structural change in ar(1) models - CUHK

Econometric Theory, 17, 2001, 87–155+ Printed in the United States of America+

STRUCTURAL CHANGE IN AR(1) MODELS TE R E N C E TA I-LE U N G CH O N G The Chinese University of Hong Kong

This paper investigates the consistency of the least squares estimators and derives their limiting distributions in an AR~1! model with a single structural break of unknown timing+ Let b1 and b2 be the preshift and postshift AR parameter, respectively+ Three cases are considered: ~i! 6 b1 6 , 1 and 6 b2 6 , 1; ~ii! 6 b1 6 , 1 and b2 5 1; and ~iii! b1 5 1 and 6 b2 6 , 1+ Cases ~ii! and ~iii! are of particular interest but are rarely discussed in the literature+ Surprising results are that, in both cases, regardless of the location of the change-point estimate, the unit root can always be consistently estimated and the residual sum of squares divided by the sample size converges to a discontinuous function of the change point+ In case ~iii!, bZ 2 does not converge to b2 whenever the change-point estimate is lower than the true change point+ Further, the limiting distribution of the break-point estimator for shrinking break is asymmetric for case ~ii!, whereas those for cases ~i! and ~iii! are symmetric+ The appropriate shrinking rate is found to be different in all cases+

1. INTRODUCTION Previous studies in the literature of structural change focus mainly on changes that take place in stationary processes, for example, a process that changes from one stationary process to another+ Recently, there also have been studies on changes in nonstationary time series ~Hansen, 1992!+ An interesting case of structural change is found in Mankiw and Miron ~1986! and Mankiw, Miron, and Weil ~1987!; these authors conclude 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+ However, the asymptotic theory on this kind of structural change still remains unexplored+ In this paper, we develop a comprehensive asymptotic theory for an AR~1! model with a single structural break of unknown timing+ Specifically, we examine the case where an AR~1! process changes from a stationary one to a nonstationary one ~or the other way around!+ In each case, the asymptotic criterion function is derived, consistency of estimators is established, and limiting This paper is based on chapter 4 of my Ph+D+ dissertation submitted to the University of Rochester+ I am grateful to my thesis adviser, Bruce Hansen, for his advice and encouragement+ My thanks also go to the editor and three anonymous referees for their helpful comments, which led to a substantial revision+ All errors are mine+ Address correspondence to: Terence Tai-Leung Chong, Department of Economics, the Chinese University of Hong Kong, Shatin, N+T+, Hong Kong; email: b792703@mailserv+cuhk+edu+hk+

© 2001 Cambridge University Press

0266-4666001 $9+50

87

88

TERENCE TAI-LEUNG CHONG

distributions of estimators are also derived+ As far as economic applications are concerned, we will restrict our discussion to the case where both the preshift and postshift parameters are in the interval ~21,1# + The plan of the paper is as follows+ Section 2 presents our basic model+ Section 3 deals with the case where the preshift and postshift parameters are both less than one in absolute value+ Section 4 studies the case where the preshift parameter is less than one in absolute value and the postshift parameter equals one+ Section 5 examines the case where the preshift parameter equals one and the postshift parameter is less than one in absolute value+ Section 6 discusses Monte Carlo experiments, and Section 7 concludes the paper+ Mathematical details are collected in Appendices A–K+ Interesting findings include the following results: ~i! in Sections 4 and 5 the asymptotic criterion function is random and has a sudden jump at the true break point; ~ii! in Section 4, the asymptotic distribution of the break-point estimator for shrinking break is asymmetric; and ~iii! in Section 5, the preshift estimator is always consistent, whereas the postshift estimator bZ 2 does not converge to b2 once the change-point estimate is lower than the true change point+

2. THE MODEL Our basic model is an AR~1! model without drift, with a structural break in the AR parameter b at an unknown time k 0 + We consider the following model: yt 5 b1 yt21 1$t # k 0 % 1 b2 yt21 1$t . k 0 % 1 «t

~t 5 1,2, + + + , T !,

(1)

where 1${% is an indicator function that equals one when the statement in the braces is true and equals zero otherwise+ We let t0 5 k 0 0T be the true break fraction and make the following assumptions+ ~A1! y0 is drawn from an independent and identical distribution with zero mean and a finite variance+ ~A2! «t ; i+i+d+~0, s 2 ! ∀t, 0 , s 2 , `, and E~«t4 ! , `+ S , ~0,1!+ ~A3! t0 [ @ st, t# We let k 5 @tT # where @{# is the greatest integer function and define sk 5 @ stT # and kN 5 @ tT S #+ Our interests are to estimate the structural parameters b1 and b2 and the time of change t0 + We estimate the following model: y[ t 5 bZ 1 yt21 1$t # @ t[ T T #% 1 bZ 2 yt21 1$t . @ t[ T T #%+

(2)

STRUCTURAL CHANGE IN AR(1) MODELS

89

The estimation method employed here is the least squares method proposed in Bai ~1994a, 1994b! and Chong ~1995!+ For any given t, the ordinary least squares ~OLS! estimators are given by @tT #

bZ 1 ~t! 5

(1 yt yt21 (1

(3)

,

@tT # 2 yt21

T

(

bZ 2 ~t! 5

@tT #11

yt yt21

(

(4)

,

T

@tT #11

2 yt21

and the change-point estimator satisfies t[ T 5 Arg min RSST ~t!,

(5)

t[~0,1!

where @tT #

RSST ~t! 5

(1

T

~ yt 2 bZ 1 ~t!yt21 ! 2 1

(

@tT #11

~ yt 2 bZ 2 ~t!yt21 ! 2+

We write kZ 5 @ t[ T T # , bZ 1 5 bZ 1~ t[ T !, and bZ 2 5 bZ 2 ~ t[ T !+ Throughout this paper, we let B1~{! and B2 ~{! be two independent Brownian motions defined on the non-negative half real line R 1 and B~{! and B~{! O be two independent standard Brownian motions on @0,1# + The symbol n denotes the weak convergence of a p d stochastic process, & represents convergence in probability, & denotes cond vergence in distribution and 5 stands for identical in distribution+ To achieve notational economy, the integral of a Brownian motion with respect to Lebesgue measure *B~r! dr is written as *B + The stochastic integral *B~r! dB~r! is written as *BdB+ 3. CASE WHERE _ b1 _ , 1 AND _ b2 _ , 1 3.1. The Asymptotic Criterion Function Consider the model in ~1! with 6 b1 6 , 1 and 6 b2 6 , 1+ A similar model has been studied by Salazar ~1982!, who provides a Bayesian analysis of structural changes in stationary AR~1! and AR~2! models with a known change point+ In our model, the change point t0 is unknown and has to be estimated by t[ T + To show the consistency of t[ T , the usual practice is to show that ~10T !RSST ~t! converges uniformly to a nonstochastic function that has a unique minimum at t 5 t0 + Thus, we will focus on the asymptotic behavior of ~10T !RSST ~t!+ The

90


following lemma is useful in deriving the limiting behavior of the criterion function ~10T !RSST ~t! and in proving Theorem 1, which follows+ LEMMA 1+ Let $ yt % Tt51 be generated according to model ~1! with 6 b1 6 , 1 and 6 b2 6 , 1+ Under Assumptions ~A1!–~A3!, we have ~3a!

1

sup

T

0#tA#tB#1

1

~3b!

@t0 T #

(1

T 1

~3c!

T

~3d!

2 yt21

*(

*

@tB T # @tA T #

p

t0 s 2

&

T

2 ( yt21 @t T #11

yt21 «t 5 op ~1!;

1 2 b12 p

&

~1 2 t0 !s 2

st#t#t0

*

1 T

;

1 2 b22

0

sup

;

@t0 T #

2 2 ( yt21 @tT #11

~t0 2 t!s 2 1 2 b12

*

5 op ~1!;

sup 6 bZ 1 ~t! 2 b1 6 5 op ~1!;

~3e!

st#t#t0

5 o ~1!; * ~t 2 t!~1 2 b ! 1 ~1 2 t !~1 2 b ! * ~t 2 t !s 1 sup * T ( y 2 1 2 b * 5 o ~1!; t ~1 2 b !b 1 ~t 2 t !~1 2 b !b sup bZ ~t! 2 5 o ~1!; * t ~1 2 b ! 1 ~t 2 t !~1 2 b ! *

~3f !

sup

st#t#t0

bZ 2 ~t! 2

~t0 2 t!~1 2 b22 !b1 1 ~1 2 t0 !~1 2 b12 !b2 2 2

0

@tT #

~3g!

t0,t#tS

2

0

2 t21

@t0 T #11

2 2

1

t0,t#tS

p

2 2

0

~3h!

p

2 1

0

1

0

2 2

0

0

2 1 2 1

2

p

sup 6 bZ 2 ~t! 2 b2 6 5 op ~1!+

~3i!

t0,t#tS

Proof+ See Appendix A+ Note from ~3e! that when st # t # t0 , bZ 1~t! converges uniformly to b1 + This should be obvious because bZ 1~t! only utilizes the data generated by the process yt 5 b1 yt21 1 «t + However, from ~3f !, bZ 2 ~t! converges uniformly to a weighted average of b1 and b2 + The weight depends on the true change point, the true preshift and postshift parameters, and the location of t+ When t0 , t # t,S ~3h! and ~3i! display similar results+ From Appendix B, the asymptotic behavior of the residual sum of squares is as follows: sup st#t#t0

*

*

~t0 2 t!~1 2 t0 !~ b2 2 b1 ! 2 s 2 1 RSST ~t! 2 s 2 2 5 op ~1!, T ~t0 2 t!~1 2 b22 ! 1 ~1 2 t0 !~1 2 b12 ! (6)

sup t0,t#tS

* T RSS ~t! 2 s 2 t ~1 2 b ! 1 ~t 2 t !~1 2 b ! * 5 o ~1!+ t0 ~t 2 t0 !~ b2 2 b1 ! s 2

1

2

T

0

2 2

0

2

2 1

p

(7)


91

Figure 1. Graph of ~10T !RSST ~t! for b1 5 0+5, b2 5 20+5, T 5 4000+

One can easily verify that ~10T !RSST ~t! converges uniformly to a piecewise concave function of t that attains a unique global minimum at t 5 t0 + A simulation of ~10T !RSST ~t! for T 5 4,000 is plotted in Figure 1+ 3.2. The Consistency and the Limiting Distributions of bZ 1 and bZ 2 If both b1 and b2 are less than one in absolute value then the process is said to be stationary throughout+ It is not difficult to show that all the OLS estimators are consistent in this case+ Bai ~1994a, 1994b! shows that, in the conventional stationary case, the change-point estimator is T-consistent+ This convergence rate is fast enough to make the limiting distributions of bZ 1 and bZ 2 behave as if the true change point t0 is known+ Theorem 1 establishes the asymptotic normality of bZ 1 and bZ 2 + THEOREM 1+ Under assumptions ~A1!–~A3!, if 6 b1 6 , 1 and 6 b2 6 , 1, the OLS estimators t[ T , bZ 1~ t[ T !, and bZ 2 ~ t[ T ! are consistent and

92


6 t[ T 2 t0 6 5 Op

!T ~ bZ 1~ t[ T ! 2 b1 !

d

!T ~ bZ 2 ~ t[ T ! 2 b2 !

d

SD

S S

1 , T

D D

&

N 0,

1 2 b12 , t0

&

N 0,

1 2 b22 + 1 2 t0

(8)

Proof+ See Appendix C+ Thus, bZ 1~ t[ T ! and bZ 2 ~ t[ T ! are asymptotically normally distributed, with variances depending on b1 , b2 , and t0 + 3.3. The Limiting Distribution of t[ T as b2 Collapses to b1 In this section, we consider the limiting distribution of t[ T + For a shift with fixed magnitude, Hinkley ~1970! showed that the limiting distribution of the changepoint estimator depends on the underlying distribution of the innovation «t in a complicated manner+ Thus, statistical inference on the change point under a break of fixed magnitude is difficult to perform+ To obtain a limiting distribution of t[ T invariant to «t , we have to let the magnitude of change go to zero at an appropriate rate+ Further, to ensure the consistency of t[ T , we should let 6 b2 2 b1 6 shrink at a rate slower than the rate of convergence of bZ 1~t0 ! and bZ 2 ~t0 ! so that there still exists a relative shift in parameters+ To achieve this, we fix b1 and let b2T be a sequence of b2 such that 6 b2T 2 b1 6 r 0 and !T 6 b2T 2 b16 r ` as T r `+ Recall from equation ~5! that the change-point estimator is defined as def

t[ T 5 Arg min RSST ~t! 5 Arg min $RSST ~t! 2 RSST ~t0 !%+ t

t

Let y [ R be a finite constant+ We first examine the asymptotic behavior of RSST ~t! 2 RSST ~t0 ! at the region t 5 t0 1 y0T ~ b2T 2 b1 ! 2 + From Appendix D, we have the following expressions+ For y # 0, RSST ~t! 2 RSST ~t0 ! n 22s 2 B1

6y6 1 2 b12

S D

1

s 2 6y6 + 1 2 b12

(9)

S D

1

s 2y 1 2 b12

(10)

For y . 0, RSST ~t! 2 RSST ~t0 ! n 22s 2 B2

y 1 2 b12

where B1~{! and B2 ~{! are defined in Section 2+


93

Let r 5 y0~1 2 b12 ! and applying the continuous mapping theorem for argmax functionals ~Kim and Pollard, 1990!, we have the following theorem+ THEOREM 2+ Under assumptions ~A1!–~A3!, suppose we fix 6 b1 6 , 1 and let b2T be a sequence of b2 such that 6 b2T 2 b1 6 r 0 and !T 6 b2T 2 b1 6 r ` as T r `+ Then the limiting distribution of t[ T is given by T ~ b2T 2 b1 ! 2 ~ t[ T 2 t0 ! 1 2 b12

d

&

S

Arg max B * ~r! 2 r[R

D

1 6r6 , 2

(11)

where B * ~r! is a two-sided Brownian motion on R defined to be B * ~r! 5 B1~2r! for r , 0 and B * ~r! 5 B2 ~r! for r $ 0+ The terms B1~{! and B2 ~{! are defined in Section 2+ Proof+ See Appendix D+ The exact distribution of the right hand side in ~11! was first derived by Picard ~1985!+ Yao ~1987! tabulates the numerical approximation of this distribution+ To understand the implication of Theorem 2, consider a Brownian motion moving along an inverted-V shape linear function with a kink at t0 + Because t[ T is the location where this motion achieves its maximum, its limiting distribution will be symmetric+ One should be careful that ~11! is not the exact distribution of the changepoint estimator for fixed shifts as we are letting the shift shrink to zero when deriving the theorem+ Rather, the theorem serves to provide a conservative confidence interval for t0 when the shift is small+ 4. CASE WHERE _ b1 _ , 1 and b2 = 1 We now consider the case where the AR~1! process shifts from a stationary process to an I ~1! process+ It is well known that the distribution theory for the least squares estimator in an AR process with a unit root or near unit root is nonstandard+ See, for example, White ~1958!; Dickey and Fuller ~1979!; Lai and Siegmund ~1983!; Ahtola and Tiao ~1984!; Chan and Wei ~1987, 1988!; Phillips ~1987, 1988!, and Perron ~1996!+ 4.1. The Asymptotic Criterion Function In our case, when b2 5 1, the criterion function ~10T !RSST ~t! behaves very differently+ The following lemma is useful in deriving its limiting behavior and in proving Theorem 3, which follows+ LEMMA 2+ Let $ yt % Tt51 be generated according to model ~1! with 6 b1 6 , 1 and b2 5 1+ Define

H

S ù t 5 t0 1 JT1 5 ~t0 , t#

j : j 5 1,2, + + + , @~1 2 t0 !T # T

and

lim Tr`

j

!T

J

5` +

94


Then under Assumptions ~A1!–~A3!, we have ~4a!

