Fuller distribution. Figure 2 agrees with our results. This figure also depicts that the larger

is the difference between β1 and β2, the better is the finite-sample performance of β1(τT ) and

β2(τT ). Note that, in the finite-sample case, the distribution of the least squares estimator

of the AR parameter should be a mixture of normal and Dickey-Fuller distributions when

the true AR parameter is smaller than but near to one. This explains why the finite-sample

performance of β1(τT ) when α = 0.7 is not very satisfactory.

3.2 Experiment 2

Second, we conduct experiments for Theorem 1.2. Part (a) of Theorem 1.2 predicts that

k is a consistent estimator of k0 when kT = o(√T ), k has a finite estimation error in

probability when kT ≍ o(√T ) and k has a larger estimation error when

√T = o(kT ). These

theoretical findings are all supported by Figure 3. Note that, as in Figure 1, there also exists

a “boundary effect” when β1 and β2 have a very small difference. Part (b) of Theorem 1.2

predicts that β1(τT ) should have a Dickey-Fuller distribution and part (c) of Theorem 1.2

predicts that β2(τT ) should have a symmetric distribution around zero that looks like a

normal distribution. These theoretical results are all supported by Figure 4. Note that

1 − 3/6000.7 is smaller than but very near to one, hence the finite-sample distribution of

β2(τT ) when α = 0.7 is a mixture of normal and Dickey-Fuller distributions.

3.3 Experiment 3

Third, we conduct experiments for Theorem 1.3. It can be shown that: (i) Part (a) of

Theorem 1.3 is supported by Figure 5, and it is interesting to find that there exists a “right-

boundary effect” when α = 0.3 (we have also conducted the experiments for the case of

α = 0.4, which is not reported here, and found that the “right-boundary effect” is reduced).

This can be explained by (i) when α = 0.3 in our finite-sample setting, the AR parameter

equals 1 + 3/6000.3 = 1.44, which is much larger than unity, and causes a huge variation

of yt when t > k0 and a retrusion to the right boundary for k; (ii) Part (b) of Theorem

1.3 predicts that β1(τT ) has a Dickey-Fuller distribution, and this is supported by the first

column in Figure 6. However, as a consequence of “right-boundary effect” of k, the finite-

15

sample distribution of β1(τT ) when α = 0.3 suffers from a shape-distortion. We have also

conducted experiments for β2(τT ) with α varying from 0.3 to 0.99995, the results also reveal

shape-distortion. We only report the results for α = 0.95, 0.97 and 0.99 here (see the second

column in Figure 6). We even increased the sample size T to 5, 000 to study the finite-sample

performance of β2(τT ) for α = 0.99 but found that the performance has no improvement.

The result is not reported here for saving space.

3.4 Experiment 4

Fourth, we conduct the experiments for Theorem 1.4. Part (a) of Theorem 1.4 predicts that

k is a consistent estimator of k0 for all kT satisfying kT = o(T ). This theoretical result is

confirmed by Figure 7. Similar to the experiment results of Theorem 1.3, the finite-sample

distributions of β1(τT ) and β2(τT ) suffer from a more or less shape-distortion, based on

experiments with α varying from 0.3 to 0.999. Only the results for α = 0.95, 0.97 and 0.99

are shown here (see Figure 8). The experiments for T = 5, 000 were also conducted for

some α’s, as is expected, the performance of β1(τT ) and β2(τT ) also has no improvement

and some of the results are even worse than the case where T = 600. The results are also

not reported for brevity.

In addition, it can be observed from Figure 1, Figure 3, Figure 5 and Figure 7 that,

as the signal of the model becoming stronger, the finite-sample performance of k improves,

although the theoretical results in Theorem 1.2 and Theorem 1.3 predict that the properties

of k under the situations of β1 = 1 and β2 = 1 − c/kT and of β1 = 1 and β2 = 1 + c/kT

should be the same as T tends to infinity.

3.5 Experiment 5

In this subsection, we conduct Monte Carlo experiments for the following mildly explosive

AR(1) model which is studied in Phillips and Magdalinos (2007a):

yt = βT yt−1 + εt, t = 1, · · · , T, (3.1)

where βT = 1+ c/kT with c > 0 and kT is a positive constant sequence increasing to infinity

such that kT = o(T ), y0 = op(√kT ), and {εt, t ≥ 1} is a sequence of i.i.d. random variables

with zero means and finite variances. Let βT stand for the least squares estimator of βT .

Phillips and Magdalinos (2007a) proved that

kTβTT

2c(βT − βT ) ⇒ ξ, (3.2)

16

where ξ is a standard Cauchy variate.

We perform the following experiments to see how well the finite sample properties of

βT match with the asymptotic results (3.2). First, we set the sample size at T = 600; the

number of replications is set at N = 50, 000; {yt}Tt=1 is generated from Model (3.1), y0 is set

at zero for simplicity and {εt}Tt=1 are generated independently from N(0, 1). In addition, we

set c = 3 and kT = Tα with α = 0.90, 0.95, 0.98 respectively. Note that the AR parameter

is always larger than unity but is close to unity for large α. The first column of Figure 9

reports the results. Note that the finite-sample performance of βT when α = 0.95 appears

to be the best in our specification. However, it should be noted that it is a mixture of

Dickey-Fuller and Cauchy distributions. The finite-sample distribution of βT for α = 0.90

obviously suffers from a shape-distortion. For smaller α’s, the results for α = 0.60, 0.70 and

0.80 which are not reported here reveal that the finite-sample distributions of βT suffer from

a more severe shape-distortion.

We also conduct an experiment for the case of T=2, 000. The second column of Figure

9 reports the corresponding results. Note that, even when T is 2, 000, the finite-sample

performance of βT under various α does not have obvious improvement. The finite-sample

performance of βT with α = 0.90 and 0.95 is even worse than the case where T = 600.

Therefore, the finite-sample performance of βT is not very satisfactory. Our numerical

studies show that the finite-sample performance is very sensitive to kT and an increase in

the sample size does not improve the peformance because the AR parameter depends on

the sample size.

4 Conclusions

In this article, we examined the asymptotic properties of the least squares estimators in

mildly integrated AR(1) model and mildly explosive AR(1) model with a structural change.

Some interesting findings are obtained: (1) the stronger the signals from the model are,

the easier it is for the change point to be located. This suggests that the estimation of

the change point in mildly explosive AR(1) model is easier than that in mildly integrated

AR(1) model. However, if the first sub-sample comes from a unit root model, then it is

more difficult to locate the change point regardless of the order of the second sub-sample.

(2) In the presence of a change point, the first sub-sample will not affect the asymptotic

properties of the least squares estimator of the second AR parameter in mildly integrated

17

Den

sity

0 100 200 300 400 500 6000.

00.

10.

20.

30.

4

(a) α=0.3

Den

sity

0 100 200 300 400 500 600

0.00

0.05

0.10

0.15

(b) α=0.5

Den

sity

0 100 200 300 400 500 600

0.00

00.

005

0.01

00.

015

0.02

00.

025

0.03

00.

035

(c) α=0.7

Figure 1: Histogram of k when β1T = 1− c/kT and β2 = 1, where c = 3, T = 600 and kT = Tα.

18

−15 −10 −5 0 5 10 15

0.0

0.1

0.2

0.3

0.4

−20 −15 −10 −5 0 5 10 15

0.00

0.05

0.10

0.15

0.20

0.25

(a) α=0.3

−15 −10 −5 0 5 10 15

0.0

0.1

0.2

0.3

0.4

−20 −15 −10 −5 0 5 10 15

0.00

0.05

0.10

0.15

0.20

0.25

(b) α=0.5

−15 −10 −5 0 5 10 15

0.0

0.1

0.2

0.3

0.4

−20 −15 −10 −5 0 5 10 15

0.00

0.05

0.10

0.15

0.20

0.25

(c) α=0.7

Figure 2: The first column is the distribution of√

τ0T1−β2

1T(β1(τT )− β1T ) while the second column is

the distribution of (1 − τ0)T (β2(τT ) − 1) under the situation of β1T = 1 − c/kT and β2 = 1, where

c = 3, T = 600 and kT = Tα, the solid lines present the graphs when T = 600 and the dashed lines

present the graphs when T = ∞.

19

Den

sity

0 100 200 300 400 500 6000.

00.

20.

40.

60.

8

(a) α=0.3

Den

sity

0 100 200 300 400 500 600

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(b) α=0.5

Den

sity

0 100 200 300 400 500 600

0.00

0.05

0.10

0.15

0.20

0.25

(c) α=0.7

Figure 3: Histogram of k when β1 = 1 and β2T = 1− c/kT , where c = 3, T = 600 and kT = Tα.

20

−20 −10 −5 0 5 10 15

0.00

0.05

0.10

0.15

0.20

0.25

−15 −10 −5 0 5 10 15

0.0

0.1

0.2

0.3

0.4

(a) α=0.3

−20 −10 −5 0 5 10 15

0.00

0.10

0.20

0.30

−15 −10 −5 0 5 10 15

0.0

0.1

0.2

0.3

0.4

(b) α=0.5

−20 −15 −10 −5 0 5 10 15

0.00

0.05

0.10

0.15

0.20

0.25

0.30

−15 −10 −5 0 5 10 15

0.0

0.1

0.2

0.3

0.4

(c) α=0.7

Figure 4: The first column is the distribution of τ0T (β1(τT ) − 1) while the second column is the

distribution of√

T1−β2

2T(β2(τT ) − β2T ) under the situation of β1 = 1 and β2T = 1 − c/kT , where

c = 3, T = 600 and kT = Tα, the solid lines present the graphs when T = 600 and the dashed lines

present the graphs when T = ∞. 21

Den

sity

0 100 200 300 400 500 6000.

00.

10.

20.

30.

40.

50.

6

(a) α=0.3

Den

sity

0 100 200 300 400 500 600

0.0

0.2

0.4

0.6

0.8

(b) α=0.5

Den

sity

0 100 200 300 400 500 600

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

(c) α=0.7

Figure 5: Histogram of k when β1 = 1 and β2T = 1 + c/kT , where c = 3, T = 600 and kT = Tα.

22

−20 −15 −10 −5 0 5 10 15

0.00

0.05

0.10

0.15

0.20

0.25

(a) α=0.3

−15 −10 −5 0 5 10 15

0.0

0.1

0.2

0.3

0.4

0.5

0.6

(b) α=0.95

−20 −15 −10 −5 0 5 10 15

0.00

0.05

0.10

0.15

0.20

0.25

(c) α=0.5

−15 −10 −5 0 5 10 15

0.0

0.1

0.2

0.3

0.4

(d) α=0.97

−20 −15 −10 −5 0 5 10 15

0.00

0.05

0.10

0.15

0.20

0.25

(e) α=0.7

−15 −10 −5 0 5 10 15

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

(f) α=0.99

Figure 6: The first column is the distribution of τ0T (β1(τT ) − 1) while the second column is the

distribution of√

τ0Tβ22T−1

βT−[τ0T ]2T (β2(τT ) − β2T ) under the situation of β1 = 1 and β2T = 1 + c/kT ,

where c = 3, T = 600 and kT = Tα. The solid lines present the graphs when T = 600 and the dashed

lines present the graphs when T = ∞.

23

Den

sity

0 100 200 300 400 500 6000.

00.

20.

40.

60.

81.

0

(a) α=0.3

Den

sity

0 100 200 300 400 500 600

0.0

0.2

0.4

0.6

0.8

1.0

(b) α=0.5

Den

sity

0 100 200 300 400 500 600

0.0

0.2

0.4

0.6

0.8

1.0

(c) α=0.7

Figure 7: Histogram of k when β1T = 1 + c/kT and β2 = 1, where c = 3, T = 600 and kT = Tα.

24

−15 −10 −5 0 5 10 15

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

−15 −10 −5 0 5 10 15

0.0

0.1

0.2

0.3

0.4

(a) α=0.95

−15 −10 −5 0 5 10 15

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

−15 −10 −5 0 5 10 15

0.0

0.1

0.2

0.3

0.4

(b) α=0.97

−15 −10 −5 0 5 10 15

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

−15 −10 −5 0 5 10 15

0.0

0.1

0.2

0.3

0.4

(c) α=0.99

Figure 8: The first column is the distribution ofβ[τ0T ]

1T

β21T−1

(β1(τT ) − β1T ) while the second column is

the distribution of√

(1−τ0)Tβ21T−1

β[τ0T ]1T (β2(τT ) − 1) under the situation of β1T = 1 + c/kT and β2 = 1,

where c = 3, T = 600 and kT = Tα. The solid lines present the graphs when T = 600 and the dashed

lines present the graphs when T = ∞.

25

−20 −10 0 10 20

0.0

0.2

0.4

0.6

0.8

−20 −10 0 10 20

0.0

0.2

0.4

0.6

0.8

1.0

(a) α=0.90

−20 −10 0 10 20

0.00

0.05

0.10

0.15

0.20

0.25

0.30

−20 −10 0 10 20

0.0

0.1

0.2

0.3

0.4

0.5

(b) α=0.95

−20 −10 0 10 20

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

−20 −10 0 10 20

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

(c) α=0.98

Figure 9: The distributions ofkT βT

T

2c (βT − βT ) with βT = Tα, where c = 3. The first column is the

distribution for T = 600 and the second column is the distribution for T = 2, 000. The solid lines

present the graphs when T = 600 or 2, 000 and the dashed lines present the graphs when T = ∞.

26

AR(1) model, while it is not the case for mildly explosive AR(1) model. This suggests that

the asymptotic properties of the least squares estimator of the second AR parameter in

mildly explosive AR(1) model with a change point are might be very different from those

in mildly explosive AR(1) model or unit root model without a change point, whereas it is

not the case in the mildly integrated AR(1) model; (3) when a unit root model switches

to a mildly integrated or mildly explosive AR(1) model, the asymptotic properties of the

least squares estimator of k0 are the same. In particular, under both situations, our results

reveal that |k − k0| = op(1) when kT = o(√T ), |k − k0| = Op(1) when kT ≍

√T and

|k − k0| = Op(k2T /T ) when

√T = o(kT ). The phase transition for the estimation error of k

occurs when kT ≍√T . (4) Our simulation results show that the finite-sample performance

of the least squares estimators of the AR parameters for the mildly explosive AR(1) model

with and without a structural change suffers from different degrees of shape-distortion. The

shape-distortion phenomenon can be explained partially by a well-known fact that the least

squares estimator of the AR parameter in the autoregression models is biased, and this bias

increases as the AR parameter gets larger (see page 209 in Phillips et al. (2011) and page

1411 in Mikusheva (2007)). The finite-sample performance of the least squares estimators

of the AR parameters coming from the mildly integrated AR(1) model with a structural

change is good.

Acknowledgements. The authors would like to thank professor Peter. C. B. Phillips,

a co-editor and three anonymous referees for their constructive and illuminating comments

that have significantly improved the present article.

References

[1] Bai, J. (1997). Estimating multiple breaks one at a time. Econometric Theory 13 (3): 315-352.

[2] Bai, J. and Perron, P. (1998). Estimating and testing linear models with multiple structural changes.

Econometrica 66 (1): 47-78.

[3] Bai, J., Chen H., Chong, T. T. L. and Wang, X. (2008). Generic consistency of the break-point

estimator under specification errors in a multiple-break model. Econometrics Journal 11: 287-307.

[4] Barsky, R. B. (1987). The fisher hypothesis and the forecastibility and persistence of inflation. Journal

of Monetary Economics 19 (1): 3-24.

27

[5] Berkes, I., Horvath, L., Ling, S. and Schauer, J. (2011). Testing for structural change of AR

model to threshold AR model. Journal of Time Series Analysis 32 (5): 547-565.

[6] Burdekin R. C. K. and Siklos, P. L. (1999). Exchange rate regimes and shifts in inflation persistence:

Does nothing else matter. Journal of Money, Credit and Banking 31: 235-247.

[7] Chan, N. H. and Wei, C. Z. (1987). Asymptotic inference for nearly nonstationary AR(1) processes.

Annals of Statistics 15 (3): 1050-1063.

[8] Chong, T. T. L. (2001). Structural change in AR(1) models. Econometric Theory 17 (1): 87-155.

[9] Chong, T. T. L. (2003). Generic consistency of the break-point estimator under specification errors.

Econometrics Journal 6: 167-192.