*

(1

T

(

T

@t0 T #11

1

~4c!

T

~4d!

2

2

0

1 T

st#t#t0

sup

*

*(

T

E

1

B 2 5 Op ~1!;

0

*

@t0 T #

yt21 «t 5 op ~1!;

@tT #11

1

~B 2 ~1! 2 1! 5 Op ~1!;

@t0 T #

2 2 ( yt21 @tT #11

~t0 2 t!s 2 1 2 b12

*

5 op ~1!;

sup 6 bZ 1 ~t! 2 b1 6 5 op ~1!;

~4f !

st#t#t0

sup 6 bZ 2 ~t! 2 16 5 Op

~4g!

st#t#t0

SD 1

T

;

sup 6 bZ 2 ~t! 2 b1 6 5 Op ~1!;

~4h!

~4j!

~1 2 t0 !s 2

T

0#t#t0

~4i!

yt21 «t n

2 n ~1 2 t0 ! 2 s 2 ( yt21 @t T #11

sup

~4e!

yt21 «t 5 op ~1!;

T

1

~4b!

*

@t0 T #

1

st#t#t0

1

*

*

@tT #

(

yt21 «t 5 Op ~1!

for t [ JT1 ;

yt21 «t * 5 Op ~1! T * @tT( #11

for t [ JT1 ;

T

@t0 T #11

1

T

S! D SD 1

~4k! 6 bZ 1 ~t! 2 16 5 Op ~4l! 6 bZ 2 ~t! 2 16 5 Op

T

1

T

for t [ JT1 ;

for t [ JT1 +

Proof+ See Appendix E+ Note that the set JT1 is different from the set ~t0 , t# S as JT1 excludes sequences of t that converge too fast to t0 + For example, JT1 contains all given constants t such that t0 , t # t+S The sequence tT 5 t0 1 ~10log T ! also belongs to JT1+ However, the sequence tT 5 t0 1 ~10T ! is not in JT1+ From Appendix F, we have sup st#t#t0

*

*

~1 2 b1 !~t0 2 t!s 2 1 RSST ~t! 2 s 2 2 5 op ~1! T 1 1 b1

(12)

and 1 RSST ~t! T

p

&

s2 1

~1 2 b1 !t0 s 2 1 1 b1

for t [ JT1 +

(13)


95

Note from ~12! and ~13! that ~10T !RSST ~t! converges uniformly to a downward sloping linear function for st # t # t0 and it converges to a flat line located above s 2 for t [ JT1+ Further, there is an asymptotic gap between ~10T !RSST ~t0 ! and ~10T !RSST ~t! for t [ JT1+ The transitional process begins at t0 and completes at the left boundary of JT1+ Thus the set JT1 establishes the rate at which plim~10T !RSST ~t! jumps from s 2 to a flat line located above s 2 once t . t0 + This rate serves as a conservative rate of consistency of t[ T + To examine the consistency of bZ 1 , we have to investigate the transitional behavior of ~10T !RSST ~t!+ It should be noted that, for any constant c . 0,

bZ 1 ~t0 1 cT

a21

1

k0

! 5 uT ~a, c! b1 1

(1 «t yt21 k0

2 (1 yt21

1

2

k 01@cT a #

(

1 ~1 2 uT ~a, c!! 1 1

k 011

«t yt21

k 01@cT a #

(

2 yt21

k 011

2

,

1@cT # 2 2 2 !0~ ( k10 yt21 1 ( kk0011 yt21 !+ where uT ~a, c! 5 ~ ( k10 yt21 1 _ For a , 2 , we have a

uT ~a, c!

p

&

1,

bZ 1 ~t0 1 cT a21 !

p

&

b1 ,

&

1,

and 1 RSST ~t0 1 cT a21 ! T For

1 _ 2

uT ~a, c!

p

&

s 2+

, a , 1, we have p

&

0,

bZ 1 ~t0 1 cT a21 !

p

and 1 RSST ~t0 1 cT a21 ! T

p

&

s2 1

~1 2 b1 !t0 s 2 + 1 1 b1

This implies that if the convergence rate of t[ T is faster than T 102 then bZ 1 will be consistent; otherwise it will be inconsistent+

96


Note that the transition is defined only when a 5 _12 , which means the speed of the transition is 10!T + For a 5 _12 , Appendix F shows that uT

S D

1 ,c n 2

t0 t0 1 ~1 2 b12 !

E

,

c

B32

0

S

bZ 1 t0 1

c

!T

D

E E

c

t0 b1 1 ~1 2 b12 !

B32

0 c

n t0 1 ~1 2 b12 !

,

B32

0

S

c 1 RSST t0 1 T !T

D

t0 s 2 ~1 2 b1 ! 2

n s2 1

SE D

21

c

t0

B32

def

5 s 2 1 w~c!,

(14)

1 1 2 b12

0

where B3 ~{! is a Brownian motion defined on R 1 + Note that when c 5 0, p p ~10T !RSST ~t0 ! & s 2 , and as c r `, ~10T !RSST ~t0 1 ~c0!T !! & s 2 1 2 @~1 2 b1 !t0 s #0~1 1 b1 !+ Because ]w~c!0]c . 0, the transition is monotonically increasing+ Figure 2 simulates the behavior of ~10T !RSST ~t! using T 5 4,000+ For t # t0 , it is linear and decreasing+ For t . t0 , it is flat and has a sudden jump near t0 + 4.2. The Consistency and the Limiting Distributions of t[ T , bZ 1 , and bZ 2 For a fixed magnitude of the break, Theorem 3, which follows, shows that bZ 1 is asymptotically normally distributed with a variance depending on b1 and t0 , whereas bZ 2 has a scaled Dickey–Fuller distribution+ Similar to Section 3+3, we should let the magnitude of the break go to zero at a certain rate to obtain the asymptotic distribution of t[ T + However, if we make 0 b1 converge to b2 ~5 1!, the process $ yt % kt51 will be a near unit-root process+ Thus, we are dealing with a structural change from a near unit-root process to an exact unit-root process+ We fix b2 at 1 and let b1T be a sequence of b1 such that b1T r 1 and T ~1 2 b1T ! r ` as T r `+ Consider the region t 5 t0 1 y0T ~1 2 b1T ! where y [ R+ From Appendix G, we have the following expressions+ For y # 0, RSST ~t! 2 RSST ~t0 ! n 22s 2 B1 ~6y6!Ba ~ 212 ! 1 6y6s 2 Ba2 ~ 212 !+


97

Figure 2. Graph of ~10T !RSST ~t! for b1 5 0+5, b2 5 1, T 5 4000+

For y . 0, RSST ~t! 2 RSST ~t0 !

ES y

n 2s 2

B2 ~s! 1 Ba

0

S DD 1 2

ES y

dB2 ~s! 1 s 2

B2 ~s! 1 Ba

0

S DD 1 2

2

ds,

where B1~{! and B2 ~{! are defined in Section 2+ The expression Ba ~ _12 ! is gener` ated by *0 exp~2s! dB1 ~s!+ Theorem 3, which follows, states that as b1 approaches one, the limiting distribution of t[ T is the argmax of a random process moving along an inverted-V shape linear function on the real line+ THEOREM 3+ Under Assumptions ~A1!–~A3!, if 6 b1 6 , 1 and b2 5 1, the OLS estimators t[ T , bZ 1~ t[ T !, and bZ 2 ~ t[ T ! are all consistent and

98


6 t[ T 2 t0 6 5 Op

!T ~ bZ 1~ t[ T ! 2 b1 !

d

SD

S

1 , T

N 0,

&

D

1 2 b12 , t0

(15)

B 2 ~1! 2 1

T ~ bZ 2 ~ t[ T ! 2 1! n

2~1 2 t0 !

E

+

1

B

2

0

Suppose we fix b2 at one and let b1T be a sequence of b1 such that 61 2 b1T 6 r 0 and T ~1 2 b1T ! r ` as T r `+ Then the limiting distribution of t[ T is given by d

~1 2 b1T !T ~ t[ T 2 t0 !

&

S SD D ES SD SD D

Arg max y[R

1 C * ~y! 2 6y6 , 1 2 Ba 2

where C * ~y! is defined to be C * ~y! 5 B1~2y! for y # 0 and

E

C * ~y! 5 2B2 ~y! 2

0

y

B2 ~s! dB2 ~s! 2 1 Ba 2

y

0

B2 ~s! 1 1 B2 ~s! ds 1 2Ba 2

for y . 0+ The terms B1~{! and B2 ~{! are defined in Section 2+ Here Ba ~ _12 ! is generated by ` *0 exp~2s! dB1 ~s!+ Proof+ See Appendix G+ An interesting and important feature of Theorem 3 is that the distribution of t[ T is asymmetric about t0 + An intuitive explanation for this result is that ~10T !RSST ~t! for breaks with fixed magnitude is asymmetric in the neighborhood of t0 as shown in Figure 2+ As we compress the size of break, this asymmetry perseveres+ Thus, t[ T 5 arg min RSST ~t! should also be asymmetric about t0 + 5. CASE WHERE b1 = 1 and _ b2 _ , 1 Consider the case where b1 5 1 and 6 b2 6 , 1+ Intuitively, it seems that the results of this case are the mirror images of the case where 6 b1 6 , 1 and b2 5 1+ However, we show that a time reversing argument cannot be applied to model ~1!+ The findings in this section turn out to be very different from those in Section 4+


99

5.1. The Asymptotic Criterion Function The following lemma is useful in deriving the limiting behavior of ~10T !RSST ~t! and in proving Theorem 4, which follows+ LEMMA 3+ Let $ yt % Tt51 be generated according to model ~1! with b1 5 1 and 6 b2 6 , 1+ Define

H

JT2 5 @ st, t0 ! ù t 5 t0 2

j : j 5 1,2, + + + , @t0 T # T

and

lim Tr`

j

!T

5`

J

and let JT1 be defined as in Lemma 2+ Under assumptions ~A1!–~A3!, we have ~5a! ~5b!

@t0 T #

1

(1

T

yt21 «t n

s2 2

~B 2 ~t0 ! 2 t0 ! 5 Op ~1!;

S

T

1

yt21 «t !T @t ( T #11

d

N 0,

&

0

~5c! ~5d!

~5e!

~5f ! ~5g! ~5h! ~5i!

~5j! ~5k! ~5l!

@t0 T #

1 T 1 T

(1

2

2 yt21 n s2

E

t0

2 n ( yt21 @t T #11

1 2 b22

* T ( y « * 5 O ~1!; 1

sup t[JT2

@tT #

t21

t

p

1

*

1 T

@t0 T #

(

@tT #11

*

yt21 «t 5 Op ~1!;

sup 6 bZ 1 ~t! 2 16 5 Op

t[JT2

SD 1

T

;

sup 6 bZ 2 ~t! 2 16 5 op ~1!;

t[JT2

sup t[JT1

sup t[JT1

*

1

@tT #

T

@t0 T #11

*T

1

(

*

yt21 «t 5 op ~1!;

( yt21 «t * 5 op ~1!; @tT #11 T

sup 6 bZ 1 ~t! 2 16 5 Op

t[JT1

SD 1

T

sup 6 bZ 2 ~t! 2 b2 6 5 op ~1!+

t[JT1

D

;

B 2 5 Op ~1!;

t 2 t0 1 B 2 ~t0 !

0

t[JT2

1 2 b22

0

@tT #

sup

s4

;

s2;

100


Proof+ See Appendix H+ The set JT2 serves a similar purpose to that of JT1 in Lemma 2+ The term JT2 excludes sequences of t that converge to t0 at a speed faster than 10!T + For example, JT2 contains all given constants t such that st # t , t0 + The sequence tT 5 t0 2 ~10log T ! also belongs to JT2 , but the sequence tT 5 t0 2 ~10T ! does not+ We now derive the asymptotic behavior of ~10T !RSST ~t!+ From Appendix I, we have the following expressions+ For t [ JT2 , ~1 2 b2 !~1 2 t0 1 B 2 ~t0 !! 2 1 RSST ~t! n s 2 1 s, T 1 1 b2 sup ta , tb[JT2

1 6RSST ~tb ! 2 RSST ~ta !6 5 op ~1!+ T

(16)

For t 5 t0 , 1 RSST ~t0 ! T

p

&

s 2+

(17)

For t [ JT1 , ~1 2 b2 !~t 2 t0 1 B 2 ~t0 !! 2 1 RSST ~t! n s 2 1 s, T 1 1 b2 sup t[JT1

*

S D*

1 1 RSST ~t! 2 RSST t 1 T T

5 op ~1!+

(18)

Note from ~16! that, for t [ JT2 , ~10T !RSST ~t! converges to a flat line located randomly above s 2 1 @~1 2 b2 !~1 2 t0 !s 2 #0~1 1 b2 ! . s 2 + For t [ JT1 , ~10T !RSST ~t! converges to a random upward-sloping linear function of t+ The lower support for this line is s 2 1 @~1 2 b2 !~t 2 t0 !s 2 #0 ~1 1 b2 ! . s 2+ In both situations, ~10T !RSST ~t! converges to a random line whose position depends on the true change point t0 , the postshift AR parameter b2 , and the realization of the standard Brownian motion B~t0 !+ Note that the larger the magnitude value of break ~1 2 b2 !, the more easily the change will be detected+ Note that we work with JT2 and JT1 in this section to reflect the fact that plim~10T !RSST ~t! is discontinuous at both sides of t0 , whereas we work with the interval @ st, t0 # and JT1 in Section 4 because plim~10T !RSST ~t! is only discontinuous at the right-hand-side neighborhood of t0 +


101

Figure 3 simulates the behavior of ~10T !RSST ~t! with T 5 4,000+ For t # t0 , it is a random flat line and has a sudden plunge near t0 + For t . t0 , it is linear, random, and increasing, which agrees with our theory+

5.2. The Monotonic Transition Note that for any positive constant c bounded away from 0, there is an asymptotic gap between ~10T !RSST ~t0 ! and ~10T !RSST ~t0 6 c!+ The speed of the transitional process is 10!T , and the transition is completed in JT2 and JT1 , respectively+ To investigate the consistency of the change-point estimator, it is sufficient to show that the asymptotic transition of ~10T !RSST ~t! from t 5 t0 to t 5 t0 6 c is monotonically increasing+

Figure 3. Graph of ~10T !RSST ~t! for b1 5 1, b2 5 0+5, T 5 4000+

102


From Appendix J, for m 5 0,1,2, + + + and ~m0T ! r 0 as T r `, we have

S

*

sup bZ 1 t0 2 m

D

*

m 2 1 5 op ~1! T

and

S

bZ 2 t0 2 1 T

D

m ~1 2 b22 !mB 2 ~t0 !~1 2 b2 ! , 2 b2 n T ~1 2 b22 !mB 2 ~t0 ! 1 1 2 t0 1 B 2 ~t0 !

S

m T

RSST t0 2

D

n s2 1

~1 2 t0 1 B 2 ~t0 !!~1 2 b2 ! 2 mB 2 ~t0 ! 2 s ~1 2 b22 !mB 2 ~t0 ! 1 1 2 t0 1 B 2 ~t0 !

def

5 s 2 1 h 1 ~m!+ If m 5 0, we have ~10T !RSST ~t0 ! m r `,

S

m 1 RSST t0 2 T T

D

(19)

n s2 1

(20) p

&

s 2 + As

~1 2 b2 !~1 2 t0 1 B 2 ~t0 !!s 2 + 1 1 b2

Thus, we bridge the gap between ~10T !RSST ~t0 ! and ~10T !RSST ~t0 2 c!+ For t 5 t0 1 ~m0T !, m 5 0,1,2, + + + , and ~m0T ! r 0 as T r `, we have

*

S D S D

sup bZ 1 t0 1 m

*

sup bZ 2 t0 1 m

* 2 b * 5 o ~1!+

m 2 1 5 op ~1!, T

m T

2

p

From Appendix J, we also have

S

m 1 RSST t0 1 T T

D

n s2 1

~1 2 b2 !B 2 ~t0 !~1 2 b22m ! 2 s 1 1 b2

def

5 s 2 1 h 2 ~m!+

(21) p

If m 5 0, we have ~10T !RSST ~t0 ! & s 2 + As m r `, ~10T !RSST ~t0 1 ~m0T !! n s 2 1 $@~1 2 b2 !B 2 ~t0 !#0~1 1 b2 !%s 2+ Thus, we bridge the gap between ~10T !RSST ~t0 ! and ~10T !RSST ~t0 1 c!+ Note from equations ~20! and ~21! that, for all m . 0, we have h 1~m 1 1! . h 1~m! . h 1~0! 5 0 and h 2 ~m 1 1! . h 2 ~m! . h 2 ~0! 5 0+ Thus the transition is monotonic+ Note also from equation ~19! that, unless m 5 0, bZ 2 ~t0 2 ~m0T !! does not converge to b2 ! Remarks+ However, because of the special behavior of ~10T !RSST ~t! described in equations ~16!–~18! and ~20!–~21!, we will have lim Tr` Pr~ kZ Þ k 0 ! 5 0+ Thus, bZ 2 should be a consistent estimator of b2 in practice+ See Theorem 4, which follows+


103

5.3. The Consistency and the Limiting Distributions of t[ T , bZ 1 , and bZ 2 Despite the fact that bZ 2 is extremely sensitive to the local behavior of k,Z Appendix K shows that for a fixed magnitude of break, lim Tr` Pr~ kZ Þ k 0 ! 5 0+ p This means T ~ t[ T 2 t0 ! & 0+ Hence the distribution of t[ T degenerates very fast for any fixed magnitude of break+ However, if we allow b2 to converge to T b1~5 1!, the process $ yt % t5k will be a near unit-root process with an initial 011 value drawn from an I ~1! process+ Thus, we are dealing with structural changes from an exact unit-root process to a near unit-root process with a nonstationary initial value+ We fix b1 at one and let b2T be a sequence of b2 such that !T ~1 2 b2T ! r 0 and T 304 ~1 2 b2T ! r ` as T r `+ Consider the region t 5 t0 1 y0~1 2 b2T ! 2 T 2, where y [ R+ From Appendix K, we have the following expressions+ For y # 0, RSST ~t! 2 RSST ~t0 ! n 22B~t0 !s 2 B1 ~6y6! 1 6y6s 2 B 2 ~t0 !+ For y . 0, RSST ~t! 2 RSST ~t0 ! n 22B~t0 !s 2 B2 ~y! 1 ys 2 B 2 ~t0 !, where B1~{!, B2 ~{!, and B~{! are defined in Section 2+ THEOREM 4+ Under Assumptions ~A1!–~A3!, if b1 5 1 and 6 b2 6 , 1 then t[ T , bZ 1~ t[ T !, and bZ 2 ~ t[ T ! are consistent, and Pr~ kZ Þ k 0 ! r 0, T ~ bZ 1 ~ t[ T ! 2 1! n

B 2 ~t0 ! 2 t0 2

E

(22)

,

t0

B2

0

!T ~ bZ 2 ~ t[ T ! 2 b2 ! n

O !1 2 b B~1! 2 2

1 2 t0 1 B 2 ~t0 !

+

Suppose we fix b1 at one and let b2T be a sequence of b2 such that !T ~1 2 b2T ! r 0 and T 304 ~1 2 b2T ! r ` as T r `+ Then the limiting distribution of t[ T is given by ~1 2 b2T ! 2 T 2 ~ t[ T 2 t0 !

d

&

Arg max y[R

S

D

B * ~y! 1 2 6y6 , B~t0 ! 2

where B * ~y! is a two-sided Brownian motion on R defined to be B * ~y! 5 B1~2y! O for y # 0 and B * ~y! 5 B2 ~y! for y . 0+ The terms B1~{!, B2 ~{!, B~{!, and B~{! are defined in Section 2+

104


Proof+ See Appendix K+ Thus bZ 1 has a scaled Dickey–Fuller distribution+ The term bZ 2 can be described as being asymptotically normally distributed with a random variance+ This is because T

(

!T ~ bZ 2 2 b2 ! 5

@t0 T #11

«t yt21 0!T 1 op ~1!,

T

(

@t0 T #11

2 yt21 0T

the numerator follows a central limit theorem, and the denominator converges to a random variable by ~5b! and ~5d! in Lemma 3+ The rapid rate of convergence of kZ in Theorem 4 is very surprising+ The rate in this case is much faster than those in the previous two cases+ In previous cases, we only have 6 kZ 2 k 0 6 5 Op ~1!; that is, the estimation of k 0 is always subject to an error of order Op ~1!+ Now, with b1 5 1 and 6 b2 6 , 1, k 0 can be precisely estimated asymptotically, despite the fact that k 0 r ` as T r `+ Further, as b2 approaches one, the limiting distribution of t[ T is the argmax of a random process moving along an inverted-V shape linear function+ 6. MONTE CARLO EXPERIMENTS For empirical applications, we perform the following experiments to see how well our asymptotic results match the small-sample properties of the estimators+ In all experiments, the sample size is set at T 5 200 and the number of replications is set at N 5 20,000; $ yt % Tt51 is generated from model ~1!; $«t % Tt51 ; nid~0,1! and y0 ; nid~0,1! independent of $«t % Tt51+ The true change point is set at t0 5 0+5+ Experiment 1a+ This experiment verifies equation ~19!, which predicts that when b1 5 1 and 6 b2 6 , 1, the postshift structural estimator bZ 2 ~t! does not converge to b2 whenever t , t0 + We let b1 5 1; b2 5 0+5, 0, 20+5+ Let bNZ 2 ~t! be the mean of bZ 2 ~t! in 20,000 replications+ We consider those t in the neighborhood of t0 ~5 0+5!+ Note that when b1 5 1 and 6 b2 6 , 1, the estimate of b2 is inaccurate for t , t0 + The strange behavior of bNZ 2 ~t! is highlighted by italic figures in Table 1+ Note that bNZ 2 ~t! is not close to b2 even for t 5 t0 2 ~10T ! 5 0+495+ Experiment 1b+ With 6 b1 6 , 1, b2 5 1, and t close to t0 , we will show that both bZ 1~t! and bZ 2 ~t! are very close to the true parameters b1 and b2 , respectively+ Let b1 5 0+5, 0, 20+5; b2 5 1+ The results in Table 2 do not image those in Table 1+ Here bNZ 1~t! is very close to b1 + Note that in experiments 1a and 1b, the estimates of the unit root are close to one in all cases+


105

Table 1. Results for experiment 1a bNZ 1~t!

bNZ 2 ~t!

bNZ 1~t!

bNZ 2 ~t!

bNZ 1~t!

bNZ 2 ~t!

bNZ

b1 5 1

b2 5 0+5

b1 5 1

b2 5 0

b1 5 1

b2 5 20+5

0+480 0+485 0+490 0+495 0+500 0+505 0+510 0+515 0+520

0+9819 0+9820 0+9822 0+9824 0+9826 0+9728 0+9702 0+9693 0+9689

0.7041 0.6773 0.6401 0.5869 0+4930 0+4915 0+4903 0+4899 0+4898

0+9817 0+9819 0+9821 0+9822 0+9824 0+9628 0+9622 0+9617 0+9612

0.4703 0.4163 0.3413 0.2249 0+0009 0+0012 0+0013 0+0013 0+0013

0+9819 0+9821 0+9823 0+9825 0+9826 0+9529 0+9450 0+9423 0+9409

0.1350 0.0541 20.0550 20.2160 20+4939 20+4923 20+4913 20+4909 20+4908

t

\

Experiment 2+ This experiment simulates the distributions of bZ 1 and bZ 2 + Case 1: b1 5 0+5, b2 5 20+5; Case 2: b1 5 0+5, b2 5 1; Case 3: b1 5 1, b2 5 0+5+

Figures 4 to 6 display the results for Cases 1 to 3, respectively+ For Case 1, Theorem 1 states that both bZ 1 and bZ 2 are asymptotically normally distributed+ The theorem is well supported by Figure 4 even for a sample size of 200+ For Case 2, Figure 5 shows that the small-sample distribution of bZ 1 is approximately normal, whereas bZ 2 appears to have a Dickey–Fuller distribution+ Both results agree with the theorem+

Table 2. Results for experiment 1b

t

\

bNZ

0+480 0+485 0+490 0+495 0+500 0+505 0+510 0+515 0+520

bNZ 1~t!

bNZ 2 ~t!

bNZ 1~t!

bNZ 2 ~t!

bNZ 1~t!

bNZ 2 ~t!

b1 5 0+5

b2 5 1

b1 5 0

b2 5 1

b1 5 20+5

b2 5 1

0+4899 0+4900 0+4900 0+4901 0+4902 0+4956 0+5044 0+5158 0+5290

0+9817 0+9820 0+9823 0+9827 0+9830 0+9830 0+9830 0+9830 0+9829

20+0001 20+0000 20+0000 20+0000 20+0000 0+0109 0+0301 0+0559 0+0862

0+9810 0+9815 0+9820 0+9825 0+9830 0+9830 0+9830 0+9830 0+9830

20+4902 20+4903 20+4904 20+4905 20+4905 20+4756 20+4512 20+4193 20+3819

0+9788 0+9799 0+9809 0+9819 0+9830 0+9830 0+9829 0+9829 0+9829

106


Figure 4. ~a! Distribution of t0 T0~1 2 b12 ! ~ bZ 1 2 b1 !; —— ~T 5 200!, {{{{{{ ~T 5 `!+ ~b! Distribution of ~1 2 t0 !T0~1 2 b22 ! ~ bZ 2 2 b2 !; —— ~T 5 200!, {{{{{{ ~T 5 `!+

!

!


107

Figure 5. ~a! Distribution of t0 T0~1 2 b12 ! ~ bZ 1 2 b1 !; —— ~T 5 200!, {{{{{{ ~T 5 `!+ ~b! Distribution of ~1 2 t0 !T0~ bZ 2 2 b2 !; —— ~T 5 200!, {{{{{{ ~T 5 `!+

!

108


Figure 6. ~a! Distribution of t0 T~ bZ 1 2 b1 !; —— ~T 5 200!, {{{{{{ ~T 5 `!+ ~b! Distribution of T0~1 2 b22 ! ~ bZ 2 2 b2 !; —— ~T 5 200!, {{{{{{ ~T 5 `!+

!


109

For Case 3, Theorem 4 predicts that bZ 1 should have a Dickey–Fuller distribution and bZ 2 should have a normal distribution+ Figure 6 agrees with Theorem 4+ Note that in a small sample, the variation of t[ T is nontrivial+ This variation is the main source of the small-sample bias of bZ 1 and bZ 2 + In Case 1 where both the pre- and postshift processes are stationary, the small-sample bias is very little+ However, in Cases 2 and 3, bZ 1 and bZ 2 consist of both stationary and nonstationary observations; thus their small-sample distributions should look like a mixture of normal and Dickey–Fuller distributions+ From Figures 5a and 6b, it is clear that the small-sample distribution has a larger variation and is more skew than a normal distribution+ From Figures 5b and 6a, the small-sample distribution has a smaller variation than and is not as skew as a Dickey–Fuller distribution+ Experiment 3+ In this experiment, we simulate the distribution of t[ T for shrinking break+ Case 1: b1 5 0+5, b2T 5 0+5 2 T 2108 ; Case 2: b1T 5 1 2 T 2104, b2 5 1; Case 3: b1 5 1, b2T 5 1 2 T 2508+

Thus, in each case, we simulate the behavior of T 304 ~ t[ T 2 t0 !+ Figures 7~a!–7~c! display the results of Cases 1 to 3, respectively+ For Cases 1 and 3, Figures 7a and 7c show that t[ T is symmetrically distributed, as predicted by Theorems 1 and 4+ For Case 2, Theorem 3 predicts that t[ T should be asymmetrically distributed+ Figure 7b agrees with our predictions+ Theoretically, as T r `, the domain for T 304 ~ t[ T 2 t0 ! should be unbounded+ However, because we fix T at 200 in this experiment, and t[ T is searched within the ~0,1! interval, hence the simulated distribution of T 304 ~ t[ T 2 t0 ! is bounded in the interval ~2200 304 3 0+5, 200 304 3 0+5! 5 ~226+6,26+6!+ Note that the distribution of t[ T has a “boundary effect,” especially for Case 3, in that there is a lot of mass near the boundary+ The clustering of mass at the boundary is due to the large value of Var~ bZ 2 ~t! 2 bZ 1~t!! when t nears zero or one+ Although 6 E ~ bZ 2 ~t! 2 bZ 1~t!!6 # 6 E ~ bZ 2 ~t0 ! 2 bZ 1~t0 !!6 for all t, actual realization of 6 bZ 2 ~t! 2 bZ 1~t!6 may be larger than 6 bZ 2 ~t0 ! 2 bZ 1~t0 !6 in finite sample, especially for the case where t is near the boundary+ For example, when t is near 0, bZ 1~t! will be calculated based only on very few observations, and therefore it has a large variance+ Note that t[ T is generated by the minimization of RSST ~t!, or in other words, by the maximization of 6 bZ 2 ~t! 2 bZ 1~t!6+ Thus, the probability that t[ T falls into the boundary is nontrivial+ This phenomenon will disappear as the sample size goes to infinity+ 7. CONCLUSION AR models have been used extensively in economics+ Many economic variables such as interest rates, real consumption, and real gross domestic product

110


Figure 7. Distribution of T 304 ~ t[ T 2 0+5!: ~a! b1 5 0+5, b2 5 0+5 2 T 2108 ; ~b! b1 5 1 2 T 2104, b2 5 1; ~c! b1 5 1, b2 5 1 2 T 2508+