[10] Csorgo, M., Szyszkowicz, B. and Wang, Q. Y. (2003). Donsker’s theorem for self-normalized

partial sums processes. Annals of Probability 31 (3): 1228-1240.

[11] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Volume 2. John Wiley

and Sons, New York.

[12] Gine, E., Gotze, F. and Mason, D. M. (1997). When is the Student t-statistic asymptotically

standard normal? Annals of Probability 25 (3): 1514–1531.

[13] Halunga, A. G. and Osborn, D. R. (2012). Ratio-based estimators for a change point in persistence.

Journal of Econometrics 171 (1): 24-31.

[14] Hansen, B. E. (2001). The new econometrics of structural change. Journal of Economic Perspertives

15 (3): 117-128.

[15] Hakkio, C. S. and Rush, M. (1991). Is the budget deficit too large? Economic Inquiry 29: 429-445.

[16] Harvey, D. I., Leybourne, S. J. and Taylor, A. M. R. (2006). Modified tests for a change in

persistence. Journal of Econometrics 134: 441-469, and Corrigendum, Journal of Econometrics 168:

407.

[17] Huang S. H., Pang, T. X. and Weng, C. G. (2014). Limit theory for moderate deviations from a

unit root under innovations with a possibly infinite variance. Methodology and Computing in Applied

Probability 16 (1): 187-206.

[18] Kejriwal, M., Perron, P. and Zhou, J. (2013). Wald tests for detecting multiple structural changes

in persistence. Econometric Theory 29: 289-323.

28

[19] Kim, J. and Pollard, D. (1990). Cube root asymptotics. Annals of Statistics 18 (1): 191-219.

[20] Lee, S., Seo, M. H. and Shin, Y. (2016). The lasso for high dimensional regression with a possible

change point. J. R. Statist. Soc. B 78 (1): 193-210.

[21] Leybourne, S. J., Kim, T., Smith, V. and Newbold, P. (2003). Tests for a change in persistence

against the null of difference stationarity. Econometrics Journal 6: 290-310.

[22] Magdalinos, T. (2012). Mildly explosive autoregression under weak and strong dependence. Journal

of Econometrics 169 (2): 179-187.

[23] Mandelbrot, B. B. (1963). The variation of certain speculative prices. Journal of Business 36: 392-

417.

[24] Mankiw, N. G. and Miron, J. A. (1986). The changing behavior of the term structure of interest

rates. Quarterly Journal of Economics 101 (2): 221-228.

[25] Mankiw, N. G., Miron, J. A. and Weil, D. N. (1987). The adjustment of expectations to a change

in regime: A study of the founding of the Federal Reserve. American Economic Review 77 (3): 358-374.

[26] Mikusheva, A. (2007). Uniform inference in autoregressive models. Econometrica 75 (5): 1411-1452.

[27] Pang, T. X. and Zhang, D. N. (2015). Asymptotic inferences for an AR(1) model with a change

point and possibly infinite variance. Communications in Statistics-Theory and Methods 44: 4848-4865.

[28] Pang, T. X., Zhang, D. N. and Chong, T. T. L. (2014). Asymptotic inferences for an AR(1) model

with a change point: stationary and nearly non-stationary cases. Journal of Time Series Analysis 35:

133-150.

[29] Phillips, P. C. B. (1987). Towards a unified asymptotic theory for autoregression. Biometrika 74 (3):

535-547.

[30] Phillips, P. C. B. and Magdalinos, T. (2007a). Limit theory for moderate deviations from a unit

root. Journal of Econometrics 136: 115–130.

[31] Phillips, P. C. B. and Magdalinos, T. (2007b). Limit theory for moderate deviations from unity

under weak dependence. In: Phillips, G.D.A., Tzavalis, E. (Eds.), The Refinement of Econometric

Estimation and Test Procedures. Cambridge: Cambridge University Press, pp.123-162.

[32] Phillips, P. C. B., Shi, S. and Yu, J. (2015a). Testing for Multiple Bubbles: Historical episodes of

exuberance and collapse in the S&P 500, International Economic Review 56 (4): 1043-1078.

29

[33] Phillips, P. C. B., Shi, S. and Yu, J. (2015b). Testing for Multiple Bubbles: Limit Theory of Real

Time Detectors. International Economic Review 56 (4): 1079-1134.

[34] Phillips, P. C. B., Shi, S. and Yu, J. (2015c). Supplement to Two Papers on Multiple Bubbles.

International Economic Review, Online Supplementary Material.

[35] Phillips, P. C. B., Wu, Y. and Yu, J. (2011). Explosive behavior in the 1990s Nasdaq: when did

exuberance escalate asset values? International Economic Review 52 (1): 201-226.

[36] Phillips, P. C. B. and Yu, J. (2011). Dating the timeline of financial bubbles during the subprime

crisis. Quantitative Economics 2: 455-491.

[37] Pitarakis, J.-Y. (2004). Least squares estimation and tests of breaks in mean and variance under

misspecification. Econometrics Jouranl 7: 32-54.

[38] Qu, Z. (2008). Testing for structural change in regression quantiles. Journal of Econometrics 146:

170-184.

[39] Stock, J. H. and Watson, M. W. (1996). Evidence on structural instability in Macroeconomic time

series relations. Journal of Business and Economic Statistics 14: 11-30.

[40] Stock, J. H. and Watson, M. W. (1999). A comparison of linear and nonlinear univariate models for

forecasting macroeconomic time series. In R. F. Engle and H. White (Eds.), Cointegration, Causality

and Forcasting: A Festschrift in Honour of C. W. J. Granger, pp. 1-44. Oxford: Oxford University

Press.

Appendix A: Proof of Theorem 1.1

In order to prove theorems when the εt’s are heavy-tailed, we employ the truncation tech-

nique in this paper. Let

l(u) = E(ε21I{|ε1| ≤ u}), b = inf{u ≥ 1 : l(u) > 0},

and

ηj = inf{s : s ≥ b+ 1,l(s)

s2≤ 1

j}, for j = 1, 2, 3, · · · .

When εt is a random variable from the DAN, then l(u) is a slowly varying function ap-

proaching a constant or infinity as u tends to infinity. It can be shown that T l(ηT ) ≤ η2T for

30

all T ≥ 1 and η2T ≈ T l(ηT ) for large T . In addition, for each T we let ε(1)t = εtI{|εt| ≤ ηT } − E(εtI{|εt| ≤ ηT })

ε(2)t = εtI{|εt| > ηT } − E(εtI{|εt| > ηT })

(A.1)

for t = 1, · · · , T . This is a well-known truncation technique for dealing with the weak conver-

gence of the random variables from the DAN with zero mean and possible infinite variance.

Huang et al. (2014) successfully extended the results in Phillips and Magdalinos (2007a) to

the DAN case by applying this truncation technique.

The following two lemmas are needed in the proof of Theorem 1.1.

Lemma A.1 Let {yt} be generated according to Model (1.1), where β1 = β1T = 1 − c/kT

for a positive constant c. Then under assumptions C1-C4, the following results hold jointly:

(a) 1√TkT l(ηT )

∑[τ0T ]t=1 yt−1εt ⇒ N(0, τ02c),

(b) 1TkT l(ηT )

∑[τ0T ]t=1 y2t−1

p→ τ02c ,

(c)y[rT ]√kT l(ηT )

⇒∫∞0 exp (−cs)dW (s) for any 0 < r ≤ τ0.

Proof. The proofs of (a) and (b) can be completed by following the proof of Theorem

3.2 in Phillips and Magdalinos (2007a) and by using the truncation technique as shown in

(A.1). The details are omitted. To prove part (c), it suffices to note that

y[rT ]√kT l(ηT )

=1√

kT l(ηT )

((1− c

kT)[rT ]y0 +

[rT ]−1∑i=0

(1− c

kT)iε[rT ]−i

)

=

[rT ]−1∑i=0

((1− c

kT)kT)i/kT ε[rT ]−i√

kT l(ηT )+ op(1)

⇒∫ ∞

0exp (−cs)dW (s)

by Theorem 1 in Csorgo et al. (2003), (3.4) in Gine et al. (1997) and assumptions C2 and

C3.

Q.E.D. �

Remark A.1 Part (c) can be written equivalently as follows: for any 0 < r ≤ τ0

y[rT ]√kT l(ηT )

⇒ N(0,1

2c). (A.2)

31

One can employ the Lindeberg-Feller central limit theorem to prove (A.2). Using Lemma 1

in Csorgo et al. (2003) and assumptions C2 and C3, one can show that

y[rT ]√kT l(ηT )

=1√

kT l(ηT )

[rT ]∑t=1

β[rT ]−t1T εt + op(1)

=1√

kT l(ηT )

[rT ]∑t=1

β[rT ]−t1T ε

(1)t · (1 + op(1)) + op(1).

Moreover, as E(εt) = 0, we have

|E(εtI{|εt| ≤ ηT })| = |E(εtI{|εt| > ηT })| = o(l(ηT )/ηT ) (A.3)

by Lemma 1 in Csorgo et al. (2003). As a result, we have

V ar

(1√

kT l(ηT )

[rT ]∑t=1

β[rT ]−t1T ε

(1)t

)

=1

kT

[rT ]∑t=1

β2([rT ]−t)1T · (1 + o(1))

=1

kT (1− β21T )

· (1 + o(1))

=1

2c· (1 + o(1))

since V ar(ε(1)t ) = E(ε

(1)t )2 = E(ε2t I{|εt| ≤ ηT })− (E(εtI{|εt| ≤ ηT }))2 = l(ηT )(1 + o(1)) by

(A.3) and recalling η2T ≈ T l(ηT ) for large T .

To prove (A.2), it suffices to verify the Lindeberg-Feller condition. For any δ > 0, we

have

[rT ]∑t=1

E

(β[rT ]−t1 ε

(1)t√

kT l(ηT )

)2

I

{∣∣∣β[rT ]−t1 ε

(1)t√

kT l(ηT )

∣∣∣ > δ

}

≤ max1≤t≤[rT ]

E

(ε(1)t√l(ηT )

)2

I

{∣∣∣β[rT ]−t1 ε

(1)t√

l(ηT )

∣∣∣ > δ√

kT

}· 1

kT

[rT ]∑t=1

β2([rT ]−t)1

≤ max1≤t≤[rT ]

E

(ε(1)t√l(ηT )

)2

I

{∣∣∣ ε(1)t√l(ηT )

∣∣∣ > δ√

kT

}· 1

2c· (1 + o(1))

= o(1)

since the variance of ε(1)1 /√

l(ηT ) exists and kT → ∞. Hence, (A.2) is proved.

Another approach to proving (A.2) is applying Ito isometry theorem to show that∫∞0 exp (−cs)dW (s)

and N(0, 12c) are identical in distribution. To this purpose, it suffices to show that the vari-

ance of∫∞0 exp (−cs)dW (s) is 1

2c . Applying Ito isometry theorem, one has

V ar

(∫ ∞

0exp (−cs)dW (s)

)= E

(∫ ∞

0exp (−cs)dW (s)

)2

=

∫ ∞

0e−2csds =

1

2c.

32

Lemma A.2 Let {yt} be generated according to Model (1.1), where β1 = β1T = 1 − c/kT

for a positive constant c and β2 = 1. Then under assumptions C1-C4, the following results

hold jointly:

(a) 1T l(ηT )

∑Tt=[τ0T ]+1 yt−1εt ⇒ 1

2(W (1)−W (τ0))2 − 1

2(1− τ0),

(b) 1T 2l(ηT )

∑Tt=[τ0T ]+1 y

2t−1 ⇒

∫ 1τ0(W (s)−W (τ0))

2ds.

Proof. To prove part (a), note that

yt−1εt =1

2(y2t − y2t−1 − ε2t ), t = [τ0T ] + 1, · · · , T,

which implies that

1

T l(ηT )

T∑t=[τ0T ]+1

yt−1εt =1

2T l(ηT )

(y2T − y2[τ0T ] −

T∑t=[τ0T ]+1

ε2t

). (A.4)

Note also thaty[τ0T ]√T l(ηT )

= op(1) (A.5)

by part (c) of Lemma A.1. Denote

ST (τ0, s) :=1√

T l(ηT )

[sT ]∑t=[τ0T ]+1

εt.

Then

ST (τ0, s) ⇒ W (s)−W (τ0) (A.6)

for any τ0 ≤ s ≤ 1 by the functional central limit theorem for the i.i.d. random variables

from the DAN (see Theorem 1 in Csorgo et al. (2003) and (3.4) in Gine et al. (1997)).

Combining this result with (A.5) immediately leads to

y2T − y2[τ0T ]

2T l(ηT )=

(y[τ0T ] +∑T

i=[τ0T ]+1 εi)2 − y2[τ0T ]

2T l(ηT )⇒ 1

2(W (1)−W (τ0))

2. (A.7)

In addition, it is trivial that ∑Tt=[τ0T ]+1 ε

2t

2T l(ηT )

p→ 1− τ02

(A.8)

when ε′ts are i.i.d. in the DAN. Now, it follows from (A.4), (A.7) and (A.8) that part (a)

holds.

33

To prove part (b), using (A.5) and (A.6), we have

1

T 2l(ηT )

T∑t=[τ0T ]+1

y2t−1

=1

T 2l(ηT )

T∑t=[τ0T ]+1

(y[τ0T ] +

t−1∑i=[τ0T ]+1

εi

)2=

1

T

T∑t=[τ0T ]+1

( y[τ0T ]√T l(ηT )

+ ST (τ0,t− 1

T))2

=

T∑t=[τ0T ]+1

∫ tT

t−1T

( y[τ0T ]√T l(ηT )

+ ST (τ0,t− 1

T))2

ds

=

T∑t=[τ0T ]+1

∫ tT

t−1T

( y[τ0T ]√T l(ηT )

+ ST (τ0, s))2

ds · (1 + op(1))

=

∫ 1

τ0

( y[τ0T ]√T l(ηT )

+ ST (τ0, s))2

ds · (1 + op(1))

⇒∫ 1

τ0

(W (s)−W (τ0))2ds. (A.9)

Finally, since part (a) and part (b) hold jointly, this completes our proof.

Q.E.D. �

Proof of Theorem 1.1. To derive the limiting distribution of τT , one can follow Appendix

G in Chong (2001) with the following two main modifications: (1) Let g(T ) = kT /c in

Appendix G in Chong (2001), (2) Replace∫∞0 exp(−s)dW1(s) by

√c∫∞0 exp (−cs)dW1(s),

which is the limiting distribution of y[τ0T ]−t−1/√

g(T )l(ηT ) for 0 ≤ t ≤ [|ν|g(T )]− 1 (where

ν is a constant) by noting that

y[τ0T ]−t−1√g(T )l(ηT )

=1√

g(T )l(ηT )

((1− c

kT)[τ0T ]−t−1y0 +

[τ0T ]−t−2∑i=0

(1− c

kT)iε[τ0T ]−t−1−i

)

=√c ·

[τ0T ]−t−2∑i=0

((1− c

kT)kT)i/kT ε[τ0T ]−t−1−i√

kT l(ηT )+ op(1)

⇒√c

∫ ∞

0exp(−cs)dW1(s) = Bc(

1

2).

Moreover, it is worth mentioning that (1.2) is a special case of the limiting distribution of

τT in Theorem 3 in Chong (2001) when the moment conditions E(y20) < ∞ and E(ε4t ) < ∞

in his paper are satisfied and c = 1. To relax these moment conditions, we only need to

apply the truncation technique (A.1). The details are omitted for brevity.

To find the limiting distribution of β1(τT ), note that |τT − τ0| = Op(kT /T ), the limit-

ing distribution in Lemma A.1(c) is irrespective of the constant r and it suffices to study

34

the magnitude of∑[τ0T ]

t=[τ0T−kT ]+1 y2t−1 in order to obtain the magnitude of

∑[τ0T ]t=[τTT ]+1 y

2t−1.

Therefore, when τT ≤ τ0, Lemma A.1(c) implies that

[τ0T ]∑t=[τ0T−kT ]+1

y2t−1 = Op(kT l(ηT )) · kT = Op(k2T l(ηT )), (A.10)

yielding[τ0T ]∑

t=[τTT ]+1

y2t−1 = Op(k2T l(ηT )). (A.11)

In addition, to obtain the magnitude of∑[τ0T ]

t=[τTT ]+1 yt−1εt, it suffices to study the magnitude

of∑[τ0T ]

t=[τ0T−kT ] yt−1εt. By squaring yt = β1T yt−1 + εt and summing over t ∈ {[τ0T − kT ] +

1, · · · , [τ0T ]} we obtain

[τ0T ]∑t=[τ0T−kT ]+1

yt−1εt =1− β2

1T

2β1T

[τ0T ]∑t=[τ0T−kT ]+1

y2t−1 −β1T2

(y2[τ0T−kT ] − y2[τ0T ])

− 1

2β1T

[τ0T ]∑t=[τ0T−kT ]+1

ε2t . (A.12)

It follows from (A.10) and Lemma A.1(c) respectively that

1− β21T

2β1T

[τ0T ]∑t=[τ0T−kT ]+1

y2t−1 = Op(kT l(ηT )), (A.13)

β1T2

(y2[τ0T−kT ] − y2[τ0T ]) =β1T2

(y[τ0T−kT ] − y[τ0T ])(y[τ0T−kT ] + y[τ0T ]) = Op(kT l(ηT )).

(A.14)

Moreover, by applying the Law of Large Numbers, we have

1

2β1T

[τ0T ]∑t=[τ0T−kT ]+1

ε2t = Op(kT l(ηT )). (A.15)

Combining (A.12)-(A.15) leads to

[τ0T ]∑t=[τ0T−kT ]+1

yt−1εt = Op(kT l(ηT )),

yielding[τ0T ]∑

t=[τTT ]+1

yt−1εt = Op(kT l(ηT )). (A.16)

35

Similarly, if τT > τ0, we have

[τ0T+kT ]∑t=[τ0T ]+1

y2t−1 =

[τ0T+kT ]∑t=[τ0T ]+1

(y[τ0T ] +

t−1−[τ0T ]∑i=1

ε[τ0T ]+i

)2

≤ 2

[τ0T+kT ]∑t=[τ0T ]+1

y2[τ0T ] + 2

[τ0T+kT ]∑t=[τ0T ]+1

( t−1−[τ0T ]∑i=1

ε[τ0T ]+i

)2= Op(k

2T l(ηT )) +Op(k

2T l(ηT ))

= Op(k2T l(ηT )),

leading to[τTT ]∑

t=[τ0T ]+1

y2t−1 = Op(k2T l(ηT )). (A.17)

In addition, by squaring yt = yt−1+ εt and summing over t ∈ {[τ0T ] + 1, · · · , [τ0T + kT ]} we

obtain

[τ0T+kT ]∑t=[τ0T ]+1

yt−1εt =1

2

(y2[τ0T+kT ] − y2[τ0T ]

)− 1

2

[τ0T+kT ]∑t=[τ0T ]+1

ε2t

=1

2

2y[τ0T ]

[τ0T+kT ]∑t=[τ0T ]+1

εt +( [τ0T+kT ]∑

t=[τ0T ]+1

εt

)2− 1

2

[τ0T+kT ]∑t=[τ0T ]+1

ε2t .

Similarly to the derivation of (A.16), we have

[τTT ]∑t=[τ0T ]+1

yt−1εt = Op(kT l(ηT )). (A.18)

Suppose kT = o(T ), applying (A.11), (A.16), (A.17), (A.18) and Lemma A.1 and following

36

Appendix G in Chong (2001), we have

√TkT (β1(τT )− β1(τ0))

=√

TkT

(∑[τTT ]t=1 ytyt−1∑[τTT ]

t=1 y2t−∑[τ0T ]

t=1 ytyt−1∑[τ0T ]t=1 y2t

)

= I{τT ≤ τ0}√

TkT

(∑[τ0T ]t=[τTT ]+1 y

2t−1∑[τTT ]

t=1 y2t−1

∑[τ0T ]t=1 yt−1εt∑[τ0T ]t=1 y2t−1

−∑[τ0T ]

t=[τTT ]+1 yt−1εt∑[τTT ]t=1 y2t−1

)

+I{τT > τ0}√

TkT

(−∑[τTT ]

t=[τ0T ]+1 y2t−1∑[τTT ]

t=1 y2t−1

∑[τ0T ]t=1 yt−1εt∑[τ0T ]t=1 y2t−1

+

∑[τTT ]t=[τ0T ]+1 yt−1εt∑[τTT ]

t=1 y2t−1

+ (β2 − β1T )

∑[τTT ]t=[τ0T ]+1 y

2t−1∑[τTT ]

t=1 y2t−1

)= I{τT ≤ τ0}

√TkT

(Op(

k2T l(ηT )

TkT l(ηT ))Op(

1√TkT

) +Op(kT l(ηT )

TkT l(ηT )))

+I{τT > τ0}√

TkT

(Op(

k2T l(ηT )

TkT l(ηT ))Op(

1√TkT

) +Op(kT l(ηT )

TkT l(ηT )) +Op(

k2T l(ηT )

Tk2T l(ηT )))

= op(1).

Thus, β1(τT ) and β1(τ0) have the same asymptotic distribution. Applying Lemma A.1, we

have √TkT (β1(τ0)− β1) =

1√TkT l(ηT )

∑[τ0T ]t=1 yt−1εt

1TkT l(ηT )

∑[τ0T ]t=1 y2t−1

⇒ N(0,2c

τ0),

which immediately implies that

√TkT (β1(τT )− β1) ⇒ N(0,

2c

τ0).

The above result is equivalent to (1.3) by the fact 1− β21T = 2c/kT (1 + o(1)).

To find the limiting distribution of β2(τT ), suppose kT = o(T ), applying (A.11), (A.16),

37

(A.17), (A.18) and Lemma A.2 and following Appendix G in Chong (2001), we have

T (β2(τT )− β2(τ0))

= T

(∑Tt=[τTT ]+1 ytyt−1∑T

t=[τTT ]+1 y2t

−∑T

t=[τ0T ]+1 ytyt−1∑Tt=[τ0T ]+1 y

2t

)

= I{τT ≤ τ0}T(−∑[τ0T ]

t=[τTT ]+1 y2t−1∑T

t=[τTT ]+1 y2t−1

∑Tt=[τ0T ]+1 yt−1εt∑Tt=[τ0T ]+1 y

2t−1

+

∑[τ0T ]t=[τTT ]+1 yt−1εt∑Tt=[τTT ]+1 y

2t−1

+ (β1T − β2)

∑[τ0T ]t=[τTT ]+1 y

2t−1∑T

t=[τTT ]+1 y2t−1

)

+I{τT > τ0}T(∑[τTT ]

t=[τ0T ]+1 y2t−1∑T

t=[τTT ]+1 y2t−1

∑Tt=[τ0T ]+1 yt−1εt∑Tt=[τ0T ]+1 y

2t−1

−∑[τTT ]

t=[τ0T ]+1 yt−1εt∑Tt=[τTT ]+1 y

2t−1

)= I{τT ≤ τ0}T

(Op(

k2T l(ηT )

T 2l(ηT ))Op(

1

T) +Op(

kT l(ηT )

T 2l(ηT )) +Op(

k2T l(ηT )

kTT 2l(ηT )))

+I{τT > τ0}T(Op(

k2T l(ηT )

T 2l(ηT ))Op(

1

T) +Op(

kT l(ηT )

T 2l(ηT )))

= op(1).

Thus, β2(τT ) and β2(τ0) also have the same asymptotic distribution. Applying Lemma A.2,

we have

T (β2(τ0)− β2) =

1T l(ηT )

∑Tt=[τ0T ]+1 yt−1εt

1T 2l(ηT )

∑Tt=[τ0T ]+1 y

2t−1

⇒ (W (1)−W (τ0))2 − (1− τ0)

2∫ 1τ0(W (s)−W (τ0))2ds

,

which immediately implies that

T (β2(τT )− β2) ⇒(W (1)−W (τ0))

2 − (1− τ0)

2∫ 1τ0(W (s)−W (τ0))2ds

.

It is trivial that

(W (1)−W (τ0))2 − (1− τ0)

2∫ 1τ0(W (s)−W (τ0))2ds

d=

W2(1)− 1

2(1− τ0)∫ 10 W

2(s)ds

by the facts

{W (s)−W (τ0), τ0 < s ≤ 1} d= {W (s− τ0), τ0 < s ≤ 1}

and {W (s− τ0)√1− τ0

, τ0 < s ≤ 1}

d={W (

s− τ01− τ0

), τ0 < s ≤ 1}

and applying the change of variables. Hence, (1.4) holds.

Q.E.D. �

38

Appendix B: Proof of Theorem 1.2

The following three lemmas are used in the proof of Theorem 1.2.

Lemma B.1 Let {yt} be generated by Model (1.1) with β1 = 1. Then under the assumptions

C1-C4, the following results hold jointly:

(a) 1T l(ηT )

∑[τ0T ]t=1 yt−1εt ⇒ 1

2(W2(τ0)− τ0),

(b) 1T 2l(ηT )

∑[τ0T ]t=1 y2t−1 ⇒

∫ τ00 W 2(s)ds.

Proof. This lemma can easily be proved by using the standard arguments in the unit root

model with finite variance and by applying the truncation technique (A.1). The details are

omitted.

Q.E.D. �

Lemma B.2 Let β2 = β2T = 1− c/kT with c > 0. Then under the assumptions C1-C4, we

have, for any τ0 < s ≤ 1

1√kT l(ηT )

[sT ]∑t=[τ0T ]+1

β[sT ]−t2T εt ⇒ N(0,

1

2c).

Proof. The proof can be completed by following the proof of part (c) of Lemma A.1 or

Remark A.1. The details are therefore omitted.

Q.E.D. �

Lemma B.3 Let {yt} be generated by Model (1.1), where β1 = 1 and β2 = β2T = 1− c/kT

with c > 0. Then, under the assumptions C1-C4, the following results hold jointly:

(a) 1√TkT l(ηT )

∑Tt=[τ0T ]+1 yt−1εt ⇒ W ( 1

2c),

(b) 1TkT l(ηT )

∑Tt=[τ0T ]+1 y

2t−1 ⇒ 1

2c(W2(τ0) + 1− τ0).

Proof. To prove part (a), note that the following decomposition holds,

1√TkT l(ηT )

T∑t=[τ0T ]+1

yt−1εt

=1√

TkT l(ηT )

T∑t=[τ0T ]+1

(βt−[τ0T ]−12T y[τ0T ] +

t−1∑i=[τ0T ]+1

βt−i−12T εi

)εt

=1√

TkT l(ηT )

((y0 +

[τ0T ]∑j=1

εj

) T∑t=[τ0T ]+1

βt−[τ0T ]−12T εt +

T∑t=[τ0T ]+1

εt

t−1∑i=[τ0T ]+1

βt−i−12T εi

):= I1 + I2, (B.1)

39

where

I1 =1√

TkT l(ηT )

(y0 +

[τ0T ]∑j=1

εj

) T∑t=[τ0T ]+1

βt−[τ0T ]−12T εt

=1 + op(1)√TkT l(ηT )

[τ0T ]∑j=1

εj

T∑t=[τ0T ]+1


=1 + op(1)√TkT l(ηT )

[τ0T ]∑j=1

ε(1)j

T∑t=[τ0T ]+1

βt−[τ0T ]−12T ε

(1)t

and

I2 =1√

TkT l(ηT )

T∑t=[τ0T ]+1

εt

t−1∑i=[τ0T ]+1

βt−i−12T εi

=1 + op(1)√TkT l(ηT )

T∑t=[τ0T ]+1

ε(1)t

t−1∑i=[τ0T ]+1

βt−i−12T ε

(1)i

by assumptions C1-C3 and Lemma 1 in Csorgo et al. (2003).

Consider the term I1. It is obvious that{ε(1)j√l(ηT )

T∑t=[τ0T ]+1

βt−[τ0T ]−12T

ε(1)t√l(ηT )

}[τ0T ]

j=1

(B.2)

is a sequence of martingale difference with respect to the filtration Fj = σ(ε1, · · · , εj) with

E( ε

(1)j√l(ηT )

T∑t=[τ0T ]+1

βt−[τ0T ]−12T

ε(1)t√l(ηT )

|Fj−1

)= 0

and

1

TkT

[τ0T ]∑j=1

E(( ε

(1)j√l(ηT )

T∑t=[τ0T ]+1

βt−[τ0T ]−12T

ε(1)t√l(ηT )

)2|Fj−1

)=

τ02c

· (1 + o(1)).

Then, applying the central limit theorem for martingale difference sequences, we have

I1 ⇒ N(0,τ02c

). (B.3)

Analogously, for I2, {ε(1)t√l(ηT )

t−1∑i=[τ0T ]+1

βt−i−12T

ε(1)i√l(ηT )

}T

t=[τ0T ]+1

(B.4)

is a sequence of martingale difference with respect to the filtration Ft = σ(ε[τ0T ], · · · , εt)

with

E( ε

(1)t√l(ηT )

t−1∑i=[τ0T ]+1

βt−i−12T

ε(1)i√l(ηT )

|Ft−1

)= 0

40

and

1

TkT

T∑t=[τ0T ]+1

E(( ε

(1)t√l(ηT )

t−1∑i=[τ0T ]+1

βt−i−12T

ε(1)i√l(ηT )

)2|Ft−1

)p→ 1− τ0

2c· (1 + o(1)).

Hence, applying the central limit theorem for martingale difference sequences leads to

I2 ⇒ N(0,1− τ02c

). (B.5)

Note that the two martingale difference sequences (B.2) and (B.4) are independent. The

proof for part (a) is completed by combining (B.1), (B.3) and (B.5).

To prove part (b), by squaring yt = (1− c/kT )yt−1 + εt and summing over t ∈ {[τ0T ] +

1, · · · , T}, we obtain

T∑t=[τ0T ]+1

y2t = (1− c

kT)2

T∑t=[τ0T ]+1

y2t−1 +

T∑t=[τ0T ]+1

ε2t + 2(1− c

kT)

T∑t=[τ0T ]+1

yt−1εt.

This implies that

(2c

kT− c2

k2T)

T∑t=[τ0T ]+1

y2t−1 = y2[τ0T ] − y2T +

T∑t=[τ0T ]+1

ε2t + 2(1− c

kT)

T∑t=[τ0T ]+1

yt−1εt. (B.6)

Note thaty[τ0T ]√T l(ηT )

⇒ W (τ0) (B.7)

since β1 = 1, it follows from Lemma B.2 and assumption C2 that

yT√T l(ηT )

=βT−[τ0T ]2T y[τ0T ] +

∑Ti=[τ0T ]+1 β

T−i2T εi√

T l(ηT )= op(1). (B.8)

In addition,

2(1− c

kT)

T∑t=[τ0T ]+1

yt−1εt = op(T l(ηT )) (B.9)

by part (a). Then, combining (B.6)-(B.9) and (A.8) together yields

1

TkT l(ηT )

T∑t=[τ0T ]+1

y2t−1 ⇒1

2c(W 2(τ0) + 1− τ0).

Note that the random part (i.e.,W 2(τ0)) of the limiting distribution of 1TkT l(ηT )

∑Tt=[τ0T ]+1 y

2t−1

is due to the limiting behavior of (∑[τ0T ]

i=1 εi)2, while the limiting distribution of 1√

TkT l(ηT )


is determined by the limiting behavior of

[τ0T ]∑j=1

εj

T∑t=[τ0T ]+1

βt−[τ0T ]−12T εt and

T∑t=[τ0T ]+1

εt

t−1∑i=[τ0T ]+1

βt−i−12T εi.

41

Note also that∑[τ0T ]

i=1 εi and∑T

t=[τ0T ]+1 εt∑t−1

i=[τ0T ]+1 βt−i−12T εi are independent of each other,

and∑[τ0T ]

i=1 εi and∑[τ0T ]

j=1 εj∑T

t=[τ0T ]+1 βt−[τ0T ]−12T εt are uncorrelated. Hence, the two limiting

distributions in part (a) and part (b) are independent. Therefore, part (a) and part (b) hold

jointly.

Q.E.D. �

Lemma B.4 Let {yt} be generated according to Model (1.1), where β1 = 1 and β2 = β2T =

1 − c/kT with c > 0. Then under assumptions C1-C4 with kT = o(√T ), we have for any

fixed integer m ≥ 0,

(a)k2T

T l(ηT )

(RSST (τ0 − m

T )−RSST (τ0))⇒ c2mW 2(τ0),

(b)k2T

T l(ηT )

(RSST (τ0 +

mT )−RSST (τ0)

)⇒ c2mW 2(τ0).