~GDP! can be well predicted by their own lags+ Some variables exhibit stationarity, whereas some variables display nonstationarity+ A shock to the production technology or a sudden change in the government policy may cause structural changes in these autoregressive models+ In particular, the changes may cause a stationary process to shift to a nonstationary one or vice versa+ In this paper, we present a structural change AR~1! model with independent and identically distributed ~i+i+d+! innovations+ We discuss three cases of struc-


111

tural changes+ Case 1 deals with the change from a stationary process to another stationary process+ Case 2 examines the situation where a stationary process shifts to an I ~1! process+ Case 3 discusses the situation where an I ~1! process shifts to a stationary process with a nonstationary initial value+ In each case, consistency of the least squares estimators is established, and their limiting distributions are derived+ Having the limiting distributions of these estimates allows us to carry out statistical inference on the parameters of interest+ Cases 2 and 3 are intriguing+ Results of these two cases do not mirrorimage each other, which is counterintuitive+ A possible explanation for this asymmetry is that for Case 2, the initial value of the postshift process is stationary, whereas Case 3 is nonstationary+ Another explanation is that we are excluding an intercept in our model+ As pointed out by Banerjee, Lumsdaine, and Stock ~1992, p+ 278!, a more proper model for Case 3 should be, in our notations, yt 5 yt21 1$t # k 0 % 1 ~ yk0 ~1 2 b2 ! 1 b2 yt21 !1$t . k 0 % 1 «t + This process avoids a spurious sharp jump to zero at the break point+ It should also be mentioned that the disturbance terms $«t % Tt51 in this paper are assumed to be i+i+d+ In the case where both 6 b1 6 and 6 b2 6 are less than one, the assumption of i+i+d+ avoids the inconsistency of b’s Z due to serial correlation of «t + This assumption is also very helpful in calculating the long run variances of processes such as («t yt21 + However, extension to the cases of heterogeneous and0or dependent «t should be possible in the nonstationary cases at the expense of a more complicated mathematical treatment+ Thus, a generalization of our model to ARMA~ p, q! models with an intercept deserves future investigation+ REFERENCES Ahtola J+ & G+C+ Tiao ~1984! Parameter inference for a nearly nonstationary first order autoregressive model+ Biometrika 71, 263–272+ Andrews, D+W+K+ ~1987! Consistency in nonlinear econometric models: A generic uniform law of large numbers+ Econometrica 55, 1465–1471+ Andrews, D+W+K+ ~1988! Laws of large numbers for dependent non-identically distributed random variables+ Econometric Theory 4, 458– 467+ Bai, J+ ~1994a! Least squares estimation of a shift in linear processes+ Journal of Time Series Analysis 15, 453– 472+ Bai, J+ ~1994b! Estimation of Structural Change Based on Wald Type Statistics+ Working paper 94-6, M+I+T+, Department of Economics+ Banerjee, A+, R+L+ Lumsdaine, & J+H+ Stock ~1992! Recursive and sequential tests of the unit root and trend break hypothesis: Theory and international evidence+ Journal of Business and Economic Statistics 10, 271–287+ Chan, N+H+ & C+Z+ Wei ~1987! Asymptotic inference for nearly nonstationary AR~1! processes+ Annals of Statistics 15, 1050–1063+ Chan, N+H+ & C+Z+ Wei ~1988! Limiting distributions of least squares estimate of unstable autoregressive processes+ Annals of Statistics 16, 367– 401+ Chong, T+T+L+ ~1995! Partial parameter consistency in a misspecified structural change model+ Economics Letters 49, 351–357+ Dickey, D+A+ & W+A+ Fuller ~1979! Distribution of the estimators for autoregressive time series with a unit root+ Journal of the American Statistical Association 74, 427– 431+

112


Hansen, B+E+ ~1992! Tests for parameter instability in regressions with I~1! processes+ Journal of Business and Economic Statistics 10, 321–335+ Hinkley, D+V+ ~1970! Inference about the change-point in a sequence of random variables+ Biometrika 57, 1–17+ Kim, J+ & D+ Pollard ~1990! Cube root asymptotics+ Annals of Statistics 18, 191–219+ Lai, T+L+ & D+ Siegmund ~1983! Fixed accuracy estimation of an autoregressive parameter+ Annals of Statistics 11, 478– 485+ Mankiw, N+G+ & J+A+ Miron ~1986! The changing behavior of the term structure of interest rates+ Quarterly Journal of Economics 101, 221–228+ Mankiw, N+G+, J+A+ Miron & D+N+ Weil ~1987! The adjustment of expectations to a change in regime: A study of the founding of the Federal Reserve+ American Economic Review 77, 358–374+ Perron, P+ ~1996! The adequacy of asymptotic approximations in the near-integrated autoregressive model with dependent errors+ Journal of Econometrics 70, 317–350+ Phillips, P+C+B+ ~1987! Time series regression with a unit root+ Econometrica 55, 277–301+ Phillips, P+C+B+ ~1988! Regression theory for near-integrated time series+ Econometrica 56, 1021– 1043+ Picard, D+ ~1985! Testing and estimating change-points in time series+ Advances in Applied Probability 17, 841–867+ Pötscher, B+M+ & I+R+ Prucha ~1989! A uniform law of large numbers for dependent and heterogeneous data process+ Econometrica 57, 675– 683+ Salazar, D+ ~1982! Structural changes in time series models+ Journal of Econometrics 19, 147–163+ White, H+ ~1984! Asymptotic Theory for Econometricians+ New York: Academic Press+ White, J+S+ ~1958! The limiting distribution of the serial correlation coefficient in the explosive case+ Annals of Mathematical Statistics 29, 1188–1197+ Yao, Y+C+ ~1987! Approximating the distribution of the maximum likelihood estimate of the changepoint in a sequence of independent random variables+ Annals of Statistics 15, 1321–1328+

APPENDIX A: PROOF OF LEMMA 1 If 6 b1 6 , 1, 6 b2 6 , 1, «t are i+i+d+, and y0 is drawn from a stationary process, then ~1! ~3a!, ~3d!, and ~3g! are consequences of the uniform law of large numbers in Andrews ~1987, Theorem 1!; ~2! ~3b! and ~3c! are consequences of the weak law of large numbers in Andrews ~1988, Theorem 2!+

* * @tT #

~3e!

sup 6 bZ 1 ~t! 2 b1 6 5 sup

st#t#t0

st#t#t0

(1 yt21 «t @tT #

2 (1 yt21

*( * @tT #

#

T @ stT #

(1

2 yt21

yt21 «t

sup st#t#t0

1

T

5 Op ~1!op ~1! 5 op ~1!+


113

To show ~3f !, utilizing the fact that T

T

(

bZ 2 ~t! 2 b1 5

@t0 T #11

~ b2 2 b1 ! 1

T

(

@tT #11

(

2 yt21

@tT #11

«t yt21 ,

T

(

2 yt21

@tT #11

2 yt21

adding and subtracting ~1 2 t0 !s 2

T

1 2 b22

~ b2 2 b1 !

,

T

(

@tT #11

2 yt21

by the triangle inequality, and last by ~3a!, ~3c!, and ~3d!, we have

sup st#t#t0

*

bZ 2 ~t! 2

5 sup

st#t#t0

*

~t0 2 t!~1 2 b22 !b1 1 ~1 2 t0 !~1 2 b12 !b2 ~t0 2 t!~1 2 b22 ! 1 ~1 2 t0 !~1 2 b12 ! T

# sup

st#t#t0

*

T

(

@t0 T #11

( @tT #11

(

2 yt21

~ b2 2 b1 ! 1

T

2

*

@tT #11

«t yt21

T

2 ( yt21 @tT #11

2 yt21

~1 2 t0 !~1 2 b12 !~ b2 2 b1 ! ~t0 2 t!~1 2 b22 ! 1 ~1 2 t0 !~1 2 b12 !

T

( «t yt21 @tT #11 T

2 ( yt21 @tT #11

1 6 b2 2 b1 6 sup

st#t#t0

*

*

T

1 6 b2 2 b1 6 sup

st#t#t0

*

*

T

2 ( yt21 @t T #11 0

T

2 ( yt21 @tT #11

T 2

~1 2 t0 !s 2 1 2 b22 T

2 ( yt21 @tT #11

*

~1 2 t0 !s 2 1 2 b22 T

(

@tT #11

2 yt21

2

~1 2 t0 !~1 2 b12 ! ~t0 2 t!~1 2 b22 ! 1 ~1 2 t0 !~1 2 b12 !

*

114


T

#

sup

T

(

2 yt21

@t0 T #11

st#t#t0

*

3 sup

st#t#t0

(

2 yt21

*

0

T

*

~1 2 b12 !~1 2 b22 !

sup 2 yt21

(

@t0 T #11

st#t#t0

*

T

( «t yt21 @tT #11 T

*

6 b2 2 b1 6T

1

T

(

@t0 T #11

1*

2 yt21

T

2 ( yt21 @t T #11 0

T

1 sup

st#t#t0

*

s 2 ~1 2 t0 !

2

1 2 b22

2 yt21

*

(

2 yt21

2

T

2

@tT #

(

@t0 T #11

1 2 b12

~ b2 2 b1 ! 1

@tT #

(1

adding and subtracting ~t 2 t0 !s 2 1 2 b22 ,

(1

2 yt21

;

@tT #

(1

2 yt21

@tT #

(1 «t yt21 2 yt21

2 yt21

T

*

*

T

0

T

2

*

s 2 ~t0 2 t!

@tT # 2 yt21

@tT #11

2 ( yt21 @t T #11

@t0 T # @tT #11

(

*2

The proof for ~3h! is analogous to ~3f !, using the fact that

~ b2 2 b1 !

1 2 b22

2 ( yt21 @tT #11

@~t0 2 t!~1 2 b22 ! 1 ~1 2 t0 !~1 2 b12 !#s 2

T

T

~1 2 t0 !s 2

T

6 b2 2 b1 6T

bZ 1 ~t! 2 b1 5

2

T

T

1

T

@~t0 2 t!~1 2 b22 ! 1 ~1 2 t0 !~1 2 b12 !#

T

(

6 b2 2 b1 6T

1

T

2 ( yt21 @t T #11

6 b2 2 b1 6~1 2 t0 !~1 2 b12 !T

st#t#t0

@t0 T #11

T

*

@t0 T #11

1 sup

#

T

( «t yt21 @tT #11

5 op ~1!+

~1 2 t0 !s 2 1 2 b22

*


115

by the triangle inequality, and last by ~3a!, ~3b!, and ~3g! we have

sup t0,t#tS

*

t0 ~1 2 b22 !b1 1 ~t 2 t0 !~1 2 b12 !b2 t0 ~1 2 b22 ! 1 ~t 2 t0 !~1 2 b12 !

bZ 1 ~t! 2

* * @tT #

(1 «t yt21

# sup

t0,t#tS

@tT #

(1

t0,t#tS

#

sup

@t0 T #

(1

2 yt21

t0,t#tS

t0,t#tS

3 sup

t0,t#tS

T

1

2 yt21

@tT #

(1 «t yt21 T

*

1

@tT #

(1

2

2 (1 yt21

~t 2 t0 !~1 2 b12 ! t0 ~1 2 b22 ! 1 ~t 2 t0 !~1 2 b12 !

6 b2 2 b1 6T @t0 T #

(1

sup t0,t#tS

2 yt21

*

@tT #

2 yt21

*

*

2

~t 2 t0 !s 2 1 2 b22

*

2

~t 2 t0 !s 2 1 2 b22

*

@tT #

(

@t0 T #11

2 yt21

T

*

2 (1 yt21

@tT #

@t0 ~1 2

t0,t#tS

@t0 T #

5 op ~1!+

2 (1 yt21

0

~t 2 t0 !s 2 1 2 b22

T

@tT #

6 b2 2 b1 6T

(1

@tT #

2

2 ( yt21 @t T #11

@t0 ~1 2 b22 ! 1 ~t 2 t0 !~1 2 b12 !#

sup

@t0 T #

(1

*

*

~t 2 t0 !s 2 1 2 b22

T

*

@tT #

6 b2 2 b1 6~t 2 t0 !~1 2 b12 !T

1 sup

#

t0,t#tS

2 yt21

1 6 b2 2 b1 6 sup

T

1 6 b2 2 b1 6 sup

*

2 yt21

*

1*

1 ~t 2 t0 !~1 2 ~1 2 b12 !~1 2 b22 !

b22 !

@tT #

(1

«t yt21 T

*

1

6 b2 2 b1 6T @t0 T #

(1

@t0 T #

t0 s 2 2 1 2 b12

(1

b12 !#s 2

2 yt21

T

*

2 yt21

2

2 (1 yt21

T

sup t0,t#tS

1 sup

t0,t#tS

*

*

*

@tT #

(

@t0 T #11

2 yt21

T

@tT #

~t 2 t0 !s 2 2 1 2 b22

(

@t0 T #11

T

2 yt21

*2

116

~3i!


sup 6 bZ 2 ~t! 2 b2 6 5 sup

t0,t#tS

t0,t#tS

*

T

( «t yt21 @tT #11 T

2 ( yt21 @tT #11

*

*( T

T

#

sup

T

(

@ tT S #11

@tT #11

2 yt21

t0,t#tS

«t yt21

T

*

5 Op ~1!op ~1! 5 op ~1!+

n

These prove Lemma 1+

APPENDIX B: DERIVATION OF EQUATIONS ~6! AND ~7! For st # t # t0 , we have @t0 T #

@tT #

RSST ~t! 5

(1 ~«t 2 ~ bZ 1~t! 2 b1 !yt21 ! 2 1 @tT(#11 ~«t 2 ~ bZ 2 ~t! 2 b1 !yt21 ! 2 T

1

( ~«t 2 ~ bZ 2 ~t! 2 b2 !yt21 ! 2 @t T #11 0

S( D @tT #

2

yt21 «t

T

5 ( «t2 2 1

1

@tT #

(1

@t0 T #

2 2~ bZ 2 ~t! 2 b1 !

2 yt21

(

@tT #11

yt21 «t

@t0 T #

1 ~ bZ 2 ~t! 2 b1 ! 2

T

2 2 2~ bZ 2 ~t! 2 b2 ! ( yt21 «t ( yt21 @tT #11 @t T #11 0

T

1 ~ bZ 2 ~t! 2 b2 ! 2

(

@t0 T #11

2 yt21 +

(B.1)

By ~B+1! and the fact that @t0 T #

bZ 2 ~t! 5 b1

2 ( yt21 @tT #11 T

(

@tT #11

2 yt21

T

1 b2

2 ( yt21 @t T #11 0

T

(

@tT #11

2 yt21

T

1

( «t yt21 @tT #11 ,

T

(

@tT #11

2 yt21


117

we have RSST ~t! 2 RSST ~t0 !

S ( D S( D @t0 T #

2

@tT #

yt21 «t

5

1

1 1 1 31

22

1

@tT #

2 (1 yt21

2 yt21 T

~ b2 2 b1 !

2 ( yt21 @t T #11 0

T

(

@tT #11

~ b2 2 b1 !