Proof. To prove part (a), from equation (B.2) in Chong (2001), we have

RSST (τ0 −m

T)−RSST (τ0)

= 2(β2T − β1)

(∑[τ0T ]t=[τ0T ]−m+1 y

2t−1

∑Tt=[τ0T ]+1 yt−1εt∑T

t=[τ0T ]−m+1 y2t−1

−∑T

t=[τ0T ]+1 y2t−1

∑[τ0T ]t=[τ0T ]−m+1 yt−1εt∑T

t=[τ0T ]−m+1 y2t−1

)

+ (β2T − β1)2

∑Tt=[τ0T ]+1 y

2t−1

∑[τ0T ]t=[τ0T ]−m+1 y

2t−1∑T

t=[τ0T ]−m+1 y2t−1

+ ΛT (τ0 −m

T)

where

ΛT (τ0 −m

T)

=

(∑[τ0T ]t=1 yt−1εt

)2∑[τ0T ]

t=1 y2t−1

−

(∑[τ0T ]−mt=1 yt−1εt

)2∑[τ0T ]−m

t=1 y2t−1

+

(∑Tt=[τ0T ]+1 yt−1εt

)2∑T

t=[τ0T ]+1 y2t−1

−

(∑Tt=[τ0T ]−m+1 yt−1εt

)2∑T

t=[τ0T ]−m+1 y2t−1

.

Obviously, ΛT (τ0 − mT ) = Op(l(ηT )). Then it follows that

k2TT l(ηT )

(RSST (τ0 −

m

T)−RSST (τ0)

)=

k2TT l(ηT )

·(− 2c

kT

)·(

Op(T l(ηT ))Op(√TkT l(ηT ))

Op(T l(ηT )) +Op(TkT l(ηT ))+

Op(TkT l(ηT ))Op(√T l(ηT ))

Op(T l(ηT )) +Op(TkT l(ηT ))

)

+k2T

T l(ηT )· c

2

k2T· Op(TkT l(ηT ))

Op(T l(ηT )) +Op(TkT l(ηT ))

[τ0T ]∑[τ0T ]−m+1

y2t−1 + op(1)

=c2(1 + op(1))

T l(ηT )

[τ0T ]∑[τ0T ]−m+1

y2t−1 + op(1)

⇒ c2mW 2(τ0)

42

by Lemma B.3, kT = o(√T ) and the observation∣∣∣∣∣∣

[τ0T ]∑t=[τ0T ]−m+1

yt−1εt

∣∣∣∣∣∣ ≤√√√√√ [τ0T ]∑

t=[τ0T ]−m+1

ε2t ·

√√√√ T∑t=[τ0T ]−m+1

y2t−1 = Op(√T l(ηT )).

To prove part (b), from equation (B.4) in Chong (2001), we have

RSST (τ0 +m

T)−RSST (τ0)

= 2(β2T − β1)

(∑[τ0T ]t=1 y2t−1

∑[τ0T ]+mt=[τ0T ]+1 yt−1εt∑[τ0T ]+m

t=1 y2t−1

−∑[τ0T ]+m

t=[τ0T ]+1 y2t−1

∑[τ0T ]t=1 yt−1εt∑[τ0T ]+m

t=1 y2t−1

)

+ (β2T − β1)2

∑[τ0T ]+mt=[τ0T ]+1 y

2t−1

∑[τ0T ]t=1 y2t−1∑[τ0T ]+m

t=1 y2t−1

+ ΛT (τ0 +m

T)

where

ΛT (τ0 +m

T)

=

(∑[τ0T ]t=1 yt−1εt

)2∑[τ0T ]

t=1 y2t−1

−

(∑[τ0T ]+mt=1 yt−1εt

)2∑[τ0T ]+m

t=1 y2t−1

+


)2∑T

t=[τ0T ]+1 y2t−1

−

(∑Tt=[τ0T ]+m+1 yt−1εt

)2∑T

t=[τ0T ]+m+1 y2t−1

.

Since ΛT (τ0 +mT ) = Op(l(ηT )) and∣∣∣∣∣∣

[τ0T ]+m∑t=[τ0T ]+1

yt−1εt

∣∣∣∣∣∣ ≤√√√√√ [τ0T ]+m∑

t=[τ0T ]+1

ε2t ·

√√√√√ [τ0T ]+m∑t=[τ0T ]+1

y2t−1 = Op(√T l(ηT )),

we have

k2TT l(ηT )

(RSST (τ0 +

m

T)−RSST (τ0)

)=

k2TT l(ηT )

·(− 2c

kT

)·(

Op(T2l(ηT ))Op(

√T l(ηT ))

Op(T 2l(ηT )) +Op(T l(ηT ))+

Op(T l(ηT ))Op(T l(ηT ))

Op(T 2l(ηT )) +Op(T l(ηT ))

)

+k2T

T l(ηT )· c

2

k2T· Op(T

2l(ηT ))

Op(T 2l(ηT )) +Op(T l(ηT ))

[τ0T ]+m∑t=[τ0T ]+1

y2t−1 + op(1)

=c2(1 + op(1))

T l(ηT )

[τ0T ]+m∑t=[τ0T ]+1

y2t−1 + op(1)

⇒ c2mW 2(τ0)

by Lemma B.1 and kT = o(√T ).

Q.E.D. �

43

Proof of Theorem 1.2. To derive the first and second parts of Theorem 1.2(a), i.e., to

prove P (k = k0) → 0 when kT = o(√T ) and |k − k0| = Op(1) when kT ≍

√T , we prove

|τT − τ0| = Op(1/T ) when kT = O(√T ) (B.10)

first. According to the proof of Theorem 3 in Chong (2001), it is sufficient to show

P (λ2T + 2λTA1 − 2λTA2 −A3 < 0) + P (λ2

T − 2λTA4 + 2λTA5 −A6 < 0) → 0,

where

λT = β2T − β1 = − c

kT,

A1 =

∑Tt=[τ0T ]+1 yt−1εt∑Tt=[τ0T ]+1 y

2t−1

,

A2 = supm∈D1T

∑[τ0T ]t=m+1 yt−1εt∑[τ0T ]t=m+1 y

2t−1

,

A3 = supm∈D1T

∣∣∣ ∑Tt=m+1 y

2t−1∑T

t=[τ0T ]+1 y2t−1

∑[τ0T ]t=m+1 y

2t−1

ΛT (m

T)∣∣∣,

A4 =

∑[τ0T ]t=1 yt−1εt∑[τ0T ]t=1 y2t−1

,

A5 = supm∈D2T

∑mt=[τ0T ]+1 yt−1εt∑mt=[τ0T ]+1 y

2t−1

,

A6 = supm∈D2T

∣∣∣ ∑mt=1 y

2t−1∑m

t=[τ0T ]+1 y2t−1

∑[τ0T ]t=1 y2t−1

ΛT (m

T)∣∣∣,

(B.11)

the definition of ΛT (mT ) can be found in the proof of Lemma B.4 and D1T = {m : m ∈ ZT ,m < [τ0T ]−MT }

D2T = {m : m ∈ ZT ,m > [τ0T ] +MT }

with MT > 0 such that MT → ∞ arbitrary slowly, where ZT denotes the set {0, 1, 2, · · · , T}.

Since λ2T > 0, it suffices to prove that, under the case of kT = O(

√T ),

Ai = op(1/kT ), i = 1, 2, 4, 5 and Aj = op(1/k2T ), j = 3, 6

in order to prove (B.10).

By applying Lemmas B.1 and B.3, it is trivial that A1 = Op(1/√TkT ) = op(1/kT ) and

A4 = Op(1/T ) = op(1/kT ). To find the order of the term A2, note that since β1 = 1, it is

44

not difficult to see that

[τ0T ]∑t=m+1

yt−1εt = y0

[τ0T ]∑t=m+1

εt +

[τ0T ]∑t=m+1

(t−1∑i=1

εi

)εt

=

y0

[τ0T ]∑t=m+1

ε(1)t +

[τ0T ]∑t=m+1

(t−1∑i=1

ε(1)i

)ε(1)t

· (1 + op(1))

= op

(√kT ([τ0T ]−m)l(ηT )

)+Op

√√√√ [τ0T ]∑

t=m+1

E

(t−1∑i=1

ε(1)i

)2

l(ηT )

= op

(√kT ([τ0T ]−m)l(ηT )

)+Op

(√([τ0T ]−m)([τ0T ] +m)l(ηT )

)= Op

(√([τ0T ]−m)([τ0T ] +m)l(ηT )

)and

[τ0T ]∑t=m+1

y2t−1 = T ([τ0T ]−m)l(ηT ) ·1

[τ0T ]−m

[τ0T ]∑t=m+1

(yt−1√T l(ηT )

)2

= Op(T ([τ0T ]−m)l(ηT )).

Consequently, we have

A2 = Op

(sup

m∈D1T

√([τ0T ]−m)([τ0T ] +m)

T ([τ0T ]−m)

)≤ Op

(1√TMT

)= op(1/kT ). (B.12)

To find the order of the term A5, note that for any small 0 < δ < 1, there exist two large

constantsN1 = N1(δ) such that βkT /N1

2T → e−c/N1 > 1−δ andN2 = N2(δ) such that βN2kT2T →

e−cN2 < δ. We then divide the set {MT +1, · · · , T − [τ0T ]}, which is the domain of m− [τ0T ]

when m ∈ D2T , into three subsets: {MT + 1, · · · , [kT /N1]}, {[kT /N1] + 1, · · · , [N2kT ]} and

{[N2kT ] + 1, · · · , T − [τ0T ]}. Note that

m∑t=[τ0T ]+1

yt−1εt =m∑

t=[τ0T ]+1

βt−1−[τ0T ]2T y[τ0T ] +

t−1∑i=[τ0T ]+1

βt−1−i2T εi

εt,

y[τ0T ]

m∑t=[τ0T ]+1

βt−1−[τ0T ]2T εt

= Op(√

T l(ηT )) ·Op

(√kT (1− β

2(m−[τ0T ])2T )l(ηT )

)= Op

(√TkT (1− β

2(m−[τ0T ])2T )l(ηT )

)

=

Op

(√T (m− [τ0T ])l(ηT )

), if MT < m− [τ0T ] ≤ [kT /N1]

Op

(√TkT l(ηT )

), if [kT /N1] < m− [τ0T ] ≤ [N2kT ]

Op

(√TkT l(ηT )

), if [N2kT ] < m− [τ0T ] ≤ T − [τ0T ]

45

since

(1− (1− c/kT )aT ) = O(aT /kT ) if aT = o(kT )

and

m∑t=[τ0T ]+1

t−1∑i=[τ0T ]+1

βt−1−i2T εi

εt

= Op

√√√√ m∑t=[τ0T ]+1

t−1∑i=[τ0T ]+1

β2(t−1−i)2T l(ηT )

= Op

(√kT (m− [τ0T ])− k2T (1− β

2(m−[τ0T ])2T )l(ηT )

)

=

Op

(√kT (m− [τ0T ])l(ηT )

), if MT < m− [τ0T ] ≤ [kT /N1]

Op (kT l(ηT )) , if [kT /N1] < m− [τ0T ] ≤ [N2kT ]

Op

(√kT (m− [τ0T ])l(ηT )

), if [N2kT ] < m− [τ0T ] ≤ T − [τ0T ]

We then have

m∑t=[τ0T ]+1

yt−1εt

=

Op

(√T (m− [τ0T ])l(ηT )

), if MT < m− [τ0T ] ≤ [kT /N1]

Op

(√TkT l(ηT )

), if [kT /N1] < m− [τ0T ] ≤ [N2kT ]

Op

(√TkT l(ηT )

), if [N2kT ] < m− [τ0T ] ≤ T − [τ0T ]

. (B.13)

Moreover, we also have

m∑t=[τ0T ]+1

y2t−1

= − 1

1− β22T

(y2m − y2[τ0T ]) +1

1− β22T

m∑t=[τ0T ]+1

ε2t +2β2T

1− β22T

m∑t=[τ0T ]+1

yt−1εt, (B.14)

2β2T1− β2

2T

m∑t=[τ0T ]+1

yt−1εt

=

Op

(kT√T (m− [τ0T ])l(ηT )

), if MT < m− [τ0T ] ≤ [kT /N1]

Op

(√Tk

3/2T l(ηT )

), if [kT /N1] < m− [τ0T ] ≤ [N2kT ]

Op

(√Tk

3/2T l(ηT )

), if [N2kT ] < m− [τ0T ] ≤ T − [τ0T ]

(B.15)

46

by (B.13),

1

1− β22T

m∑t=[τ0T ]+1

ε2t

= Op (kT (m− [τ0T ])l(ηT ))

=

Op (kT (m− [τ0T ])l(ηT )) , if MT < m− [τ0T ] ≤ [kT /N1]

Op

(k2T l(ηT )

), if [kT /N1] < m− [τ0T ] ≤ [N2kT ]

Op (kT (m− [τ0T ])l(ηT )) , if [N2kT ] < m− [τ0T ] ≤ T − [τ0T ]

(B.16)

and

− 1

1− β22T

(y2m − y2[τ0T ])

= − 1

1− β22T

(ym + y[τ0T ])(ym − y[τ0T ])

= − 1

1− β22T

m∑i=[τ0T ]+1

βm−i2T εi +

(βm−[τ0T ]2T + 1

)y[τ0T ]

·

m∑i=[τ0T ]+1

βm−i2T εi +

(βm−[τ0T ]2T − 1

)y[τ0T ]

= kT ·

(Op

(√kT (1− β

2(m−[τ0T ])2T )l(ηT )

)+Op(

√T l(ηT ))

)·(Op

(√kT (1− β

2(m−[τ0T ])2T )l(ηT )

)+Op

((1− β

m−[τ0T ]2T )

√T l(ηT )

))= Op(kT

√T l(ηT )) ·

(Op

(√kT (1− β

2(m−[τ0T ])2T )l(ηT )

)+Op

((1− β

m−[τ0T ]2T )

√T l(ηT )

))

=

Op(kT

√T l(ηT )) ·Op

(m−[τ0T ]

kT

√T l(ηT )

), if MT < m− [τ0T ] ≤ [kT /N1]

Op(kT√

T l(ηT )) ·Op(√T l(ηT )), if [kT /N1] < m− [τ0T ] ≤ [N2kT ]

Op(kT√

T l(ηT )) ·Op(√T l(ηT )), if [N2kT ] < m− [τ0T ] ≤ T − [τ0T ]

=

Op (T (m− [τ0T ])l(ηT )) , if MT < m− [τ0T ] ≤ [kT /N1]

Op(TkT l(ηT )), if [kT /N1] < m− [τ0T ] ≤ [N2kT ]

Op(TkT l(ηT )), if [N2kT ] < m− [τ0T ] ≤ T − [τ0T ]

. (B.17)

47

Combining (B.14)-(B.17), we have

m∑t=[τ0T ]+1

y2t−1

=

Op

(max

{kT√

T (m− [τ0T ])l(ηT ), T (m− [τ0T ])l(ηT )})

,

if MT < m− [τ0T ] ≤ [kT /N1]


Op (max {kT (m− [τ0T ])l(ηT ), TkT l(ηT )}) , if [N2kT ] < m− [τ0T ] ≤ T − [τ0T ]

=



Op(TkT l(ηT )), if [N2kT ] < m− [τ0T ] ≤ T − [τ0T ]

.

(B.18)

Hence, it follows from (B.13) and (B.18) that, when MT < m− [τ0T ] ≤ [kT /N1], we have∑mt=[τ0T ]+1 yt−1εt∑mt=[τ0T ]+1 y

2t−1

= Op

(√T (m− [τ0T ])l(ηT )

T (m− [τ0T ])l(ηT )

)≤ Op

(1√TMT

)= op(1/kT )

since kT = O(√T ) and MT → ∞. When [kT /N1] < m − [τ0T ] ≤ [N2kT ] or [N2kT ] <

m− [τ0T ] ≤ T − [τ0T ], we have∑mt=[τ0T ]+1 yt−1εt∑mt=[τ0T ]+1 y

2t−1

= Op

(√TkT l(ηT )

TkT l(ηT )

)= op(1/kT ).

Using the above arguments, we have

A5 = supm∈D2T

∑mt=[τ0T ]+1 yt−1εt∑mt=[τ0T ]+1 y

2t−1

= op(1/kT ).

Note also that

A3 = supm∈D1T

∣∣∣ ∑Tt=m+1 y

2t−1∑T

t=[τ0T ]+1 y2t−1

∑[τ0T ]t=m+1 y

2t−1

ΛT (m

T)∣∣∣

≤(

1∑Tt=[τ0T ]+1 y

2t−1

+1∑[τ0T ]

t=[τ0T ]−MTy2t−1

)(sup

m∈D1T

∣∣∣(∑[τ0T ]t=1 yt−1εt)

2∑[τ0T ]t=1 y2t−1

−(∑m

t=1 yt−1εt)2∑m

t=1 y2t−1

∣∣∣+ sup

m∈D1T

∣∣∣(∑Tt=[τ0T ]+1 yt−1εt)

2∑Tt=[τ0T ]+1 y

2t−1

−(∑T

t=m+1 yt−1εt)2∑T

t=m+1 y2t−1

∣∣∣)=

(Op

( 1

TkT l(ηT )

)+Op

( 1

TMT l(ηT )

))·Op(l(ηT ))

= op(1/k2T ),

48

and

A6 = supm∈D2T

∣∣∣ ∑mt=1 y

2t−1∑m

t=[τ0T ]+1 y2t−1

∑[τ0T ]t=1 y2t−1

ΛT (m

T)∣∣∣

≤(

1∑[τ0T ]t=1 y2t−1

+1∑[τ0T ]+MT

t=[τ0T ]+1 y2t−1

)(sup

m∈D2T

∣∣∣(∑[τ0T ]t=1 yt−1εt)

2∑[τ0T ]t=1 y2t−1

−(∑m

t=1 yt−1εt)2∑m

t=1 y2t−1

∣∣∣+ sup

m∈D2T

∣∣∣(∑Tt=[τ0T ]+1 yt−1εt)

2∑Tt=[τ0T ]+1 y

2t−1

−(∑T


t=m+1 y2t−1

∣∣∣)≤

(Op(

1

T 2l(ηT )) +Op(

1

TMT l(ηT )))·Op(l(ηT ))

= op(1/k2T ).

Hence, |τT − τ0| = Op(1/T ) when kT = O(√T ) is proved, implying the second part of

Theorem 1.2(a).

To prove the first part of Theorem 1.2(a), i.e., to prove P (k = k0) → 0 when kT = o(√T ),

note that we have proven that |τT − τ0| = Op(1/T ). Thus, for any η > 0, there exists a

positive integer M such that

P (k = k0) = P (|k − k0| > M) + P (|k − k0| ≤ M, k = k0)

≤ η +M∑

m=1

P( k2TT l(ηT )

(RSST (τ0 −

m

T)−RSST (τ0)

)< 0)

+

M∑m=1

P( k2TT l(ηT )

(RSST (τ0 +

m

T)−RSST (τ0)

)< 0).

Applying Lemma B.4 to the above inequality, one immediately has P (k = k0) → 0 due to

the finiteness of M and arbitrariness of η.

We turn to the proof of the third part of Theorem 1.2(a), i.e., the limiting distribution

of τT when√T = o(kT ). We follow Appendix K in Chong (2001) and let g(T ) = k2T /(c

2T ).

Note that (1.5) is a special case of the limiting distribution of τT in Theorem 4 in Chong

(2001) when√T = o(kT )

∗, the moment conditions E(y20) < ∞ and E(ε4t ) < ∞ in his

paper are satisfied and c = 1. One can apply the truncation technique (A.1) to weaken the

∗In Theorem 4 in Chong (2001), the condition T 3/4(1 − β2T ) → ∞, that is kT = o(T 3/4) in

our case, is imposed so as to derive the limiting distribution of τT . By checking his proof carefully,

this condition is imposed only for showing that g(T )√T

∑[|v|g(T )]−1t=0

(1√g(T )

∑ti=0 ε[τ0T ]−iε[τ0T ]−t

)1

g(T )=

op(1) when εt’s have finite 4-th moments (see page 153 in Chong (2001)). However, since∑[|v|g(T )]−1t=0

(1√g(T )

∑ti=0 ε[τ0T ]−iε[τ0T ]−t

)1√g(T )

=∑[|v|g(T )]−1

t=0

(1√g(T )

∑t−1i=0 ε[τ0T ]−i

)ε[τ0T ]−t√

g(T )+ Op(1) =

Op(1), the condition√

g(T )T

= o(1) is sufficient to achieve the goal. Note that g(T ) ≍ k2T /T , hence, the

condition kT = o(T ) is sufficient. Therefore, the condition T 3/4(1− β2T ) → ∞ is not needed in Theorem 4

in Chong (2001).

49

conditions E(y20) < ∞ and E(ε4t ) < ∞ and accommodate assumptions C1 and C3. The

details are omitted for brevity.

To find the limiting distribution of β1(τT ). Note that we have proven that |τT − τ0| =

op(1/T ) when kT = o(√T ), |τT − τ0| = Op(1/T ) when kT ≍

√T and |τT − τ0| = Op(k

2T /T

2)

when√T = o(kT ), we consider the limiting distribution of β1(τT ) under the above three

cases separately.

Consider the case where√T = o(kT ) first. In this case, note that |τT −τ0| = Op(k

2T /T

2),

then, for τT ≤ τ0, we have

[τ0T ]∑t=[τ0T−k2T /T ]+1

y2t−1 =

[τ0T ]∑t=[τ0T−k2T /T ]+1

(y[τ0T ] −

[τ0T ]∑i=t

εi

)2

=

[τ0T ]∑t=[τ0T−k2T /T ]+1

y2[τ0T ] · (1 + op(1))

= Op(k2T l(ηT )), (B.19)

implying[τ0T ]∑

t=[τTT ]+1

y2t−1 = Op(k2T l(ηT )). (B.20)

In addition, squaring yt = yt−1+ εt and sum over t ∈ {[τ0T −k2T /T ]+1, · · · , [τ0T ]}, we have

[τ0T ]∑t=[τ0T−k2T /T ]+1

yt−1εt

=1

2

y2[τ0T ] − y2[τ0T−k2T /T ] −[τ0T ]∑

t=[τ0T−k2T /T ]+1

ε2t

=

1

2

(y[τ0T ] + y[τ0T−k2T /T ]

)(y[τ0T ] − y[τ0T−k2T /T ]

)− 1

2

[τ0T ]∑t=[τ0T−k2T /T ]+1

ε2t

= Op(√

T l(ηT ))Op(√

k2T l(ηT )/T ) +Op(k2T l(ηT )/T )

= Op(kT l(ηT )).

As a result,[τ0T ]∑

t=[τTT ]+1

yt−1εt = Op(kT l(ηT )). (B.21)

50

Similarly, if τT > τ0, it follows from Lemma B.2 that

[τ0T+k2T /T ]∑t=[τ0T ]+1

y2t−1 =

[τ0T+k2T /T ]∑t=[τ0T ]+1

(βt−1−[τ0T ]2T y[τ0T ] +

t−1∑i=[τ0T ]+1

βt−1−i2T εi

)2

=

[τ0T+k2T /T ]∑t=[τ0T ]+1

β2(t−1−[τ0T ])2T y2[τ0T ] · (1 + op(1))

= k2T /T ·Op(T l(ηT ))

= Op(k2T l(ηT )), (B.22)

implying that[τTT ]∑

t=[τ0T ]+1

y2t−1 = Op(k2T l(ηT )). (B.23)

By squaring yt = β2T yt−1 + εt and summing over t ∈ {[τ0T ] + 1, · · · , [τ0T + k2T /T ]}, we

obtain

[τ0T+k2T /T ]∑t=[τ0T ]+1

yt−1εt =1− β2

2T

2β2T

[τ0T+k2T /T ]∑t=[τ0T ]+1

y2t−1 +y2[τ0T+k2T /T ]

− y2[τ0T ]

2β2T− 1

2β2T

[τ0T+k2T /T ]∑t=[τ0T ]+1

ε2t .

(B.24)

First, applying (B.22) immediately yields

1− β22T

2β2T

[τ0T+k2T /T ]∑t=[τ0T ]+1

y2t−1 = Op(kT l(ηT )). (B.25)

Secondly, since√T = o(kT ), we have

1

2β2T

[τ0T+k2T /T ]∑t=[τ0T ]+1

ε2t = Op(k2T l(ηT )/T ). (B.26)

Thirdly, since |β2T | < 1, we have

[k2T /T ]−1∑i=0

β2i2T = O(k2T /T )

which implies that

[k2T /T ]−1∑i=0

βi2T ε[τ0T+k2T /T ]−i = Op(kT

√l(ηT )/

√T ).

51

We have

y2[τ0T+k2T /T ] − y2[τ0T ]

=(y[τ0T+k2T /T ] + y[τ0T ]

)(y[τ0T+k2T /T ] − y[τ0T ]

)=

[k2T /T ]−1∑i=0

βi2T ε[τ0T+k2T /T ]−i + (β

[k2T /T ]2T + 1)y[τ0T ]

·

[k2T /T ]−1∑i=0

βi2T ε[τ0T+k2T /T ]−i + (β

[k2T /T ]2T − 1)y[τ0T ]

=

(Op(kT

√l(ηT )/

√T ) +Op(

√T l(ηT ))

)(Op(kT

√l(ηT )/

√T ) +

k2T /T

kTOp(

√T l(ηT )

)= Op(

√T l(ηT ))Op(kT

√l(ηT )/

√T )

= Op(kT l(ηT )). (B.27)

Combining (B.24)-(B.27) together leads to

[τ0T+k2T /T ]∑t=[τ0T ]+1

yt−1εt = Op(kT l(ηT )), (B.28)

yielding[τTT ]∑

t=[τ0T ]+1

yt−1εt = Op(kT l(ηT )). (B.29)

Now, applying (B.20), (B.21), (B.23), (B.29) and Lemma B.1 and following Appendix

G in Chong (2001), we have

T (β1(τT )− β1(τ0))

= T


t=1 y2t−∑[τ0T ]


)

= I{τT ≤ τ0}T(∑[τ0T ]

t=[τTT ]+1 y2t−1∑[τTT ]

t=1 y2t−1

∑[τ0T ]t=1 yt−1εt∑[τ0T ]t=1 y2t−1

−∑[τ0T ]


)

+I{τT > τ0}T(−∑[τTT ]

t=[τ0T ]+1 y2t−1∑[τTT ]

t=1 y2t−1

∑[τ0T ]t=1 yt−1εt∑[τ0T ]t=1 y2t−1

+


t=1 y2t−1

+ (β2T − β1)

∑[τTT ]t=[τ0T ]+1 y

2t−1∑[τTT ]

t=1 y2t−1

)= I{τT ≤ τ0}T

(Op(

k2T l(ηT )

T 2l(ηT ))Op(

1

T) +Op(

kT l(ηT )

T 2l(ηT )))

+I{τT > τ0}T(Op(

k2T l(ηT )

T 2l(ηT ))Op(

1

T) +Op(

kT l(ηT )

T 2l(ηT )) +Op(

k2T l(ηT )

kTT 2l(ηT )))

= op(1). (B.30)

52

Thus, β1(τT ) and β1(τ0) have the same asymptotic distribution. We then invoke Lemma

B.1 to obtain

T (β1(τ0)− β1) =

1T l(ηT )


1T 2l(ηT )

∑[τ0T ]t=1 y2t−1

⇒ W 2(τ0)− τ02∫ τ00 W 2(s)ds

,

which immediately leads to

T (β1(τT )− β1) ⇒W 2(τ0)− τ02∫ τ00 W 2(s)ds

. (B.31)

It is trivial thatW 2(τ0)− τ02∫ τ00 W 2(s)ds

d=

W 2(1)− 1

2τ0∫ 10 W 2(s)ds

by the fact {W (s)√τ0

, 0 < s ≤ τ0

}d={W (

s

τ0), 0 < s ≤ τ0

}and applying the change of variables. Hence, (B.31) is equivalent to (1.6).

Consider the case where kT ≍√T . Note that |τT − τ0| = Op(1/T ) under this situation.

It is trivial that

[τ0T ]∑t=[τTT ]+1

y2t−1 = Op(T l(ηT )),

[τ0T ]∑t=[τTT ]+1

yt−1εt = Op(√T l(ηT )) (B.32)

and[τTT ]∑

t=[τ0T ]+1

y2t−1 = Op(T l(ηT )),

[τTT ]∑t=[τ0T ]+1

yt−1εt = Op(√T l(ηT )). (B.33)

Consequently, similar to (B.30), we have

T (β1(τT )− β1(τ0))

= I{τT ≤ τ0}T(∑[τ0T ]

t=[τTT ]+1 y2t−1∑[τTT ]

t=1 y2t−1

∑[τ0T ]t=1 yt−1εt∑[τ0T ]t=1 y2t−1

−∑[τ0T ]


)

+I{τT > τ0}T(−∑[τTT ]

t=[τ0T ]+1 y2t−1∑[τTT ]

t=1 y2t−1

∑[τ0T ]t=1 yt−1εt∑[τ0T ]t=1 y2t−1

+


t=1 y2t−1

+ (β2T − β1)

∑[τTT ]t=[τ0T ]+1 y

2t−1∑[τTT ]

t=1 y2t−1

)

= I{τT ≤ τ0}T(Op(

T l(ηT )

T 2l(ηT ))Op(

1

T) +Op(

√T l(ηT )

T 2l(ηT )))

+I{τT > τ0}T(Op(

T l(ηT )

T 2l(ηT ))Op(

1

T) +Op(

√T l(ηT )

T 2l(ηT )) +Op(

T l(ηT )

kTT 2l(ηT )))

= op(1),

which means (B.31) still holds when kT ≍√T .

53

Consider the case where kT = o(√T ). Note that we have proven that P (k = k0) → 0

under this situation. Hence, following the proof of Theorem 4 in Chong (2001), one can

show that (B.31) still holds. The details are omitted.

To find the limiting distribution of β2(τT ), we also consider the following three cases

where√T = o(kT ), kT ≍

√T and kT = o(

√T ) separately.

When√T = o(kT ), applying (B.20), (B.21), (B.23) and (B.29) and Lemma B.3 and

following Appendix G in Chong (2001), we have

√TkT (β2(τT )− β2(τ0))

=√

TkT


t=[τTT ]+1 y2t

−∑T

t=[τ0T ]+1 ytyt−1∑Tt=[τ0T ]+1 y

2t

)

= I{τT ≤ τ0}√

TkT

(−∑[τ0T ]

t=[τTT ]+1 y2t−1∑T

t=[τTT ]+1 y2t−1

∑Tt=[τ0T ]+1 yt−1εt∑Tt=[τ0T ]+1 y

2t−1

+


2t−1

+ (β1 − β2T )

∑[τ0T ]t=[τTT ]+1 y

2t−1∑T

t=[τTT ]+1 y2t−1

)

+I{τT > τ0}√TkT

(∑[τTT ]t=[τ0T ]+1 y

2t−1∑T

t=[τTT ]+1 y2t−1

∑Tt=[τ0T ]+1 yt−1εt∑Tt=[τ0T ]+1 y

2t−1

−∑[τTT ]


2t−1

)= I{τT ≤ τ0}

√TkT

(Op(

k2T l(ηT )

TkT l(ηT ))Op(

1√TkT

) +Op(kT l(ηT )

TkT l(ηT )) +Op(

k2T l(ηT )

Tk2T l(ηT )))

+I{τT > τ0}√TkT

(Op(

k2T l(ηT )

TkT l(ηT ))Op(

1√TkT

) +Op(kT l(ηT )

TkT l(ηT )))

= op(1).

Thus, β2(τT ) and β2(τ0) have the same asymptotic distribution. Applying Lemma B.3, we

have √TkT (β2(τ0)− β2T ) =

1√TkT l(ηT )


1TkT l(ηT )

∑Tt=[τ0T ]+1 y

2t−1

⇒ W (2c)

W 2(τ0) + 1− τ0,

which immediately leads to

√TkT (β2(τT )− β2T ) ⇒

W (2c)

W 2(τ0) + 1− τ0. (B.34)

(B.34) is equivalent to (1.7) by noting that 1− β22T = 2c/kT (1 + o(1)).