(

@tT #11

(

T

(

2 yt21

@tT #11

@t0 T #

1

T

T

1

T

T

0

T

@t0 T #

(

@tT #11

T

2

0

T

21 2

2

2 yt21

(

(

@tT #11

(1

T

2 ( yt21 @t T #11 0

@t0 T #

24 2

T

2 ( yt21 @t T #11 0

2 ( yt21 ( yt21 «t @t T #11 @tT #11 0

T

2 yt21

1 L T ~t!,

(B.2)

S ( D S( D S ( 2

2

@tT # 1

1

@tT #

(1

2 yt21

@t0 T #11

«t yt21

@t0 T #11

2 yt21

@tT #11

«t yt21

T

(

D

2

T

2

T

(

D S( 2

T

yt21 «t

2

2 yt21

0

2

2 yt21

yt21 «t

@t0 T #

T

2 ( yt21 @tT #11

T

@tT #11

1

0

( «t yt21 @t T #11

T

2

2

T

( yt21 «t @t T #11

@t0 T #

@t0 T #11

@t0 T #

2 yt21

( «t yt21 @t T #11

2 ( yt21 @tT #11

(

1 ~ b2 2 b1 ! 2

2 ( yt21 @tT #11 T

yt21 «t

0

( «t yt21 @tT #11

2 ( yt21 ( yt21 «t @tT #11 @t T #11

(

@tT #11

2 ( yt21 @t T #11

T

2 ( yt21 @tT #11

@t0 T #

2

2 ( yt21 @tT #11

2 ( yt21 @tT #11

1

2 yt21

( «t yt21 @tT #11

@t0 T #

@t0 T #

2 2

T

2 ( yt21 @tT #11 2 ( yt21 @tT #11

~ b1 2 b2 !

2 yt21

( «t yt21 @tT #11

1

T

@tT #11

T

T

0

T

L T ~t! 5

1

T

5 2~ b2 2 b1 !

where

T

( «t yt21 @tT #11

2 yt21

2 ( yt21 @t T #11

2 2 ~ b1 2 b2 !

1

1

2

@t0 T #

(1

2

yt21 «t

@tT #11

2 yt21

+

118


Let

l1 5

l2 5

S S

~1 2 t0 !~1 2 b12 !~ b2 2 b1 ! ~t0 2 t!~1 2

b22 !

1 ~1 2 t0 !~1 2

D D

2

b12 !

~t0 2 t!~1 2 b22 !~ b2 2 b1 !

and

2

~t0 2 t!~1 2 b22 ! 1 ~1 2 t0 !~1 2 b12 !

+

By ~B+1!, the fact that ~ b2 2 b1 ! 2 s 2 ~t0 2 t!~1 2 t0 ! ~t0 2 t!~1 2

b22 !

1 ~1 2 t0 !~1 2

b12 !

5 l1

~t0 2 t!s 2 12

b12

1 l2

~1 2 t0 !s 2 1 2 b22

,

and the triangle inequality, we have

sup st#t#t0

*T

1

RSST ~t! 2 s 2 2

~ b2 2 b1 ! 2 s 2 ~t0 2 t!~1 2 t0 ! ~t0 2 t!~1 2 b22 ! 1 ~1 2 t0 !~1 2 b12 !

S( D @tT #

#

*T ( « 2s * 1 1

2

1

1 sup

*

1 sup

*

1 sup

*

st#t#t0

st#t#t0

st#t#t0

1 sup

st#t#t0

1

sup

@tT #

st#t#t0

2~ bZ 2 ~t! 2 b1 !

~ bZ 2 ~t! 2 b1 ! 2

2~ bZ 2 ~t! 2 b2 !

*

2

yt21 «t

T

2 t

*

~ bZ 2 ~t! 2 b2 ! 2

2 (1 yt21

T

1 T 1 T 1 T 1 T

@t0 T #

(

@tT #11

yt21 «t

*

@t0 T #

2 2 l1 ( yt21 @tT #11

T

( yt21 «t @t T #11 0

T

~t0 2 t!s 2

*

*

2 2 l2 ( yt21 @t T #11 0

1 2 b12

~1 2 t0 !s 2 1 2 b22

*

+

We add and subtract ~ bZ 2 ~t! 2 b1 ! 2 @~t0 2 t!s 2 #0~1 2 b12 ! in the fourth term and ~ bZ 2 ~t! 2 b2 ! 2 @~1 2 t0 !s 20~1 2 b22 !# in the sixth term+ By the triangle inequality, the fact that sup6a 2 2 b 2 6 # sup6a 2 b6sup6a 1 b6, and Lemma 1, the preceding expression is bounded by

STRUCTURAL CHANGE IN AR(1) MODELS #

*T

sup 6 bZ 1 ~t! 2 b1 6 sup * ( yt21 «t * (1 «t2 2 s 2 * 1 st#t#t T 1 st#t#t T

1

1

0

*

1 sup ~ bZ 2 ~t! 2 b1 ! 2 sup

*

st#t#t0

st#t#t0

st#t#t0

st#t#t0

~t0 2 t!s 2

1 sup

1 2 b12

st#t#t0

T 1 T

@t0 T #

(

@tT #11

yt21 «t

@t0 T #

*

2 2 ( yt21 @tT #11

~t0 2 t!s 2 1 2 b12

*

sup 6~ bZ 2 ~t! 2 b1 ! 2 2 l 1 6

st#t#t0

1 sup ~ bZ 2 ~t! 2 b2 ! 2 st#t#t0

1 2 b22

1

st#t#t0

1 2 sup 6 bZ 2 ~t! 2 b2 6

~1 2 t0 !s 2

@tT #

0

1 2 sup 6 bZ 2 ~t! 2 b1 6 sup

1

119

*

*

T

( yt21 «t @t T #11 0

T

*

T

2 ( yt21 @t T #11 0

T

2

~1 2 t0 !s 2 1 2 b22

*

sup 6~ bZ 2 ~t! 2 b2 ! 2 2 l 2 6

st#t#t0

5 op ~1!+

n

This derives equation ~6!+ For t0 , t # t,S we can write the residual sum of squares as follows: @t0 T #

RSST ~t! 5

(1

@tT #

~«t 2 ~ bZ 1 ~t! 2 b1 !yt21 ! 2 1

(

@t0 T #11

~«t 2 ~ bZ 1 ~t! 2 b2 !yt21 ! 2

T

1

( ~«t 2 ~ bZ 2 ~t! 2 b2 !yt21 ! 2 @tT #11

T

5 ( «t2 2 2~ bZ 1 ~t! 2 b1 ! 1

2 2~ bZ 1 ~t! 2 b2 !

S(

@tT #11

«t yt21

T

(

(1

@t0 T #

yt21 «t 1 ~ bZ 1 ~t! 2 b1 ! 2

(1

@tT #

@tT #

0

0

2 yt21

2 yt21 ( yt21 «t 1 ~ bZ 1~t! 2 b2 ! 2 @t ( @t T #11 T #11

D

2

T

2

@t0 T #

@tT #11

2 yt21

+

(B.3)

120


By ~B+3! and the fact that @t0 T #

@tT #

(1

bZ 1 ~t! 5

2 yt21

b1 1

@tT #

(1

@tT #

2 ( yt21 @t T #11 @tT #

(1

2 yt21

(1 «t yt21

b2 1

0

,

@tT #

(1

2 yt21

2 yt21

we have

RSST ~t! 2 RSST ~t0 ! 5 2~ b2 2 b1 !

1

@t0 T #

@tT #

( yt21 «t (1 @t T #11 @tT #

(1

(

~t 2 t0 !~1 2 b12 !~ b2 2 b1 !

D

and

2

2 yt21

1 L T ~t!+

(B.4)

2 yt21

2

t0 ~1 2 b22 ! 1 ~t 2 t0 !~1 2 b12 !

2 yt21

2 (1 yt21

(1

2 yt21

(1

S

(

@t0 T #11

@tT #

2 yt21

@tT #

Let

@tT #

yt21 «t

@t0 T #

@t0 T #11

1 ~ b2 2 b1 ! 2

(1

2

0

@tT #

l3 5

@t0 T # 2 yt21

l4 5

S

t0 ~1 2 b22 !~ b2 2 b1 !

D

2

t0 ~1 2 b22 ! 1 ~t 2 t0 !~1 2 b12 !

+

By ~B+3!, the fact that t0 ~ b2 2 b1 ! 2 s 2 ~t 2 t0 ! t0 ~1 2

b22 !

1 ~t 2 t0 !~1 2

5 l3

b12 !

t0 s 2 12

b12

1 l4

~t 2 t0 !s 2 ,

1 2 b22

and the triangle inequality, we have

t0,t#tS

#

* T RSS ~t! 2 s 2 t ~1 2 b ! 1 ~t 2 t !~1 2 b ! * 1

sup

*

t0 ~ b2 2 b1 ! 2 s 2 ~t 2 t0 !

2

T

2 2

0

1 T

T

(1 «t2 2 s 2

1 sup

t0,t#tS

1 sup

t0,t#tS

1 sup

t0,t#tS

* *

*

*

*

1 sup

t0,t#tS

~ bZ 1 ~t! 2 b1 ! 2 ~ bZ 1 ~t! 2 b2 !

1 T

~ bZ 1 ~t! 2 b1 !

2 T

@t0 T #

(1

2 yt21 2 l3

2

@tT #

T

@t0 T #11

(

2 1

0

yt21 «t

@t0 T #

(1

yt21 «t

t0 s 2 1 2 b12

*

*

@tT #

(

~ bZ 1 ~t! 2 b2 ! 2

@t0 T #11

T

*

2 yt21

2 l4

~t 2 t0 !s 2 1 2 b22

*

S(

1 sup

t0,t#tS

D

2

T

@tT #11

«t yt21

T

T

(

@tT #11

2 yt21

+


121

We add and subtract ~ bZ 1~t! 2 b1 ! 2 @t0 s 2 0~1 2 b12 !# in the third term and ~ bZ 1~t! 2 b2 ! 2 @~t 2 t0 !s 2 #0~1 2 b22 ! in the sixth term+ Using sup6a 2 2 b 2 6 # sup6a 2 b6sup6a 1 b6, sup~a 2 b! 2 # ~sup6a 2 b6! 2 , and Lemma 1, the preceding expression is bounded by #

*

( «t2 2 s 2 1

T

1 1

* *

T

1

1

*

1

@t0 T #

T

(1

2 yt21 2

t0 s 2 1 2 b12

2 T

@t0 T #

(1

t0 s 2 1 2 b12

*

yt21 «t

sup ~ bZ 1 ~t! 2 b1 ! 2

sup 6~ bZ 1 ~t! 2 b1 ! 2 2 l 3 6

t0,t#tS

*

1 sup ~ bZ 1 ~t! 2 b2 ! 2 sup

*

t0,t#tS

t0,t#tS

t0,t#tS

t0,t#tS

sup 6 bZ 1 ~t! 2 b1 6

t0,t#tS

t0,t#tS

1 2 sup 6 bZ 1 ~t! 2 b2 6 sup

1 sup

*

t0,t#tS

~t 2 t0 !s 2 1 2 b22

1

@tT #

T

@t0 T #11

1

@tT #

T

(

yt21 «t

*

2 2 ( yt21 @t T #11

~t 2 t0 !s 2

0

1 2 b22

*

sup 6~ bZ 1 ~t! 2 b2 ! 2 2 l 4 6

t0,t#tS

1 sup 6 bZ 2 ~t! 2 b2 6 sup t0,t#tS

t0,t#tS

*T

1

( yt21 «t * @tT #11 T

5 op ~1!+ This shows equation ~7!+

n

APPENDIX C: PROOF OF THEOREM 1 If 6 t[ T 2 t0 6 5 Op ~10T !, or equivalently 6 kZ 2 k 0 6 5 Op ~1!, then for any h . 0, there exists an M , ` such that Pr~6 kZ 2 k 0 6 . M ! , h+ We shall prove this by using the contradiction argument+ Suppose t[ T is not T-consistent; then there exists a sequence MT . 0 such that MT r `, ~MT 0T ! r 0 as T r `, and lim Pr~6 kZ 2 k 0 6 . MT ! 5 a,

Tr`

where a is a positive constant in ~0,1# + Now, let l 5 b2 2 b1 , Z T 5 $1,2, + + + + , T %, D1T 5 $m : m [ Z T , m , k 0 2 MT %, D2T 5 $m : m [ Z T , m . k 0 1 MT %, D3T 5 $m : m [ Z T , k 0 2 MT # m # k 0 1 MT %+

122


Note that

S

Pr~6 kZ 2 k 0 6 . MT ! 5 Pr infm[D1T øD2T RSST

SD m

T

, infm[D3T RSST

S DD m

T

+

Because k 0 [ D3T , we have infm[D3T RSST ~m0T ! # RSST ~t0 !, and the preceding probability is bounded by

S

SD F SD m

# Pr infm[D1T øD2T RSST 2

#

( Pr i51

S

T

m

infm[DiT RSST

T

D G D

, RSST ~t0 !

2 RSST ~t0 ! , 0 +

By using ~B+2! and ~B+4! in Appendix B, the right-hand side equals

1 51

5 Pr infm[D1T

2l

k0

T

( yt21 «t k 11 0

T

( k 11

2

( yt21 «t m11 k0

2 ( yt21 m11

2 yt21

0

2

T

2 ( yt21 m11

1 l2 1

(

2 yt21

k 011

1 51

1 Pr infm[D2T

2l

LT

k0

T

(

2 yt21

m

T

m11

k0

m

( yt21 «t (1 yt21 «t k 11 2

0

m

( k 11

k0

2 (1 yt21

2 yt21

0

(

k 011

LT

k0

m

2 yt21

(1

2 yt21

,0

2

m

2 ( yt21

1 l2 1

62

SD

62

SD m

T

,0 +

Using 2inf x 5 sup~2x!, sup~x 1 y! # sup x 1 sup y, sup xy # sup x sup y, and sup x # sup6 x6, the preceding item is bounded by # Pr~22lA 1 1 2lA 2 1 A 3 . l2 ! 1 Pr~2lA 4 2 2lA 5 1 A 6 . l2 !,

STRUCTURAL CHANGE IN AR(1) MODELS where T

A1 5

( yt21 «t k 11 0

T

(

5 op ~1!