54

When kT ≍√T , applying (B.32) and (B.33), we have√

TkT (β2(τT )− β2(τ0))

= I{τT ≤ τ0}√

TkT

(−∑[τ0T ]

t=[τTT ]+1 y2t−1∑T

t=[τTT ]+1 y2t−1

∑Tt=[τ0T ]+1 yt−1εt∑Tt=[τ0T ]+1 y

2t−1

+


2t−1

+ (β1 − β2T )

∑[τ0T ]t=[τTT ]+1 y

2t−1∑T

t=[τTT ]+1 y2t−1

)

+I{τT > τ0}√TkT

(∑[τTT ]t=[τ0T ]+1 y

2t−1∑T

t=[τTT ]+1 y2t−1

∑Tt=[τ0T ]+1 yt−1εt∑Tt=[τ0T ]+1 y

2t−1

−∑[τTT ]


2t−1

)

= I{τT ≤ τ0}√

TkT

(Op(

T l(ηT )

TkT l(ηT ))Op(

1√TkT

) +Op(

√T l(ηT )

TkT l(ηT )) +Op(

T l(ηT )

Tk2T l(ηT )))

+I{τT > τ0}√TkT

(Op(

T l(ηT )

TkT l(ηT ))Op(

1√TkT

) +Op(

√T l(ηT )

TkT l(ηT )))

= op(1),

which means that (B.34) still holds.

When kT = o(√T ), following the proof of Theorem 4 in Chong (2001), one can show

that (B.34) holds. The details are omitted.

Q.E.D. �

Appendix C: Proof of Theorem 1.3

Lemma B.1 and the following two lemmas are key ingredients in the proof of Theorem 1.3.

Lemma C.1 Suppose assumptions C1 and C2 are fulfilled and c > 0, then

(a) for any 0 < τ ≤ 1, 1√kT l(ηT )

∑[τT ]t=1 (1 +

ckT

)t−1−[τT ]εt ⇒ X,

(b) for any 0 < τ ≤ 1, 1√kT l(ηT )

∑[τT ]t=1 (1 +

ckT

)−tεt ⇒ Y,

where, X and Y are independent N(0, 12c) random variables.

Proof. The proof can be completed by following the proof of Lemma 4.2 in Phillips and

Magdalinos (2007a) and the truncation technique (A.1). Thus the details are omitted.

Q.E.D. �

Lemma C.2 Let {yt} be generated according to Model (1.1), where β1 = 1 and β2 = β2T =

1 + c/kT with c > 0. Then under assumptions C1-C4, the following results hold jointly:

55

(a)β−(T−[τ0T ])2T√TkT l(ηT )

∑Tt=[τ0T ]+1 yt−1εt ⇒ XW (τ0),

(b)β−2(T−[τ0T ])2TTkT l(ηT )

∑Tt=[τ0T ]+1 y

2t−1 ⇒ 1

2cW2(τ0),

where X is a random variable obeying N(0, 12c) and is independent of W (τ0).

Proof. To prove part (a), note that the following decomposition holds:

1

βT−[τ0T ]2T

√TkT l(ηT )

T∑t=[τ0T ]+1

yt−1εt

=1

βT−[τ0T ]2T

√TkT l(ηT )

T∑t=[τ0T ]+1

(βt−[τ0T ]−12T y[τ0T ] +

t−1∑i=[τ0T ]+1

βt−i−12T εi

)εt

=1

βT−[τ0T ]2T

√TkT l(ηT )

(y0 + [τ0T ]∑j=1

εj

) T∑t=[τ0T ]+1

βt−[τ0T ]−12T εt +

T∑t=[τ0T ]+1

εt

t−1∑i=[τ0T ]+1

βt−i−12T εi

:= II1 + II2, (C.1)

where

II1 =1

βT−[τ0T ]2T

√TkT l(ηT )

(y0 +

[τ0T ]∑j=1

εj

) T∑t=[τ0T ]+1


=1

βT−[τ0T ]2T

√TkT l(ηT )

[τ0T ]∑j=1

εj

T∑t=[τ0T ]+1

βt−[τ0T ]−12T εt · (1 + op(1))

and

II2 =1

βT−[τ0T ]2T

√TkT l(ηT )

T∑t=[τ0T ]+1

εt

t−1∑i=[τ0T ]+1

βt−i−12T εi

=1

βT−[τ0T ]2T

√TkT l(ηT )

T∑t=[τ0T ]+1

ε(1)t

t−1∑i=[τ0T ]+1

βt−i−12T ε

(1)i · (1 + op(1))

by the truncation technique (A.1) and some simple calculus.

To handle the term II1. Applying part (a) of Lemma C.1 and observing the fact∑[τ0T ]t=1 εt√T l(ηT )

⇒ W (τ0), it can be shown that

II1 ⇒ XW (τ0) (C.2)

by noting that

1

βT−[τ0T ]2T

√kT l(ηT )

T∑t=[τ0T ]+1


=1

βT−[τ0T ]2T

√kT l(ηT )

T−[τ0T ]∑s=1

βs−12T ε[τ0T ]+s

=1√

kT l(ηT )

[(1−τ0)T ]∑s=1

βs−1−[(1−τ0)T ]2T ε[τ0T ]+s · (1 + op(1))

56

and

1√kT l(ηT )

[(1−τ0)T ]∑s=1

βs−1−[(1−τ0)T ]2T ε[τ0T ]+s

d=

1√kT l(ηT )

[(1−τ0)T ]∑s=1

βs−1−[(1−τ0)T ]2T εs ⇒ X.

Consider the term II2. It is trivial that

E( 1

βT−[τ0T ]2T

√TkT l(ηT )

T∑t=[τ0T ]+1

ε(1)t

t−1∑i=[τ0T ]+1

βt−i−12T ε

(1)i

)= 0

and

V ar( 1

βT−[τ0T ]2T

√TkT l(ηT )

T∑t=[τ0T ]+1

ε(1)t

t−1∑i=[τ0T ]+1

βt−i−12T ε

(1)i

)

=1

β2(T−[τ0T ])2T TkT

T∑t=[τ0T ]+1

t−1∑i=[τ0T ]+1

β2(t−i−1)2T · (1 + o(1))

=1

β2(T−[τ0T ])2T T

· 1

kT (β22T − 1)

·(β2(T−[τ0T ])

2T − 1

β22T − 1

− (T − [τ0T ]))· (1 + o(1))

= o(1) (C.3)

by kT = o(T ). Consequently,

II2 = op(1). (C.4)

Obviously, combining (C.1), (C.2) and (C.4) yields part (a).

To prove part (b). Observe that

1

β2(T−[τ0T ])2T TkT l(ηT )

T∑t=[τ0T ]+1

y2t−1

=1


T∑t=[τ0T ]+1

(βt−[τ0T ]−12T y[τ0T ] +

t−1∑i=[τ0T ]+1

βt−i−12T εi

)2=

y2[τ0T ]


T∑t=[τ0T ]+1

β2(t−[τ0T ]−1)2T +

1


T∑t=[τ0T ]+1

( t−1∑i=[τ0T ]+1

βt−i−12T εi

)2+

2y[τ0T ]


T∑t=[τ0T ]+1

(βt−[τ0T ]−12T

t−1∑i=[τ0T ]+1

βt−i−12T εi

):= III1 + III2 + III3, (C.5)

where

III1 =y2[τ0T ]


T∑t=[τ0T ]+1

β2(t−[τ0T ]−1)2T ,

57

III2 =1


T∑t=[τ0T ]+1

( t−1∑i=[τ0T ]+1

βt−i−12T εi

)2and

III3 =2y[τ0T ]


T∑t=[τ0T ]+1

(βt−[τ0T ]−12T

t−1∑i=[τ0T ]+1

βt−i−12T εi

).

Obviously,

III1 ⇒1

2cW 2(τ0). (C.6)

Consider the term III2. Note that

III2 =1


T∑t=[τ0T ]+1

( t−1∑i=[τ0T ]+1

βt−i−12T ε

(1)i

)2· (1 + op(1))

and

E

1


T∑t=[τ0T ]+1

( t−1∑i=[τ0T ]+1

βt−i−12T ε

(1)i

)2=

1

β2(T−[τ0T ])2T TkT

T∑t=[τ0T ]+1

t−1∑i=[τ0T ]+1

β2(t−i−1)2T · (1 + o(1))

= o(1)

by similar arguments as in (C.3). Hence, we have

III2 = op(1). (C.7)

For the term III3, using (C.6) and (C.7) and applying Cauchy-Schwarz inequality im-

mediately leads to

III3 = op(1). (C.8)

Combining (C.5)-(C.8), we show part (b).

Moreover, by checking the above proofs carefully, one can find that parts (a) and (b)

hold jointly and X and W (τ0) are mutually independent.

Q.E.D. �

Lemma C.3 Let {yt} be generated according to Model (1.1), where β1 = 1 and β2 = β2T =

1 + c/kT , where c is a fixed positive constant. Then under the assumptions C1-C4 with

kT = o(√T ), we have for any fixed integer m ≥ 0,

(a)k2T

T l(ηT )

(RSST (τ0 − m

T )−RSST (τ0))⇒ c2mW 2(τ0),

(b)k2T

T l(ηT )

(RSST (τ0 +

mT )−RSST (τ0)

)⇒ c2mW 2(τ0).

58

Proof. The proof is similar to that of Lemma B.4, with the use of the result in (C.16)

below. The details are omitted.

Q.E.D. �

Proof of Theorem 1.3. We first prove the first and second parts of Theorem 1.3(a).

Similarly to the proof of Theorem 1.2(a), we first prove (B.10) when kT = O(√T ). The

definitions of Ai, i = 1, · · · , 6 can be found in (B.11). First, it is trivial that

A1 = op(1/kT ), A4 = op(1/kT ) (C.9)

by Lemma B.1 and Lemma C.2. Secondly, note that the same reasoning in (B.12) also

implies

A2 = op(1/kT ). (C.10)

For the term A5, following the proof of Theorem 1.2, we have

y[τ0T ]

m∑t=[τ0T ]+1

βt−1−[τ0T ]2T εt

= Op

(√TkT (β

2(m−[τ0T ])2T − 1)l(ηT )

)

=

Op

(√T (m− [τ0T ])l(ηT )

), if MT < m− [τ0T ] ≤ [kT /N1]

Op

(√TkT l(ηT )

), if [kT /N1] < m− [τ0T ] ≤ [N2kT ]

Op

(√TkTβ

(m−[τ0T ])2T l(ηT )

), if [N2kT ] < m− [τ0T ] ≤ T − [τ0T ]

and

m∑t=[τ0T ]+1

t−1∑i=[τ0T ]+1

βt−1−i2T εi

εt

= Op

(√kT (m− [τ0T ]) + k2T (β

2(m−[τ0T ])2T − 1)l(ηT )

)

=

Op

(√kT (m− [τ0T ])l(ηT )

), if MT < m− [τ0T ] ≤ [kT /N1]

Op (kT l(ηT )) , if [kT /N1] < m− [τ0T ] ≤ [N2kT ]

Op

(kTβ

m−[τ0T ]2T l(ηT )

), if [N2kT ] < m− [τ0T ] ≤ T − [τ0T ]

.

59

Hence,

m∑t=[τ0T ]+1

yt−1εt

= y[τ0T ]

m∑t=[τ0T ]+1

βt−1−[τ0T ]2T εt +

m∑t=[τ0T ]+1

t−1∑i=[τ0T ]+1

βt−1−i2T εi

εt

=

Op

(√T (m− [τ0T ])l(ηT )

), if MT < m− [τ0T ] ≤ [kT /N1]

Op

(√TkT l(ηT )

), if [kT /N1] < m− [τ0T ] ≤ [N2kT ]

Op

(√TkTβ

(m−[τ0T ])2T l(ηT )

), if [N2kT ] < m− [τ0T ] ≤ T − [τ0T ]

. (C.11)

Moreover, we also have

2β2T1− β2

2T

m∑t=[τ0T ]+1

yt−1εt

=

Op

(kT√

T (m− [τ0T ])l(ηT )), if MT < m− [τ0T ] ≤ [kT /N1]

Op

(√Tk

3/2T l(ηT )

), if [kT /N1] < m− [τ0T ] ≤ [N2kT ]

Op

(√Tk

3/2T β

(m−[τ0T ])2T l(ηT )

), if [N2kT ] < m− [τ0T ] ≤ T − [τ0T ]

,

1

1− β22T

m∑t=[τ0T ]+1

ε2t

= Op (kT (m− [τ0T ])l(ηT ))

=

Op (kT (m− [τ0T ])l(ηT )) , if MT < m− [τ0T ] ≤ [kT /N1]

Op

(k2T l(ηT )

), if [kT /N1] < m− [τ0T ] ≤ [N2kT ]

Op (kT (m− [τ0T ])l(ηT )) , if [N2kT ] < m− [τ0T ] ≤ T − [τ0T ]

60

and

− 1

1− β22T

(y2m − y2[τ0T ])

= − 1

1− β22T

(ym + y[τ0T ])(ym − y[τ0T ])

= − 1

1− β22T

m∑i=[τ0T ]+1

βm−i2T εi +

(βm−[τ0T ]2T + 1

)y[τ0T ]

·

m∑i=[τ0T ]+1

βm−i2T εi +

(βm−[τ0T ]2T − 1

)y[τ0T ]

= kT ·

(Op

(√kT (β

2(m−[τ0T ])2T − 1)l(ηT )

)+Op(β

m−[τ0T ]2T

√T l(ηT ))

)·(Op

(√kT (β

2(m−[τ0T ])2T − 1)l(ηT )

)+Op

((β

m−[τ0T ]2T − 1)

√T l(ηT )

))

=

Op(kT

√T l(ηT )) ·Op

(m−[τ0T ]

kT

√T l(ηT )

), if MT < m− [τ0T ] ≤ [kT /N1]

Op(kT√

T l(ηT )) ·Op(√T l(ηT )), if [kT /N1] < m− [τ0T ] ≤ [N2kT ]

Op(kTβm−[τ0T ]2T

√T l(ηT )) ·Op(β

m−[τ0T ]2T

√T l(ηT )), if [N2kT ] < m− [τ0T ] ≤ T − [τ0T ]

=



Op(TkTβ2(m−[τ0T ])2T l(ηT )), if [N2kT ] < m− [τ0T ] ≤ T − [τ0T ]

.

Thus,

m∑t=[τ0T ]+1

y2t−1

= − 1

1− β22T

(y2m − y2[τ0T ]) +1

1− β22T

m∑t=[τ0T ]+1

ε2t +2β2T

1− β22T

m∑t=[τ0T ]+1

yt−1εt

=



Op(TkTβ2(m−[τ0T ])2T l(ηT )), if [N2kT ] < m− [τ0T ] ≤ T − [τ0T ]

. (C.12)

Consequently, it follows from (C.11) and (C.12) that∑mt=[τ0T ]+1 yt−1εt∑mt=[τ0T ]+1 y

2t−1

= Op

(√T (m− [τ0T ])l(ηT )

T (m− [τ0T ])l(ηT )

)≤ Op(1/

√TMT ) = op(1/kT )

by kT = O(√T ) and MT → ∞ when MT < m− [τ0T ] ≤ [kT /N1],∑m

t=[τ0T ]+1 yt−1εt∑mt=[τ0T ]+1 y

2t−1

= Op

(√TkT l(ηT )

TkT l(ηT )

)= op(1/kT )

61

when [kT /N1] < m− [τ0T ] ≤ [N2kT ] and∑mt=[τ0T ]+1 yt−1εt∑mt=[τ0T ]+1 y

2t−1

= Op

(√TkTβ

m−[τ0T ]2T l(ηT )

TkTβ2(m−[τ0T ])2T l(ηT )

)= op(1/kT )

when [N2kT ] < m− [τ0T ] ≤ T − [τ0T ], yielding

A5 = supm∈D2T

∑mt=[τ0T ]+1 yt−1εt∑mt=[τ0T ]+1 y

2t−1

= op(1/kT ). (C.13)

Thirdly, note that

A3 ≤(Op

( 1

TkTβ2(T−[τ0T ])2T l(ηT )

)+Op

( 1

TMT l(ηT )

))·Op(l(ηT )) = op(1/k

2T ) (C.14)

and

A6 ≤(Op(

1

T 2l(ηT )) +Op(

1

TMT l(ηT )))·Op(l(ηT )) = op(1/k

2T ). (C.15)

The readers are referred to the proof of Theorem 1.2(a) for more details. Hence, (B.10)

is proved by combining (C.9), (C.10), (C.13), (C.14) and (C.15). Therefore, |τT − τ0| =

Op(1/T ) under kT = O(√T ) is proved. This implies the second part of Theorem 1.3(a).

To prove the first part of Theorem 1.3(a), i.e., to prove P (k = k0) → 0 when kT = o(√T ),

one can refer to the proof of Theorem 1.2(a) and applying Lemma C.3. The details are not

provided here for brevity.

To derive the third part of Theorem 1.3(a), we follow Appendix K in Chong (2001).

Since the asymptotic analysis in mildly explosive model is more complicated than that in

mildly integrated model, we offer more details for the proof. We write β2T = 1+ c/kT = 1+

1/(√

Tg(T )) with g(T ) = k2T /(c2T ). Then, g(T ) → ∞ and g(T ) = o(kT ) since

√T = o(kT )

and kT = o(T ). Denote

ΛT (τ) =

(∑[τ0T ]t=1 yt−1εt

)2∑[τ0T ]

t=1 y2t−1

−

(∑[τT ]t=1 yt−1εt

)2∑[τT ]

t=1 y2t−1

+


)2∑T

t=[τ0T ]+1 y2t−1

−

(∑Tt=[τT ]+1 yt−1εt

)2∑T

t=[τT ]+1 y2t−1

.

The following observation is a simple generalization of Proposition A.1 in Phillips and Mag-

dalinos (2007), hence the proof is omitted:(T

kT

)a

= o((1 + c/kT )bT ), for any a > 0 and b > 0. (C.16)

For τ = τ0 + νg(T )/T and ν ≤ 0, by applying (C.16) and Lemmas B.1 and C.2 and by

the proof of (B.19), we have the following results:

ΛT (τ)

l(ηT )= Op(1);

62

∣∣∣∣(β2T − 1)

l(ηT )

∑[τ0T ]t=[τT ]+1 y

2t−1


t=[τT ]+1 y2t−1

∣∣∣∣= Op

(Tg(T )

√TkTβ

T−[τ0T ]2T

kT (Tg(T ) + TkTβ2(T−[τ0T ])2T )

)= Op

(Tg(T )

√TkT

Tk2Tβ(T−[τ0T ])2T

)

= op

((T

kT

)1/2

· 1

β(T−[τ0T ])2T

)= op(1); ∑T

t=[τ0T ]+1 y2t−1∑T

t=[τT ]+1 y2t−1

p→ 1;

1√Tg(T )l(ηT )

[|ν|g(T )]−1∑t=0

y[τ0T ]−t−1ε[τ0T ]−t

= −y[τ0T ]√T l(ηT )

· 1√g(T )l(ηT )

[|ν|g(T )]−1∑t=0

(−ε[τ0T ]−t) · (1 + op(1))

⇒ −W1(τ0)W1(|ν|);

and

1

Tg(T )l(ηT )

[|ν|g(T )]−1∑t=0

y2[τ0T ]−t−1 ⇒ |ν|W 21 (τ0).

One is referred to Appendix K in Chong (2001) for more details.

Then, using equation (B.2) in Chong (2001), we have

RSST (τ)−RSST (τ0)

l(ηT )

= −2(β2T − 1)

l(ηT )

[τ0T ]∑t=[τT ]+1

yt−1εt · (1 + op(1)) +(β2T − 1)2

l(ηT )

[τ0T ]∑t=[τT ]+1

y2t−1 · (1 + op(1)) + op(1)

= − 2(1 + op(1))√Tg(T )l(ηT )

[|ν|g(T )]−1∑t=0

y[τ0T ]−t−1ε[τ0T ]−t +1 + op(1)

Tg(T )l(ηT )

[|ν|g(T )]−1∑t=0

y2[τ0T ]−t−1 + op(1)

⇒ −2W1(τ0)W1(|ν|) + |ν|W 21 (τ0). (C.17)

For τ = τ0 + νg(T )/T and ν > 0, note that

[τ0T+νg(T )]∑t=[τ0T ]+1

y2t−1 =

[τ0T+νg(T )]∑t=[τ0T ]+1

(βt−1−[τ0T ]2T y[τ0T ] +

t−1∑i=[τ0T ]+1

βt−1−i2T εi

)2

= y2[τ0T ]

[τ0T+νg(T )]∑t=[τ0T ]+1

β2(t−1−[τ0T ])2T +

[τ0T+νg(T )]∑t=[τ0T ]+1

( t−1∑i=[τ0T ]+1

βt−1−i2T εi

)2

+2y[τ0T ]

[τ0T+νg(T )]∑t=[τ0T ]+1

βt−1−[τ0T ]2T

t−1∑i=[τ0T ]+1

βt−1−i2T εi. (C.18)

63

Note that

[τ0T+νg(T )]∑t=[τ0T ]+1

β2(t−1−[τ0T ])2T = O

( ∣∣∣∣∣kT (1− β2[νg(T )]2T )

kT (1− β22T )

∣∣∣∣∣ )= O

( ∣∣∣∣kT · g(T )/kTkT (1− β2

2T )

∣∣∣∣ )= O(g(T )), (C.19)

yielding

y2[τ0T ]

[τ0T+νg(T )]∑t=[τ0T ]+1

β2(t−1−[τ0T ])2T = Op(Tg(T )l(ηT )). (C.20)

In addition, it can be shown that

E

[τ0T+νg(T )]∑t=[τ0T ]+1

( t−1∑i=[τ0T ]+1

βt−1−i2T ε

(1)i

)2=

[τ0T+νg(T )]∑t=[τ0T ]+1

t−1∑i=[τ0T ]+1

β2(t−1−i)2T l(ηT ) · (1 + o(1))

= O

(g(T )

β22T − 1

l(ηT )

)+O

((1− β

νg(T )2T )l(ηT )

(1− β22T )

2

)= O(kT g(T )l(ηT )) +O(k2T l(ηT ) · g(T )/kT )

= O(kT g(T )l(ηT )),

implying

[τ0T+νg(T )]∑t=[τ0T ]+1

( t−1∑i=[τ0T ]+1

βt−1−i2T εi

)2= Op(kT g(T )l(ηT )) = op(Tg(T )l(ηT )). (C.21)

Combining (C.18), (C.20) and (C.21) and applying Cauchy-Schwarz inequality, we have

[τ0T+νg(T )]∑t=[τ0T ]+1

y2t−1 = Op(Tg(T )l(ηT )). (C.22)

Note that the above arguments also imply that

max1≤t≤[νg(T )]−1

|y[τ0T ]+t − y[τ0T ]| = op(√

T l(ηT )).

Then, applying (C.22), we have the following results:

ΛT (τ)

l(ηT )= Op(1);

∣∣∣∣(β2T − 1)

l(ηT )·−∑[τ0T ]

t=1 yt−1εt∑[τT ]

t=[τ0T ]+1 y2t−1∑[τT ]

t=1 y2t−1

∣∣∣∣ = Op

(T 2g(T )

kT (T 2 + Tg(T ))

)= Op

(g(T )kT

)= op(1)

64

by Lemma B.1 and g(T ) = o(kT ); and∑[τ0T ]t=1 y2t−1∑[τT ]t=1 y2t−1

p→ 1.

Now, making use of equation (B.4) in Chong (2001), we have

RSST (τ)−RSST (τ0)

l(ηT )

=2(β2T − 1)

l(ηT )

[τT ]∑t=[τ0T ]+1

yt−1εt · (1 + op(1)) +(β2T − 1)2

l(ηT )

[τT ]∑t=[τ0T ]+1

y2t−1 · (1 + op(1)) + op(1)

=2(1 + op(1))√Tg(T )l(ηT )

[|ν|g(T )]−1∑t=0

y[τ0T ]+tε[τ0T ]+t+1 +1 + op(1)

Tg(T )l(ηT )

[|ν|g(T )]−1∑t=0

y2[τ0T ]+t+1 + op(1)

=−2y[τ0T ]√T l(ηT )

· 1 + op(1)√g(T )l(ηT )

[|ν|g(T )]−1∑t=0

(−ε[τ0T ]+t+1) + νy2[τ0T ]

T l(ηT )· (1 + op(1)) + op(1)

⇒ −2W1(τ0)W2(ν) + νW 21 (τ0). (C.23)

Finally, applying the continuous mapping theorem for argmax/argmin functionals (cf.

Kim and Pollard (1990)), it follows from (C.17) and (C.23) that

c2T 2

k2T(τT − τ0) =

T

g(T )(τT − τ0) = ν

= argminν∈R

{RSST (τ)−RSST (τ0)}

= argminν∈R

{RSST (τ)−RSST (τ0)

l(ηT )

}⇒ argmin

ν∈R

{− 2W 2

1 (τ0) ·(W ∗(ν)

W1(τ0)− |ν|

2

)}= argmax

ν∈R

{W ∗(ν)

W1(τ0)− |ν|

2

},

where W ∗(ν) is a two-sided Brownian motion on R defined to be W ∗(ν) = W1(−ν) for ν ≤ 0

and W ∗(ν) = W2(ν) for ν > 0.

To prove Theorem 1.3(b), since we have shown that |τT − τ0| = op(1/T ) when kT =

o(√T ), |τT − τ0| = Op(1/T ) when kT ≍

√T and |τT − τ0| = Op(k

2T /T

2) when√T = o(kT ),

we study the limiting distribution of β1(τT ) under the above three cases separately.

Consider the case where√T = o(kT ) first. In this case, note that |τT −τ0| = Op(k

2T /T

2).

Then, for τT ≤ τ0, we have

[τ0T ]∑t=[τTT ]+1

y2t−1 = Op(k2T l(ηT )) (C.24)

and[τ0T ]∑

t=[τTT ]+1

yt−1εt = Op(kT l(ηT )) (C.25)

65

by (B.20) and (B.21). For τT > τ0, it follows from the similar arguments in the proofs of

(B.22) and (B.28) respectively that

[τ0T+k2T /T ]∑t=[τ0T ]+1

y2t−1 =

[τ0T+k2T /T ]∑t=[τ0T ]+1

(βt−1−[τ0T ]2T y[τ0T ] +

t−1∑i=[τ0T ]+1

βt−1−i2T εi

)2

=

[τ0T+k2T /T ]∑t=[τ0T ]+1

(βt−1−[τ0T ]2T y[τ0T ] +Op(

√kTβ

t−1−[τ0T ]2T l(ηT ))

)2

=

[τ0T+k2T /T ]∑t=[τ0T ]+1

β2(t−1−[τ0T ])2T y2[τ0T ] · (1 + op(1))

= k2T /T ·Op(T l(ηT ))

= Op(k2T l(ηT )),

yielding[τTT ]∑

t=[τ0T ]+1

y2t−1 = Op(k2T l(ηT )) (C.26)

and

[τ0T+k2T /T ]∑t=[τ0T ]+1

yt−1εt =1− β2

2T

2β2T

[τ0T+k2T /T ]∑t=[τ0T ]+1

y2t−1 +y2[τ0T+k2T /T ]

− y2[τ0T ]

2β2T− 1

2β2T

[τ0T+k2T /T ]∑t=[τ0T ]+1

ε2t

= Op(kT l(ηT )) +Op(√

T l(ηT )) ·Op(kT√

l(ηT )/√T ) +Op(k

2T l(ηT )/T )

= Op(kT l(ηT )),

yielding[τTT ]∑

t=[τ0T ]+1

yt−1εt = Op(kT l(ηT )). (C.27)

66

Applying (C.24)-(C.27) and following Appendix G in Chong (2001), we have

T (β1(τT )− β1(τ0))

= T


t=1 y2t−∑[τ0T ]


)

= I{τT ≤ τ0}T(∑[τ0T ]

t=[τTT ]+1 y2t−1∑[τTT ]

t=1 y2t−1

∑[τ0T ]t=1 yt−1εt∑[τ0T ]t=1 y2t−1

−∑[τ0T ]


)

+I{τT > τ0}T(−∑[τTT ]

t=[τ0T ]+1 y2t−1∑[τTT ]

t=1 y2t−1

∑[τ0T ]t=1 yt−1εt∑[τ0T ]t=1 y2t−1

+


t=1 y2t−1

+ (β2T − β1)

∑[τTT ]t=[τ0T ]+1 y

2t−1∑[τTT ]

t=1 y2t−1

)= I{τT ≤ τ0}T

(Op(

k2T l(ηT )

T 2l(ηT ))Op(

1

T) +Op(

kT l(ηT )

T 2l(ηT )))

+I{τT > τ0}T(Op(

k2T l(ηT )

T 2l(ηT ))Op(

1

T) +Op(

kT l(ηT )

T 2l(ηT )) +Op(

k2T l(ηT )

kTT 2l(ηT )))

= op(1). (C.28)

Thus, β1(τT ) and β1(τ0) have the same asymptotic distribution. Then, applying Lemma

B.1 we have

T (β1(τ0)− β1) =

1T l(ηT )


1T 2l(ηT )

∑[τ0T ]t=1 y2t−1

⇒ W 2(τ0)− τ02∫ τ00 W 2(t)dt

,

implying

T (β1(τT )− β1) ⇒W 2(τ0)− τ02∫ τ00 W 2(t)dt

. (C.29)

This is equivalent to (1.8).

Consider the case where kT ≍√T . In this case, since |k−k0| = Op(1), one can follow the

proofs in Theorem 1.2 to show that T (β1(τT ) − β1(τ0)) = op(1) still holds by using (B.32)

and (B.33). The details are omitted here. Hence, (C.29) still holds when kT ≍√T .

Consider the case where kT = o(√T ). In this case, since P (k = k0) → 0, one can follow

the lines in the proof of Theorem 4 in Chong (2001) to show that (C.29) still holds. The

details are also omitted.

67

To prove Theorem 1.3(c). Similarly, when√T = o(kT ), making use of (C.24)-(C.27)

and following Appendix G in Chong (2001), we have

βT−[τ0T ]2T

√TkT (β2(τT )− β2(τ0))

= βT−[τ0T ]2T

√TkT


t=[τTT ]+1 y2t

−∑T

t=[τ0T ]+1 ytyt−1∑Tt=[τ0T ]+1 y

2t

)

= I{τT ≤ τ0}βT−[τ0T ]2T

√TkT

(−∑[τ0T ]

t=[τTT ]+1 y2t−1∑T

t=[τTT ]+1 y2t−1

∑Tt=[τ0T ]+1 yt−1εt∑Tt=[τ0T ]+1 y

2t−1

+


2t−1

+ (β1 − β2T )

∑[τ0T ]t=[τTT ]+1 y

2t−1∑T

t=[τTT ]+1 y2t−1

)

+I{τT > τ0}βT−[τ0T ]2T

√TkT

(∑[τTT ]t=[τ0T ]+1 y

2t−1∑T

t=[τTT ]+1 y2t−1

∑Tt=[τ0T ]+1 yt−1εt∑Tt=[τ0T ]+1 y

2t−1

−∑[τTT ]


2t−1

)= I{τT ≤ τ0}βT−[τ0T ]

2T

√TkT

(Op(

k2T l(ηT )


)Op(1

βT−[τ0T ]2T

√TkT l(ηT )

)

+Op(kT l(ηT )


) +Op(k2T l(ηT )

β2(T−[τ0T ])2T Tk2T l(ηT )

))

+I{τT > τ0}βT−[τ0T ]2T

√TkT

(Op(

k2T l(ηT )


)Op(1

βT−[τ0T ]2T

√TkT

)

+Op(kT l(ηT )


))

= op(1)

by using (C.16). Thus, β2(τT ) and β2(τ0) also have the same asymptotic distribution.

Applying Lemma C.2, we have

βT−[τ0T ]2T

√TkT (β2(τ0)− β2T ) =

β−(T−[τ0T ])2T√TkT l(ηT )


β−2(T−[τ0T ])2TTkT l(ηT )

∑Tt=[τ0T ]+1 y

2t−1

⇒ 2cX

W (τ0),

implying

βT−[τ0T ]2T

√TkT (β2(τT )− β2T ) ⇒

2cX

W (τ0). (C.30)

This is equivalent to (1.9) since β22T − 1 = 2c/kT (1 + o(1)).

βT−[τ0T ]2T

√TkT (β2(τT ) − β2(τ0)) = op(1) when kT ≍

√T or kT = o(

√T ) can also be

proved by similar arguments in the proof of Theorem 1.2, and the details are omitted.