2 yt21

k 011

by ~3a! and ~3c! of Lemma 1; k0

( yt21 «t m11

A 2 5 sup

# sup

k0

m[D1T

(

m[D1T

2 yt21

m11

*

k0

1

yt21 «t

( k 0 2 m m11

k0

(

2 yt21 0~k 0 2 m!

m11

*

5 op ~1!

by the uniform law of large numbers in Andrews ~1987, Theorem 1!;

A 3 5 sup m[D1T

3

*

T

2 ( yt21 m11

LT

k0

T

2 2 ( yt21 ( yt21 k 11 m11 0

1 * sup

k0

2

(1

*

k 011

2 ( yt21 k 11 0

1

MT

m

( yt21 «t k 11 0

m

(

2 yt21

5 op ~1!

k0

(

k 02MT

2 yt21

*

2

«t yt21

m11

T

2 ( yt21 m11

~op ~1! 1 Op ~1!! 5 Op

5 op ~1!

k 011

2 yt21

T

2

T

by ~3a! and ~3b! of Lemma 1;

m[D2T

(

k 011

1

1

S( D S( D 2

«t yt21

2 yt21

A 5 5 sup

m

2 (1 yt21

T

1 Op

k0

1

2 (1 yt21

k0

A4 5

1 T

2

m

2

k0

S S D S DD (1 yt21 «t

*1 #

yt21 «t

1

m[D1T

1

T

yt21 «t

1 sup

T

m

S( D S( D

m[D1T

5 Op

SD

*2

S D 1

MT

5 op ~1!;

2

123

124


by a similar argument as in A 2 ;

A 6 5 sup m[D2T

3

*

m

2 (1 yt21 m

(

k 011

1 * sup

1

m[D2T

2

k0

2 (1 yt21 2

yt21 «t

1

2 (1 yt21

*

S

1 Op

1

2

k0

m

2 (1 yt21

T

( «t yt21 k 11 0

D S 2

2

T

T

(

D

2

T

~Op ~1! 1 op ~1!! 5 Op

1 k 01MT k 011

2 ( yt21 m11

0

1

*

( «t yt21 m11

2 ( yt21 k 11

MT

1

k0

m

yt21 «t

S S D S DD 1

*1

1

#

S( D S( D

1 sup

T

m

T

2 2 yt21 ( yt21

m[D2T

5 Op

SD

LT

k0

2 yt21

2

*2

S D 1

MT

5 op ~1!+

Because A 1 –A 6 are op ~1! and l2 is a positive constant, we have Pr~6 kZ 2 k 0 6 . MT ! # Pr~op ~1! . l2 ! 1 Pr~op ~1! . l2 ! r 2 Pr~l2 , 0! 5 0+ This means a 5 0, which contradicts the original argument that a . 0+ Thus t[ T is T-consistent+ To find the limiting distributions of bZ 1~ t[ T ! and bZ 2 ~ t[ T !, note that t[ T 2 t0 5 Op ~10T ! and

1

!T ~ bZ 1~ t[ T ! 2 bZ 1~t0 !! 5 !T

5 1$ t[ T # t0 %!T

1

@ t[ T T #

(1

@t0 T #

yt yt21 2

@ t[ T T #

(1

1 1$ t[ T . t0 %!T

(1

1

(1

2

@ t[ T T #

(1

@t0 T #

@t0 T #

«t yt21

«t yt21

@t0 T #

(1

2 yt21

yt yt21

(1

2 yt21

@ t[ T T #

(1

2 yt21

2 yt21

2 @ t[ T T #

@t0 T #

(1

(

~ b1 yt21 1 «t !yt21 1

@ t[ T T #

(1

(1

@t0 T #11

~ b1 yt21 1 «t !yt21 @t0 T #

(1

2 yt21

~ b2 yt21 1 «t !yt21 @ t[ T T #

(1

2 yt21

@t0 T #

2

2

2

2 yt21


5 1$ t[ T # t0 %!T

1

@t0 T #

@t0 T #

@t0 T #

2 ( yt21 (1 @ t[ T #11

(1

1

@t0 T #

(1

2 yt21 @ t[ T T #

1 1$ t[ T . t0 %!T 2

@ t[ T T #11

2

T

@ t[ T T #

(

«t yt21

@ t[ T T #

(1

2 yt21

1

@t0 T #

(1

2 yt21

(

2 yt21

@t0 T #11 @ t[ T T #

(1

2

2 yt21

«t yt21

@ t[ T T #

(1

2 yt21

2

S S D S ! D S DD ! S S D S D S D S DD !

5 1$ t[ T # t0 %!T Op 1 1$ t[ T . t0 %

@t0 T #11

2 yt21

@ t[ T T #

1 ~ b2 2 b1 !

(

«t yt21

0

(1

2 yt21

@ t[ T T #

@t0 T #

2 ( yt21 (1 @t T #11 @ t[ T T #

«t yt21

125

1

T

T Op

1

Op

1

T

2 Op

T

Op

1

1

T

1

1 Op

T

T

1

1 Op

T

5 op ~1!+

Thus, bZ 1~ t[ T ! and bZ 1~t0 ! have the same asymptotic distribution+ Similarly, bZ 2 ~ t[ T ! and bZ 2 ~t0 ! have the same asymptotic distribution+ Define It 5 s~«i : i # t ! as the sigma field generated by the past history of $«t %+ @t0 T # T Because $«t yt21 , It % t51 and $«t yt21 , It % t5@t are martingale difference sequences 0 T #11 with E~«t yt21 6It21 ! 5 0

~t 5 1,2, + + + , T !,

@t0 T #

1 @t0 T # 1

( t51

E~~«t yt21 ! 2 6It21 !

p

&

s4

, `,

1 2 b12

T

E~~«t yt21 ! 2 6It21 ! T 2 @t0 T # t5@t(T #11

p

&

0

s4 1 2 b22

, `+

Applying the central limit theorem for martingale difference sequences ~see, e+g+, White, 1984, ch+ V! and by ~3b! and ~3c! in Lemma 1, we have @t0 T #

!T ~ bZ 1~ t[ T ! 2 b1 ! 5d !T ~ bZ 1~t0 ! 2 b1 ! 5

(1

S

«t yt21 0!T d

@t0 T #

(1

&

N 0,

2 yt21 0T

T

!T ~ bZ 2 ~ t[ T ! 2 b2 ! 5d !T ~ bZ 2 ~t0 ! 2 b1 ! 5

( «t yt21 0!T @t T #11 0

T

(

@t0 T #11

This proves Theorem 1+

2 yt21 0T

d

&

1 2 b12

S

N 0,

t0

D

,

1 2 b22 1 2 t0

D

+

n

126


APPENDIX D: PROOF OF THEOREM 2 To derive the limiting distribution of t[ T for shrinking shift, let b2T 5 b1 1 ~10!g~T !!, where g~T ! . 0, with g~T ! r ` and @ g~T !0T # r 0 as T r ` + For t 5 t0 1 y@ g~T !0T # and y # 0, by ~B+2! in Appendix B and the facts that

S( D @t0 T #

2

yt21 «t

~i! L T ~t! 5

1

@t0 T #

(1

S( T

12

2 (1 yt21

T

( yt21 «t @tT #11

12

2

T

2 ( yt21 @t T #11 0

T

2 ( yt21 @t T #11 0

2 yt21

@tT #

yt21 «t

2

T

(1

@t0 T #

«t yt21

@t0 T #11

1

11 2 2 D 11 2 2 (1 yt21 «t (1

2 yt21

2 @t0 T #

@tT #

T

( yt21 «t @t T #11

2 ( yt21 @tT #11

0

5 Op ~1!~1 2 ~1 2 op ~1!! ~1 1 op ~1!!! 2

1 Op ~1!~1 2 ~1 1 op ~1!! 2 ~1 2 op ~1!!! 5 op ~1!; @t0 T #

T

2 ( yt21 ( yt21 «t @tT #11 @t T #11

~ii! ~ b2T 2 b1 !

0

T

5

Op ~g~T !!Op ~!T !

!g~T !Op ~T !

2 ( yt21 @tT #11

5 Op

S! D g~T ! T

5 op ~1!; T

~iii!

2 ( yt21 @t T #11 0

p

&

T

(

@tT #11

1,

2 yt21

we have RSST ~t! 2 RSST ~t0 ! @t0 T #

5 22~ b2T 2 b1 ! 52

(

@t0 T #

@tT #11

yt21 «t 1 ~ b2T 2 b1 ! 2

@6 y6 g~T !#21

2

!g~T !

(0

yk02t21 «k02t 1

SS D

n 22s 2 B1

6y6

1 2 b12

2

1

1

6y6

D

2 yt21 1 op ~1!

@6 y6 g~T !#21

g~T !

2 ~1 2 b12 !

(

@tT #11

,

where B1~{! is a Brownian motion defined on R 1 +

(0

yk202t21 1 op ~1!


127

Similarly, for t 5 t0 1 y@ g~T !0T # and y . 0, from ~B+4! in Appendix B and the fact 2 that @y0~1 2 b2T !# r @y0~1 2 b12 !# as T r `, we have RSST ~t! 2 RSST ~t0 ! @tT #

5 2~ b2T 2 b1 !

@tT #

2 yt21 1 op ~1! ( yt21 «t 1 ~ b2T 2 b1 ! 2 @t ( @t T #11 T #11 0

52

0

@yg~T !#21

2

!g~T !

(0

~2yk01t «k01t11 ! 1

SS D

n 22s 2 B2

y

1 2 b12

2

y 2~1 2 b12 !

D

@yg~T !#21

1

(0

g~T !

yk201t 1 op ~1!

,

where B2 ~{! is another Brownian motion defined on R 1 independent of B1~{!+ Define r 5 y0~1 2 b12 ! and apply the continuous mapping theorem for argmax functionals ~see Kim and Pollard, 1990!+ We have T ~ b2T 2 b1 ! 2 1 2 b12

~ t[ T 2 t0 ! 5 r[ 5 Arg min $RSST ~t! 2 RSST ~t0 !% r

d

&

H S S D

Arg min 22s 2 B * ~r! 2 r

6r6

DJ

1

5 Arg max B * ~r! 2 r

1 2

2

6r6 ,

where B * ~r! is a two-sided Brownian motion on R defined to be B * ~r! 5 B1~2r! for r # 0 and B * ~r! 5 B2 ~r! for r . 0+ This proves Theorem 2+ n

APPENDIX E: PROOF OF LEMMA 2 Conditions ~4a! and ~4d! are special cases of ~3a!+ Conditions ~4e! and ~4f ! are identical to ~3d! and ~3e!, respectively+ ~4b!

1 T

T

T

1

2 ~ yt2 2 yt21 2 «t2 ! ( yt21 «t 5 2T @t ( @t T #11 T #11 0

0

5 5

T 2 k0 2T 1 2 t0 2

1

T

yT2 T 2 k0

SS ! 2

yk20

2

T 2 k0

yk 0 T 2 k0 yk20

T 2 k0

1

2

2

( «t2 @t T #11 0

T 2 k0

2

T

1

!T 2 k

0 T

1

( «j j5k 11 0

«t2 T 2 k 0 @t ( T #11 0

D

+

D

2

128


Because yk 0

p

!T 2 k

&

0

T

1

and

!T 2 k

0

( «j n sB~1!, j5k 11

0

0

by the continuous mapping theorem, we have

S!

yk 0 T 2 k0

1

T

1

!T 2 k

( «j j5k 11

0

0

D

2

n s 2 B 2 ~1!+

Therefore 1 T

T

( yt21 «t n @t T #11

~1 2 t0 !s 2 2

0

~4c!

1 T

2

T

2 5 ( yt21 @t T #11

~B 2 ~1! 2 1! 5 Op ~1!+

~T 2 k 0 ! 2 T

0

2

S

T

0

T

5 ~1 2 t0 ! 2

yk 0

(

@t0 T #11

n ~1 2 t0 ! 2 s 2

E

«j

011

S

op ~1! 1

t2k 022

1

!

D

2

t21

1

( !T 2 k 1 !T 2 k j5k( @t T #11 0 0 T 2 k0

(

D

1 T 2 k0

2

«k01j11

j50

1 T 2 k0

1

B 2 ~s! ds 5 Op ~1!,

0

where s 5 ~t 2 k 0 2 2!0~T 2 k 0 !+ To prove ~4g!, we use the triangle inequality, ~4b!–~4e!, and the facts that @t0 T #

sup st#t#t0

@t0 T #

2 # ( ( yt21 @tT #11 1

2 yt21 +

Then we have

sup 6 bZ 2 ~t! 2 16 5 sup

st#t#t0

st#t#t0

#

*

@t0 T #

~ b1 2 1!

T

(

(

@tT #11

(

@t0 T #11

«t yt21

2 yt21

S

2 ( yt21 @tT #11 @t0 T #

6 b1 2 16

1

5

@tT #11

T

«t yt21 1

T

1

@t0 T #11

(

@t0 T # 2 yt21 1

*

(1

T

( «t yt21 @t T #11 0

Op ~T ! 1 op ~T ! 1 Op ~T ! Op ~T 2 !

*(

@t0 T #

2 yt21 1 sup

5 Op

st#t#t0

*D

SD 1

T

+

@tT #11

«t yt21

*

*


129

~4h! By the triangle inequality and ~4g!, sup 6 bZ 2 ~t! 2 b1 6 # sup 6 bZ 2 ~t! 2 16 1 61 2 b1 6 5 Op ~1!+

st#t#t0

st#t#t0

Conditions ~4i! and ~4j! can be proved by using an argument similar to that in ~4b!+ ~4k! For t [ JT1 ,

6 bZ 1 ~t! 2 16 5

*

@t0 T #

(1

~ b1 2 1!

@t0 T #

(1

2 yt21 1

@tT #

«t yt21 1

@t0 T # 2 yt21 1

@t0 T #

# 6 b1 2 16

(

@t0 T #11

2 yt21

* ** @t0 T #

2 yt21

1

@tT #

( @t T #11

«t yt21

@tT #

(1

(1

(

@t0 T #11

(1

0

1

@t0 T #

(1

2 yt21

«t yt21 2 yt21

*

@tT #

(

@t0 T #11

«t yt21

@tT #

*

(

2 yt21

T

*( *

@t0 T #11

@t0 T #

# 6 b1 2 16

@t0 T #

1

(1

T

2 yt21

T

1

@~t 2 t0 !T #

2

AT

*

T @~t 2 t0 !T # 2

@tT #

( @t T #11 0

yt21 «t T

AT 1

@t0 T #

(1

*

2 yt21

«t yt21

1

T

,

where AT 5

@~t2t0 !T #

(0

S!