Hence, (C.30) still holds.

Q.E.D. �

68

Appendix D: Proof of Theorem 1.4

The following three lemmas are needed in the proof of Theorem 1.4.

Lemma D.1 Let {yt} be generated according to Model (1.1), where β1 = β1T = 1 + c/kT

with c > 0. Then under the assumptions C1-C4, the following results hold jointly:

(a)β−[τ0T ]1T

kT l(ηT )

∑[τ0T ]t=1 yt−1εt ⇒ XY,

(b)β−2[τ0T ]1T

k2T l(ηT )

∑[τ0T ]t=1 y2t−1 ⇒ 1

2cY2,

where, X and Y are independent N(0, 12c) random variables.

Proof. Following the proof of Theorem 4.3 in Phillips and Magdalinos (2007a) and applying

the truncation technique as shown in (A.1). The details are omitted.

Q.E.D. �


with c > 0 and β2 = 1. Then under the assumptions C1-C4, the following results hold

jointly:

(a)β−[τ0T ]1T√

TkT l(ηT )

∑Tt=[τ0T ]+1 yt−1εt ⇒ Y (W (1)−W (τ0)),

(b)β−2[τ0T ]1T

TkT l(ηT )

∑Tt=[τ0T ]+1 y

2t−1 ⇒ (1− τ0)Y

2,

where Y is that as in Lemma D.1 and is independent of W (1)−W (τ0).

Proof. To prove part (a), note that

β−[τ0T ]1T√

TkT l(ηT )

T∑t=[τ0T ]+1

yt−1εt

=β−[τ0T ]1T√

TkT l(ηT )

T∑t=[τ0T ]+1

(y[τ0T ] +

t−1∑i=[τ0T ]+1

εi

)εt

=β−[τ0T ]1T√

TkT l(ηT )

(β[τ0T ]1T y0 +

[τ0T ]∑j=1

β[τ0T ]−j1T εj

) T∑t=[τ0T ]+1

εt +

T∑t=[τ0T ]+1

εt

t−1∑i=[τ0T ]+1

εi

:= IV1 + IV2, (D.1)

where

IV1 =β−[τ0T ]1T√

TkT l(ηT )

(β[τ0T ]1T y0 +

[τ0T ]∑j=1

β[τ0T ]−j1T εj

) T∑t=[τ0T ]+1

εt

and

IV2 =β−[τ0T ]1T√

TkT l(ηT )

T∑t=[τ0T ]+1

εt

t−1∑i=[τ0T ]+1

εi.

69

Applying part (b) of Lemma C.1 and assumption C3, it can be shown that

IV1 =1√

kT l(ηT )

[τ0T ]∑j=1

β−j1T εj ·

1√T l(ηT )

T∑t=[τ0T ]+1

εt + op(1) ⇒ Y (W (1)−W (τ0)). (D.2)

In addition, it is not difficult to show that

IV2 = Op

( β−[τ0T ]1T√

TkT l(ηT )· T l(ηT )

)= op(1) (D.3)

by (C.16).

Combining (D.1), (D.2) and (D.3) leads to part (a).

To prove part (b), note that

β−2[τ0T ]1T

TkT l(ηT )

T∑t=[τ0T ]+1

y2t−1

=β−2[τ0T ]1T

TkT l(ηT )

T∑t=[τ0T ]+1

(y[τ0T ] +

t−1∑i=[τ0T ]+1

εi

)2

=β−2[τ0T ]1T

TkT l(ηT )

T∑t=[τ0T ]+1

( [τ0T ]∑j=1

β[τ0T ]−j1T εj +

t−1∑i=[τ0T ]+1

εi

)2· (1 + op(1))

:= (V1 + V2 + V3) · (1 + op(1)), (D.4)

where

V1 =1

TkT l(ηT )

T∑t=[τ0T ]+1

( [τ0T ]∑j=1

β−j1T εj

)2,

V2 =β−2[τ0T ]1T

TkT l(ηT )

T∑t=[τ0T ]+1

( t−1∑i=[τ0T ]+1

εi

)2and

V3 =2β

−2[τ0T ]1T

TkT l(ηT )·[τ0T ]∑j=1

β[τ0T ]−j1T εj ·

T∑t=[τ0T ]+1

t−1∑i=[τ0T ]+1

εi

Obviously,

V1 ⇒ (1− τ0)Y2 (D.5)

by applying part (b) of Lemma C.1.

For the term V2, we have

V2 = Op

( β−2[τ0T ]1T

TkT l(ηT )· T 2l(ηT )

)= op(1) (D.6)

by making use of (C.16).

70

Finally, applying Cauchy-Schwarz inequality, it follows from (D.5) and (D.6) that

V3 = op(1). (D.7)

It is ready to see that part (b) is proved by combining (D.4)-(D.7).

One can also verfiy that parts (a) and (b) hold jointly and Y and W (1) − W (τ0) are

mutually independent.

Q.E.D. �


and β2 = 1 with c > 0. Then under the assumptions C1-C4, we have for any fixed integer

m ≥ 0,

(a) kT

β2[τ0T ]1T l(ηT )

(RSST (τ0 − m

T )−RSST (τ0))⇒ c2mY 2,

(b) kT


(RSST (τ0 +

mT )−RSST (τ0)

)⇒ c2mY 2.

Proof. To prove part (a), from equation (B.2) in Chong (2001), we have

RSST (τ0 −m

T)−RSST (τ0)

= 2(β2 − β1T )

(∑[τ0T ]t=[τ0T ]−m+1 y

2t−1


t=[τ0T ]−m+1 y2t−1

−∑T

t=[τ0T ]+1 y2t−1

∑[τ0T ]t=[τ0T ]−m+1 yt−1εt∑T

t=[τ0T ]−m+1 y2t−1

)

+ (β2 − β1T )2

∑Tt=[τ0T ]+1 y

2t−1

∑[τ0T ]t=[τ0T ]−m+1 y

2t−1∑T

t=[τ0T ]−m+1 y2t−1

+ ΛT (τ0 −m

T)

where

ΛT (τ0 −m

T)

=

(∑[τ0T ]t=1 yt−1εt

)2∑[τ0T ]

t=1 y2t−1

−

(∑[τ0T ]−mt=1 yt−1εt

)2∑[τ0T ]−m

t=1 y2t−1

+


)2∑T

t=[τ0T ]+1 y2t−1

−

(∑Tt=[τ0T ]−m+1 yt−1εt

)2∑T

t=[τ0T ]−m+1 y2t−1

.

Obviously, ΛT (τ0 − mT ) = Op(l(ηT )). It follows that

kT


(RSST (τ0 −

m

T)−RSST (τ0)

)=

kT


·(− 2c

kT

)( Op(mkTβ2[τ0T ]1T l(ηT ))Op(

√TkTβ

[τ0T ]1T l(ηT ))

Op(mkTβ2[τ0T ]1T l(ηT )) +Op(TkTβ

2[τ0T ]1T l(ηT ))

+Op(TkTβ

2[τ0T ]1T l(ηT ))

Op(√kTβ

[τ0T ]1T )

)

+c2

kTβ2[τ0T ]1T l(ηT )

·Op(TkTβ

2[τ0T ]1T l(ηT ))

Op(mkTβ2[τ0T ]1T l(ηT )) +Op(TkTβ

2[τ0T ]1T l(ηT ))

[τ0T ]∑[τ0T ]−m+1

y2t−1 + op(1)

=c2(1 + op(1))


[τ0T ]∑[τ0T ]−m+1

y2t−1 + op(1)

⇒ c2mY 2

71

by Lemmas C.1 and D.2 and the observations

∣∣∣∣∑[τ0T ]

t=[τ0T ]−m+1 yt−1εt∑Tt=[τ0T ]−m+1 y

2t−1

∣∣∣∣ ≤√∑[τ0T ]

t=[τ0T ]−m+1 ε2t√∑T

t=[τ0T ]−m+1 y2t−1

= Op(1

√kTβ

[τ0T ]1T

)

and ∑[τ0T ]t=[τ0T ]−m+1 y

2t−1

my2[τ0T ]

p→ 1.

To prove part (b), from equation (B.4) in Chong (2001), we have

RSST (τ0 +m

T)−RSST (τ0)

= 2(β2 − β1T )

(∑[τ0T ]t=1 y2t−1

∑[τ0T ]+mt=[τ0T ]+1 yt−1εt∑[τ0T ]+m

t=1 y2t−1

−∑[τ0T ]+m

t=[τ0T ]+1 y2t−1

∑[τ0T ]t=1 yt−1εt∑[τ0T ]+m

t=1 y2t−1

)

+ (β2 − β1T )2

∑[τ0T ]+mt=[τ0T ]+1 y

2t−1

∑[τ0T ]t=1 y2t−1∑[τ0T ]+m

t=1 y2t−1

+ ΛT (τ0 +m

T)

where

ΛT (τ0 +m

T)

=

(∑[τ0T ]t=1 yt−1εt

)2∑[τ0T ]

t=1 y2t−1

−

(∑[τ0T ]+mt=1 yt−1εt

)2∑[τ0T ]+m

t=1 y2t−1

+


)2∑T

t=[τ0T ]+1 y2t−1

−

(∑Tt=[τ0T ]+m+1 yt−1εt

)2∑T

t=[τ0T ]+m+1 y2t−1

.

Then, using the facts ΛT (τ0 +mT ) = Op(l(ηT )) and

∣∣∣∣∑[τ0T ]+m

t=[τ0T ]+1 yt−1εt∑[τ0T ]+mt=[τ0T ]+1 y

2t−1

∣∣∣∣ ≤√∑[τ0T ]+m

t=[τ0T ]+1 ε2t√∑[τ0T ]+m

t=[τ0T ]+1 y2t−1

= Op(1

√kTβ

[τ0T ]1T

),

we have

kT


(RSST (τ0 +

m

T)−RSST (τ0)

)=

kT


·(− 2c

kT

)(Op(k2Tβ

2[τ0T ]1T l(ηT ))

√kTβ

[τ0T ]1T

+Op(mkTβ

2[τ0T ]1T l(ηT ))Op(kTβ

[τ0T ]1T l(ηT ))

Op(k2Tβ2[τ0T ]1T l(ηT )) +Op(mkTβ

2[τ0T ]1T l(ηT ))

)

+c2


·Op(k

2Tβ

2[τ0T ]1T l(ηT ))

Op(k2Tβ2[τ0T ]1T l(ηT )) +Op(mkTβ

2[τ0T ]1T l(ηT ))

[τ0T ]+m∑t=[τ0T ]+1

y2t−1 + op(1)

=c2(1 + op(1))


[τ0T ]+m∑t=[τ0T ]+1

y2t−1 + op(1)

⇒ c2mY 2

by Lemmas C.1 and D.1.

72

Q.E.D. �

Proof of Theorem 1.4. Consider the first part of (1.4). We prove |τT − τ0| = Op(1/T )

first, and we only need to show that

Ai = op(1/kT ), i = 1, 2, 4, 5 and Aj = op(1/k2T ), j = 3, 6.

The definition of Ai’s can be found in equation (B.11).

First, by applying Lemmas D.1 and D.2, it can be shown that

A1 = op(1/kT ) and A4 = op(1/kT ).

Secondly, we investigate the magnitude of the terms A2 and A5 in probability. For A2, note

that Cauchy-Schwarz inequality implies

supm∈D1T

∣∣∣∣∣∑[τ0T ]

t=m+1 yt−1εt∑[τ0T ]t=m+1 y

2t−1

∣∣∣∣∣ ≤ supm∈D1T

√√√√ ∑[τ0T ]t=m+1 ε

2t∑[τ0T ]

t=m+1 y2t−1

,

it suffices to prove that the magnitude of the right hand side of the above inequality is

op(1/kT ). Applying Lemma C.1, we immediately have

supm∈D1T

√√√√∑[τ0T ]t=m+1 ε

2t∑[τ0T ]

t=1 y2t−1

≤

√√√√∑[τ0T ]t=1 ε2t

y2[τ0T ]−1

≤ Op

(√T l(ηT )


)= op(1/kT )

by (C.16), which implies that

A2 ≤ supm∈D1T

√√√√∑[τ0T ]t=m+1 ε

2t∑[τ0T ]

t=1 y2t−1

= op(1/kT ).

Similarly, for the term A5, we have

supm∈D2T

∣∣∣∣∣∑m

t=[τ0T ]+1 yt−1εt∑mt=[τ0T ]+1 y

2t−1

∣∣∣∣∣ ≤ supm∈D2T

√√√√ ∑mt=[τ0T ]+1 ε

2t∑m

t=[τ0T ]+1 y2t−1

≤

√√√√∑Tt=[τ0T ]+1 ε

2t

y2[τ0T ]

= Op

(√T l(ηT )


)= op(1/kT ),

which implies that

A5 = op(1/kT ).

73

Note also that

A3 = supm∈D1T

∣∣∣ ∑Tt=m+1 y

2t−1∑T

t=[τ0T ]+1 y2t−1

∑[τ0T ]t=m+1 y

2t−1

ΛT (m

T)∣∣∣

≤(

1∑Tt=[τ0T ]+1 y

2t−1

+1∑[τ0T ]

t=[τ0T ]−MTy2t−1

)(sup

m∈D1T

∣∣∣(∑[τ0T ]t=1 yt−1εt)

2∑[τ0T ]t=1 y2t−1

−(∑m

t=1 yt−1εt)2∑m

t=1 y2t−1

∣∣∣+ sup

m∈D1T

∣∣∣(∑Tt=[τ0T ]+1 yt−1εt)

2∑Tt=[τ0T ]+1 y

2t−1

−(∑T


t=m+1 y2t−1

∣∣∣)≤

(Op

( 1

TkTβ2[τ0T ]1T l(ηT )

)+Op

( 1


))·Op(l(ηT ))

= op(1/k2T ),

and

A6 = supm∈D2T

∣∣∣ ∑mt=1 y

2t−1∑m

t=[τ0T ]+1 y2t−1

∑[τ0T ]t=1 y2t−1

ΛT (m

T)∣∣∣

≤(

1∑[τ0T ]t=1 y2t−1

+1∑[τ0T ]+MT

t=[τ0T ]+1 y2t−1

)(sup

m∈D2T

∣∣∣(∑[τ0T ]t=1 yt−1εt)

2∑[τ0T ]t=1 y2t−1

−(∑m

t=1 yt−1εt)2∑m

t=1 y2t−1

∣∣∣+ sup

m∈D2T

∣∣∣(∑Tt=[τ0T ]+1 yt−1εt)

2∑Tt=[τ0T ]+1 y

2t−1

−(∑T


t=m+1 y2t−1

∣∣∣)≤

(Op(

1

k2Tβ2[τ0T ]1T l(ηT )

) +Op(1


))·Op(l(ηT ))

= op(1/k2T ).

Hence, |τT − τ0| = Op(1/T ) is proved, and the proof of P (k = k0) → 0 can be completed by

using Lemma D.3. The details are omitted.

To find the limiting distribution of β1(τT ), note that the limiting distributions of β1(τT )

and β1(τ0) are the same by recalling the fact P (k = k0) → 0 and the similar arguments in

the proof of Theorem 4 in Chong (2001). The details are omitted. Then, invoke Lemma

D.1, we have

kTβ[τ0T ]1T (β1(τ0)− β1T ) =

β−[τ0T ]1T

kT l(ηT )


β−2[τ0T ]1T

k2T l(ηT )

∑[τ0T ]t=1 y2t−1

⇒ 2cX/Yd= 2cζ,

which implies

kTβ[τ0T ]1T (β1(τT )− β1T ) ⇒ 2cζ.

This is equivalent to (1.11) since β21T − 1 = 2c/kT (1 + o(1)).

74

Analogously, β2(τT ) and β2(τ0) have the same asymptotic distribution. Then, applying

Lemma D.2 leads to

√TkTβ

[τ0T ]1T (β2(τ0)− β2) =

β−[τ0T ]1T√

TkT l(ηT )


β−2[τ0T ]1T

TkT l(ηT )

∑Tt=[τ0T ]+1 y

2t−1

⇒ W (1)−W (τ0)

(1− τ0)Y,

which implies

√TkTβ

[τ0T ]1T (β2(τT )− β2) ⇒

W (1)−W (τ0)

(1− τ0)Y.

This is equivalent to (1.12).

Q.E.D. �

75

structural change in non-stationary ar(1)...

Documents