1

D

2

yk01t @~t 2 t0 !T #

1

+

@~t 2 t0 !T #

The proof is completed by utilizing ~4a!, ~4e!, ~4i! and the following facts+ ~i! By the definition of JT1 , T0@~t 2 t0 !T # 2 5 o~1! for t [ JT1 + ~ii! By the definition of JT1 and by using the continuous mapping theorem, we have 1

AT n s2

E

5 Op ~1!+

1

B2

0

~4l! By the fact that sup t[JT1

* ( * *( * T2

T

@tT #11

2 yt21

#

T2

T

k11 N

2 yt21

5 Op ~1!

130


and by ~4j!,

sup 6 bZ 2 ~t! 2 16 5 sup

t[JT1

t[JT1

*

T

( yt21 «t @tT #11 T

2 ( yt21 @tT #11

*

*( T

#

1 T

*( * T2

T

k11 N

2 yt21

@tT #11

sup t[JT1

yt21 «t

T

*

5 Op

SD 1

T

+

n

These prove Lemma 2+

APPENDIX F: DERIVATION OF EQUATIONS ~12!–~14! For 6 b1 6 , 1, b2 5 1, by ~B+1! in Appendix B, the triangle inequality, and the fact that sup6ab6 # sup6a6sup6b6, we have

*T 1 #* (« 2s * T

sup

1

st#t#t0

RSST ~t! 2 s 2 2

~1 2 b1 !~t0 2 t!s 2 1 1 b1

*

T

2 t

2

1

st#t#t0

*T ( y « * 1

1 sup 6 bZ 1 ~t! 2 b1 6 sup

st#t#t0

t21

1 sup

st#t#t0

*

st#t#t0

~ bZ 2 ~t! 2 b1 ! 2

1 2 sup 6 bZ 2 ~t! 2 16 st#t#t0

*

1 T

*

@t0 T #

1

(

T

@tT #11

1 T

S

2 2 ( yt21 @tT #11

T

( yt21 «t @t T #11 0

5 op ~1!+

sup 6T ~ bZ 2 ~t! 2 1!6

st#t#t0

yt21 «t

@t0 T #

T

*

T

1

t

1

1 2 sup 6 bZ 2 ~t! 2 b1 6 sup st#t#t0

@tT #

2 ( yt21 2 @t T #11

D

0

T2

*

~1 2 b1 !~t0 2 t!s 2 1 1 b1

*


131

All six preceding terms above are op ~1! by Lemma 2+ To show that the fourth term is also op ~1!, we use the fact that ~1 2 b1 !0~1 1 b1 ! 5 ~1 2 b1 ! 20~1 2 b12 ! and the triangle inequality that sup6a 2 b6 5 sup6a 2 c 1 c 2 b6 # sup6a 2 c6 1 sup6c 2 b6, with @t0 T #

(

a 5 ~ bZ 2 ~t! 2 b1 ! 2 c 5 ~ bZ 2 ~t! 2 b1 ! 2

2 yt21

@tT #11

b5

,

T

~1 2 b1 !~t0 2 t!s 2

and

,

1 1 b1

~t0 2 t!s 2 +

1 2 b12

Conditions ~4e! and ~4h! in Lemma 2 imply that sup6a 2 c6 and sup6c 2 b6 are both op ~1!+ This derives ~12!+ For t [ JT1 , by ~B+3! in Appendix B and the triangle inequality, we have

*T

1

~1 2 b1 !t0 s 2

RSST ~t! 2 s 2 2 #

*

1 T

1 1 b1

T

(1 «t2 2 s 2

*

* *

1 2~ bZ 1 ~t! 2 b1 !

*

1

@t0 T #

T

(1

yt21 «t

@t0 T #

(1

1 ~ bZ 1 ~t! 2 b1 ! 2 1 ~ bZ 1 ~t! 2 1! 2

2 yt21

2

T

~1 2 b1 !t0 s 2 1 1 b1

*

**

@tT #

(

1 2~ bZ 1 ~t! 2 1!

@t0 T #11

yt21 «t

T

*

2 1 * ~ bZ 2 ~t! 2 1! yt21 «t *+ ( yt21 T @tT( @t T #11 #11 @tT #

1 T

T

1

0

In the third term, we use the facts that ~1 2 b1 !0~1 1 b1 ! 5 @~1 2 b1 ! 2 #0~1 2 b12 ! and the triangle inequality that 6a 2 b6 5 6a 2 c 1 c 2 b6 # 6a 2 c6 1 6c 2 b6, with @t0 T #

a 5 ~ bZ 1 ~t! 2 b1 !

(1

2

2 yt21

b5

,

T

~1 2 b1 !t0 s 2

and

1 1 b1

c 5 ~ bZ 1 ~t! 2 b1 ! 2

t0 s 2 1 2 b12

Thus, the preceding equation is bounded by #

*

1 T

* * ( * (

T 1

1

1 ~ bZ 1 ~t! 2 b1 ! 2 1 1

1

( «t2 2 s 2 1 2

t0 s 2 12

b12

*

@t0 T #

T

yt21 «t 6 bZ 1 ~t! 2 b1 6

1

@t0 T #

2 yt21 2

T

1

t0 s 2 1 2 b12

*

6 bZ 1 ~t! 1 1 2 2b1 66 bZ 1 ~t! 2 16 1 26 bZ 1 ~t! 2 16

@~t 2 t0 !T # 2 T

1 6 bZ 2 ~t! 2 16

~ bZ 1 ~t! 2 1! 2

*T

1

@~t 2 t0 !T #

( yt21 «t * + @tT #11 T

@tT #

1 2

(

@t0 T #11

2 yt21

*

1

@tT #

T

@t0 T #11

(

yt21 «t

*

+

132


All seven preceding terms above are op ~1! by Lemma 2 and by the definition of JT1+ This derives equation ~13!+ To derive ~14!, first note that for any positive constant c,

S

bZ 1 t0 1

1

D S D

c

1

5 uT

!T

2

b1 1

,c

(1 «t yt21 k0

2 (1 yt21

S S DD

1 1 2 uT

2

k0

1

2

,c

1

@k 01c!T #

(

«t yt21

k 011

11

k 01c! T

(

k 011

2 yt21

2

,

where k0

uT

2 (1 yt21

S D 1

2

,c 5

(1

+

@k 01c!T #

k0

2 yt21

(

1

2 yt21

k 011

By ~3b! and the fact that yk20 0T 1 T

@k 01c! T #

( k 11

2 yt21 5

0

1

@c! T #

T

(1

p

&

0, plus the invariance principle that

2 yt1k n s2 021

E

c

B32 ,

0

where B3 ~{! is a Brownian motion defined on R 1 , we have ~i! uT

S D 1

2

t0

,c n

t0 1 ~1 2 b12 !

E

;

c

B32

0

S

~ii! bZ 1 t0 1

c

!T

D

E E

n

c

t0 1 ~1 2 b12 !

B32

Additionally, we have sup c[R 11

~iv!

sup c[R 11

*bZ St 1 !T D 2 1* 5 o ~1!; c

2

1 T

*

0

@k 01c! T #

(

k 011

B32

0

0

~iii!

c

t0 b1 1 ~1 2 b12 !

p

*

yt21 «t 5 op ~1!;

+

STRUCTURAL CHANGE IN AR(1) MODELS ~v!

sup c[R 11

1 T

*

T

( yt21 «t @k 1c! T #11 0

*

133

5 Op ~1!+

From ~B+3! in Appendix B, ~4e! of Lemma 2, and ~i!–~v! given immediately preceding, we have 1 T

S

RSST t0 1

c

!T

D

5

1 T

SS !D D SS !D D ( SS !D D ( T

c

c

2 2 bZ 1 t0 1

SS SS

1 bZ 1 t0 1

2 bZ 2 t0 1

1 T

T

!T c

T

2 yt21

@k 01c! T #

1

( T

1

( yt21 «t @k 1c! T #11

T

SS SS !D D c

0

D ! D

n s2 1

S

1

2

T

(

T

E E

B32

B32

0

t0 1 ~1 2

b12 !

E

D

2

0

B32

s2

SE D

21

t0

B32

0

+ 11 2

E

b12

t0 s 2 1 2 b12

c

0

t0 s 2 ~1 2 b1 ! 2 c

2

2

c

0

c

2

1 T

k0

2 1 op ~1! (1 yt21

2 yt21 1 op ~1!

k 011

c

t0 1 ~1 2 b12 !

2 b1

@k 01c! T #

1

~1 2 b1 !~1 2 b12 !

t0 ~ b1 2 1!

5 s2 1

This derives equation ~14!+

T

c

bZ 1 t0 1

21

yt21

k 011

T

(1 «t2 1 op ~1! 1

yt21 «t

k 011

T

21

(1 yt21 «t

1

T

2

T

k0

@k 01c! T #

1

D D ! D D 21

1

k0

1

T

21

c

1 bZ 1 t0 1

1

2

2 b1

T

2 b1

T

1

1 bZ 1 t0 1

5

c

( «t2 2 2 bZ 1 t0 1

B32

134


APPENDIX G: PROOF OF THEOREM 3 To show 6 t[ T 2 t0 6 5 Op ~10T !, it is sufficient to prove that A1–A6 defined in Appendix C are all op ~1! in the case where 6 b1 6 , 1 and b2 5 1+ Now T

A1 5

( yt21 «t k 11

5 op ~1!

0

T

(

2 yt21

k 011

by ~4b! and ~4c! of Lemma 2; k0

(

yt21 «t

m11

A 2 5 sup

5 op ~1!

k0

m[D1T

(

2 yt21

m11

by the uniform law of large numbers in Andrews ~1987, Theorem 1!;

A 3 5 sup m[D1T

3

*

T

2 ( yt21 m11

LT

k0

T

2 2 ( yt21 ( yt21 k 11 m11

m

T

0

1 * sup

S

1 sup m[D1T

k0

D S 2

( yt21 «t 1

2

k0

m[D1T

*

1

T2

*1 #

1

(

k 011

( yt21 «t 1

D

m

2 (1 yt21

2

k 011

1 Op

2

(

k 011

1

MT

«t yt21

2

T

2 yt21

m11

T

(

k 02MT

2 yt21

*

T

«t yt21

k0

(

2 yt21

S( D S( D T

1

1

T

2

m

2 (1 yt21

S S D S DD

5 Op

SD

2 yt21

m11

~op ~1! 1 Op ~1!! 5 Op

*2

S D 1

MT

5 op ~1!;

2


135

k0

A4 5

(1 yt21 «t

5 op ~1!

k0

(1

2 yt21

by ~4a! and ~4e! of Lemma 2; m

A 5 5 sup

ym2 2 yk20 2 5 sup

0

m

m[D2T

(

k 011

A 6 5 sup m[D2T

3

m

( yt21 «t k 11

*

2 ( yt21 k 11

2

m

m

(

k 011

1 *

LT

k0

2 2 yt21 ( yt21

S

1

k0

1 sup m[D2T

1

T

m

T

D S 2

k0

*

*1

1

#

k0

D

2

( yt21 «t 1

m

1

2 (1 yt21

m

2 (1 yt21

2 (1 yt21

(

k 011

1 Op

2

«t yt21

2

T

m11

T

2 ( yt21 k 11

2 ( yt21 m11

0

1

MT2

2 yt21

2

*

T

«t yt21

1 k 01MT k 011

S( D S( D 2

T

S S D S DD 1

SD

2

( yt21 «t

m[D2T

5 op ~1!+

0

2 (1 yt21

sup

5 Op

0

m

m[D2T

2 yt21

( «t2 k 11

*2

~Op ~1! 1 Op ~1!! 5 op ~1!+

~To prove that A 5 5 op ~1!, explicit formulae would be needed+ I am not able to provide a general proof because of the unknown asymptotic properties of 2 supm[D2T ~ ( km011 yt21 «t !0~ ( km011 yt21 ! under nonstationarity of yt + No study on the uniform convergence under this kind of nonstationarity has been done in the literature+ However, the referees and I believe that the result is true and intuitive+! Because A 1 –A 6 are op ~1!, thus t[ T is T-consistent+ To find the limiting distribution of bZ 1~ t[ T !, note that t[ T 2 t0 5 Op ~10T ! and

!T ~ bZ 1~ t[ T ! 2 bZ 1~t0 !! 5 !T

1

@ t[ T T #

(1

@t0 T #

yt yt21 2

@ t[ T T #

(1

2 yt21

(1

yt yt21

@t0 T #

(1

2 yt21

2

136


5 1$ t[ T # t0 %!T

1

@t0 T #

@t0 T #

(

@ t[ T T #11

1 1$ t[ T . t0 %!T

2 yt21

(1

@ t[ T T #

(1

1

(

@ t[ T T #11

2

@t0 T #

(1

2 yt21

@ t[ T T #

2

@t0 T #

«t yt21

@ t[ T T #

(1

2 yt21

1

(1

@t0 T #

(1

2 yt21

(

@t0 T #11

2

«t yt21

@ t[ T T #

(1

2 yt21

2

2 yt21

@ t[ T T #

(1

@t0 T #11

2 yt21

@ t[ T T #

1 ~ b2 2 b1 !

(

«t yt21

0

@ t[ T T #

2 yt21

@ t[ T T #

@t0 T #

2 ( yt21 (1 @t T #11

«t yt21

2 yt21

S S D S ! D S DD ! S S D S D S D S DD !

5 1$ t[ T # t0 %!T Op

1

T

1 1$ t[ T . t0 % T Op

1

Op

1

T

2 Op

T

Op

1

1

T

1 Op

T

1

T

1 Op

1

T

5 op ~1!+ Thus, bZ 1~ t[ T ! and bZ 1~t0 ! have the same asymptotic distribution+ @t0 T # Because $«t yt21 ,It % t51 is a martingale difference sequence, with E~«t yt21 6It21 ! 5 0, @t0 T #

(1

p

E~~«t yt21 ! 2 6It21 !

&

s4 1 2 b12

, `+

Applying the central limit theorem for martingale difference sequences and the fact that 1 T

@t0 T #

(1

2 yt21

p

&

t0 s 2 1 2 b12

,

we have @t0 T #

d

!T ~ bZ 1~ t[ T ! 2 b1 ! 5 !T ~ bZ 1~t0 ! 2 b1 ! 5

(1

«t yt21 0!T d

@t0 T #

(1

2 yt21 0T

&

S

N 0,

1 2 b12 t0

D

+


137

To find the limiting distribution of bZ 2 ~ t[ T !, note that t[ T 2 t0 5 Op ~10T ! and T ~ bZ 2 ~ t[ T ! 2 bZ 2 ~t0 !!

5T

1

T

( yt yt21 @ t[ T #11 T

T

(

@ t[ T T #11

T

( yt yt21 @t T #11

2

0

T

(

2 yt21

1

2 yt21

@t0 T #11 @t0 T #

5 1$ t[ T # t0 %T 2

2

T

0

T

T

(

@ t[ T T #11

(

2 yt21

@t0 T #11

(

1 ~ b1 2 b2 !

@ t[ T T #11

(

1

2 yt21

T

@ t[ T T #11 @ t[ T T #

(

1

@ t[ T T #11

2 yt21

0

T

(

@ t[ T T #11

(

2 yt21

@t0 T #11

2 yt21

2 @ t[ T T #

T

T

T

@ t[ T T #11

2 ( yt21 ( «t yt21 @t T #11 @t T #11 0

«t yt21

(

2 yt21

@t0 T #

1 1$ t[ T . t0 %T

@t0 T #

T

2 ( yt21 ( «t yt21 @ t[ T #11 @t T #11

(

2

@t0 T #11 T

(

2 yt21

«t yt21

@ t[ T T #11

2 yt21

S S D S D S D S DD S S D S D S DD

5 1$ t[ T # t0 %T Op

1

T

1 1$ t[ T . t0 %T Op

Op

2

1

T

2

1

1 Op

T

1

Op

1

T

2 Op

T

1 Op

2

+

1

T2

1

T2

5 op ~1!+ Thus, bZ 2 ~ t[ T ! and bZ 2 ~t0 ! have the same asymptotic distribution+ Applying ~4b! and ~4c! in Lemma 2, we get 1 d

T ~ bZ 2 ~ t[ T ! 2 1! 5 T ~ bZ 2 ~t0 ! 2 b2 ! 5

T

T

( «t yt21 @t T #11

1 T2

0

T

(

@t0 T #11

2 yt21

n

B 2 ~1! 2 1 2~1 2 t0 !

E

+

1

B

2

0

These derive the limiting distribution of bZ 1~ t[ T ! and bZ 2 ~ t[ T ! for fixed magnitude of break+ To derive the limiting distribution of t[ T for shrinking shift, we fix b2 at one and let b1T 5 1 2 @10g~T !# , where g~T ! . 0, with g~T ! r ` and @ g~T !0T # r 0 as T r `+ Let y be a finite constant, and B1~{! and B2 ~{! be defined as in Section 2+ For t 5 t0 1 y@ g~T !0T # and y # 0, we have L T ~t! 5 op ~1! where L T ~t! is defined in Appendix B, and

138


yk02t21

!g~T !

5

FS

1

!g~T ! k 02t22

( i50

5

ns

SS

12

E

`

D

k 02t22

1

12

g~T !

k 02t22

D D

g~T !

S

12

i50

g~T ! i0g~T !

1

(

y0 1

«k02t212i

!g~T !

d

exp~2s! dB1 ~s! 5 sBa

0

SD

1 g~T !

D

i

«k02t212i

G

1 op ~1!

1

2

,

where Ba ~ _12 ! is a Brownian motion defined on R 1 + Because yk02t21

!g~T !

n sBa

SD 1

2

and

@6 y6 g~T !#21

1

!g~T !

(0

«k02t n sB1 ~6y6!,

we have @6 y6 g~T !#21

1

(0

!g~T !

yk02t21 «k02t

!g~T !

n s 2 Ba

@6 y6 g~T !#21

1

(0

g ~T ! 2

SD SD 1

2

B1 ~6y6!

and

1

yk202t21 n 6y6s 2 Ba2

2

+

Further, because @t0 T #

~1 2 b1T !

T

2 ( yt21 ( yt21 «t @tT #11 @t T #11 0

T

(

@tT #11

5

Op ~g 2 ~T !!Op ~T !

2 yt21

g~T !Op ~T ! 2

5 Op

S D g~T ! T

5 op ~1!

and T

2 ( yt21 @t T #11

p

0

&

T

(

@tT #11

1,

2 yt21

equation ~B+2! in Appendix B becomes RSST ~t! 2 RSST ~t0 ! @t0 T #

5 22~1 2 b1T ! 52

2

(

@tT #11

@t0 T #

yt21 «t 1 ~1 2 b1T ! 2

@6 y6 g~T !#21

g~T !

n 22s 2 Ba

(0

SD 1

2

yk02t21 «k02t 1 B1 ~6y6! 1 6y6s 2 Ba2

g ~T ! 1

2

2 yt21 1 op ~1!

@6 y6 g~T !#21

1 2

SD

(

@tT #11

+

(0

yk202t21 1 op ~1!


139

Similarly, for t 5 t0 1 y@ g~T !0T # and y . 0, from ~B+4! in Appendix B, we have L T ~t! 5 op ~1!, @t0 T #

@tT #

(1

~1 2 b1T !

(

yt21 «t

@t0 T #11

2 yt21

Op ~T !Op ~g 2 ~T !!

5

@tT #

g~T !Op ~T 2 !

2 (1 yt21

yk 0

!g~T !

S

ns

E

g~T !

`

D

k0

1

5 12

5 Op

S D g~T ! T

k 021

y0

!g~T !

( i50

1

d

@yg~T !#21

1

(0

g~T !

S! D

2

yk01t

g~T !

5

@yg~T !#21

1

(0

g~T !

S!

@yg~T !#21

yk01t

(0 !g~T ! «k 1t11 n s 2

!g~T !

0

B2 ~s! 1 Ba

y

B2 ~s! 1 Ba

0

i

«k02i

g~T !

S DD S DD 1

g~T !

1

,

2

yk 0

0

ES

ES

1

( «k 1t2i 1 !g~T ! g~T ! i50

0

1

12

t21

1

y

n s2

S D! SD

exp~2s! dB1 ~s! 5 sBa

0

5 op ~1!,

D

2

2

2

1

2

ds,

dB2 ~s!+

Hence, equation ~B+4! in Appendix B becomes RSST ~t! 2 RSST ~t0 !

5

@tT #

@tT #

0

0

2 yt21 1 op ~1! ( yt21 «t 1 ~1 2 b1T ! 2 @t ( @t T #11 T #11

5 2~1 2 b1T !

@yg~T !#21

2

(0

g~T !

yk01t «k01t11 1

ES y

n 2s 2

B2 ~s! 1 Ba

0

5 22s 2 Ba2

F

3 2

S DD 1

2

1

@yg~T !#21

(0

g ~T ! 2

yk201t 1 op ~1!

ES y

dB2 ~s! 1 s 2

B2 ~s! 1 B

0

SD

S DD 1

2

2

ds

1

2

E

B2 ~y! 2 1 Ba 2

SD

y

0

E

B2 ~s! dB2 ~s! 2 1 Ba2 2

SD

y

0

S

D

G

1 B2 ~s! B2 ~s! 11 ds 2 y + 1 1 2 2Ba Ba 2 2

SD

SD

140


By the continuous mapping theorem for argmax functionals, we have ~1 2 b1T !T ~ t[ T 2 t0 ! 5 y[ 5 Arg max $RSST ~t! 2 RSST ~t0 !% y[R

d

&

H

Arg max 22s 2 Ba2 y[R

5 Arg max y[R

SD 1

2

S S D DJ 1 C * ~y! 2 6y6 1 2 Ba 2

S SD D

1 C * ~y! 2 6y6 + 1 2 Ba 2

n

This proves Theorem 3+

APPENDIX H: PROOF OF LEMMA 3 ~5a! By the fact that ~ yk0 0!T ! n sB~t0 ! and ~ y02 0T ! 5 op ~1!, we have 1 T

@t0 T #

(1

yt21 «t 5

n

~5b!

1 2T s2 2

@t0 T #

(1

1

2 ~ yt2 2 yt21 2 «t2 ! 5

2T

S

@t0 T #

yk20 2 y02 2

(1

«t2

D

~B 2 ~t0 ! 2 t0 ! 5 Op ~1!+

T

1

yt21 «t !T @t ( T #11 0

5

T

1

!T t5k(11 0

5

5

1

!T 1

SS

S

k0

y0 1 ( «j

k0

( !T j51

j51

S

1 op ~1!+

«j

D

t21

b2t2k021 yk0 1

D(

(

b2t2i21 «i «t

T

k 011

b2t2k021 «t 1

T

k 011

(

i5k 011

b2t2k021 «t

D

1

1

T

t21

0

0

( ( b2t2i21 «i «t t5k 11 i5k 11 T

!T t5k(11 0

S

«t

t21

(

i5k 011

b2t2i21 «i

D

D


141

T k0 By Assumption ~A2! that «’s are i+i+d+, $«j ( t5k b2t2k021 «t , Ij % j51 and 011 t21 t2i21 T $«t ( i5k011 b2 «i , It %t5k011 will both be martingale difference sequences with

S

T

( b2t2k 21 «t 6Ij21 t5k 11

E «j 1 T

0

0

k0

(E j51

S

(

T

T

( E t5k 11 0

~ j 5 1,2, + + + , k 0 !,

D D

b2t2i21 «i 6It21

50

0

0

i5k 011

50

( b2t2k 21 «t t5k 11

T

«j

t21

E «t 1

SS

D

SS

«t

D

2

p

6Ij21

t21

( b2t2i21 «i i5k 11 0

t0 s 4

&

, `,

1 2 b22

~t 5 k 0 1 1, k 0 1 2, + + + , T !,

D D 2

p

6It21

~1 2 t0 !s 4

&

+

1 2 b22

By the central limit theorem for martingale difference sequences and by the independence of the two martingale difference sequences given previously, we have

S

T

1

yt21 «t !T @t ( T #11

d

N 0,

&

0

~5c! ~5d!

@t0 T #

1 T 1 T

(1

2

@t0 T #

1 2 b22

1

y0

@tT #

1

2 5 ( yt21 T t5k(11 @t T #11 0

S! D 0

5

1

1 T

yk 0

( «i i51

1

!T

S

T

T

;

2

2

1

n s2

T

E

t0

B 2 5 Op ~1!;

0

t21

( b2t2i21 «i i5k 11

b2t2k021 yk0 1

0

@tT #

2

yk 0

12

D

t21

(1 !T

2 yt21 5

@tT #

s4

(

t5k 011 @tT #

( t5k 11

S

0

@tT #

( t5k 11 0

D

2

b22~t2k021!

S

t21

b2t2k021

(

i5k 011

t21

( b2t2i21 «i i5k 11 0

b2t2i21 «i

D

D

2

+

The first term converges weakly to @s 2 B 2 ~t0 !#0~1 2 b22 !+ Because 6 yk0 0!T 6 5 Op ~1! and ~10!T !sup t.k0 6«t 6 5 op ~1!, the second term is bounded by #2 #2 #2

*! * * ( * ! *S ( * * yk 0

@tT #

sup

T

t[JT1

yk 0

`

T

k 011

k 011

b2t2k021

b2t2k021 1 2 b2

* !T *S ~1 2 6 b 6! yk 0

1

2

2

1

1 2 b2t2k021 1 2 b2 `

1

( k 11 0

*

*!

1

b22~t2k021! 1 2 b2

T

sup 6«t 6 t.k 0

*D !

1 ~1 2 6 b2 6!~1 2 b22 !

1

D!

T

1 T

sup 6«t 6 t.k 0

sup 6«t 6 5 op ~1!+ t.k 0

142


Last, because 6 b2 6 , 1 and «i are i+i+d+ with finite fourth moment, we have lim Tr`

1

@tT #

T

t5k 011

(

S(

Var

SS

D DD S (

~t 2 t0 !s 2

2

t21

E

i5k 011

5

b2t2i21 «i

t21

( b2t2i21 «i i5k 11 0

,

1 2 b22

2

D

4

t21

,E

i5k 011

b2t2i21 «i

, `+

By the uniform weak law of large numbers for dependent and heterogeneous processes, sup t[JT1

*

1 T

@tT #

( t5k 11 0

S

t21

( b2t2i21 «i i5k 11 0

D

~t 2 t0 !s 2

2

2

1 2 b22

*

5 op ~1!+

Thus, 1 T

@tT #

2 n ( yt21 @t T #11

t 2 t0 1 B 2 ~t0 !

~5e! ~5f !

sup t[JT2

sup t[JT2

*T

1

*

1 T

s 2+

1 2 b22

0

sup * ( (1 yt21 «t * 5 t[J * 1 !T !T

@tT #

@tT #

d

( @tT #11

*

t[JT2

d

0

yt21 «t T

@tT #11

t0

sup s 2

&

t[JT2

*( ! !* * E * @t0 T #

yt21 «t 5 sup

t[JT2

t

* *

T

BdB 5 Op ~1!;

@tT #

~5g!

sup 6 bZ 1 ~t! 2 16 5 sup

t[JT2

5

~5h!

1 T

@tT #

(1

#

2 yt21

t[JT2

*

T

yt21 «t

T2

1

(1

SD

t[JT2

2 yt21

1

T

; @t0 T #

T

2 1 ( «t yt21 1 ( «t yt21 ( yt21 @t T #11 @tT #11 @t T #11 0

@t0 T #

(

@tT #11

T

2 yt21 1

# sup

2 ( yt21 @t T #11 0

@t0 T #

t[JT2

( @tT #11 T

2 yt21

( «t yt21 @t T #11 0

T

(

@t0 T #11

2 yt21

*

+

1 sup

(

@t0 T #11

T

*

T

0

6 b2 2 16

1

1

sup

@ stT #

T

~ b2 2 1!

*( * @tT #

Op ~1!Op ~1! 5 Op

sup 6 bZ 2 ~t! 2 16 5 sup

t[JT2

(1 yt21 «t

t[JT2

BdB 5 Op ~1!;

sup s 2

&

T 2

@t0 T #

* E * t

yt21 «t

t[JT2

*

2 yt21

@t0 T #

(

@tT #11

«t yt21

@t0 T #

(

@tT #11

2 yt21

*

*


143

Using the fact that @t0 T #

S D

2

yt21

( @tT #11 ! T

1 @~t0 2 t!T #

n s2

E

t0

t

B 2 5 Op ~1!,

the definition of JT2 , ~5d!, and ~5f !, the first term is bounded by T

6 b2 2 16

sup

k0 2 k

t[JT2

2 ( yt21 @t T #11

T ~k 0 2 k!

sup t[JT2

0

@t0 T #

(

@tT #11

T

2 yt21

5 Op ~T 2102 !Op ~1!Op ~1! 5 op ~1!+

The second term is bounded by

*( * @t0 T #

1

sup t[JT2

k0 2 k

*( * T ~k 0 2 k!

sup t[JT2

@t0 T #

@tT #11

2 yt21

sup t[JT2

@tT #11

«t yt21

T

5 Op ~T 2102 !Op ~1!Op ~1! 5 op ~1!+ The last term is op ~1! by ~5b! and ~5d! of this lemma+ Thus, sup t[JT2 6 bZ 2 ~t! 2 16 5 op ~1!+

~5i!

sup t[JT1

*

1 T

@tT #

( yt21 «t @t T #11 0

*

5 sup

t[JT1

# sup

t[JT1

* *

1 sup

1 T

@tT #

( @t T #11 0

1

@tT #

T

@t0 T #11

t[JT1

*

S

(

1 T

b2t2k021 yk0 «t

@tT #

(

t21

b2t2k021 yk0 1

t21

(

@t0 T #11 i5k 011

(

i5k 011

D*

b2t2i21 «i «t

* *

b2t2i21 «i «t +

Because 6 yk0 0!T 6 5 Op ~1! and ~10!T ! sup t.k0 6«t 6 5 op ~1!, the first term is bounded by @10~1 2 b2 !#6 yk0 0 k 0 6~10!T !sup t.k0 6«t 6 5 op ~1!+ t21 t21 Further, because E~ ( i5k b2t2i21 «i «t ! 5 0 and Var~ ( i5k b2t2i21 «i «t ! # 011 011 2 4 s 0~1 2 b2 ! , `, by the uniform weak law of large numbers for dependent and heterogeneous processes ~Pötscher and Prucha, 1989; Andrews, 1987!, the second term converges to zero+ This shows ~5i!+

!

144


To show ~5j!, by ~5b!, ~5i!, and the triangle inequality, we have sup t[JT1

*

~5k!

1 T

T

( yt21 «t @tT #11

* * #

1 T

T

( yt21 «t @t T #11 0

*

1 sup

t[JT1

*

1

@tT #

T

@t0 T #11

*

(

yt21 «t 5 op ~1!+

sup 6 bZ 1 ~t! 2 16

t[JT1

5 sup

t[JT1

*

(

@t0 T #11

2 yt21 1

(

@t0 T #11

«t yt21

2 (1 yt21

*

0

*

@t0 T #

2 1 ( ( yt21 @t T #11 1

«t yt21 1 sup

t[JT1

*

*

@tT #

(

@t0 T #11

«t yt21

*

@t0 T #

(1

5

(1

@tT #

«t yt21 1

@tT #

T

6 b2 2 16 #

@t0 T #

@tT #

~ b2 2 1!

Op ~T ! 1 Op ~T ! 1 op ~T ! Op ~T ! 2

5 Op

2 yt21

SD 1

T

by ~5a!, ~5c!, ~5d!, and ~5i!+

~5l!

sup 6 bZ 2 ~t! 2 b2 6 5 sup

t[JT1

t[JT1

*

T

( «t yt21 @tT #11 T

(

@tT #11

2 yt21

*

#

T

sup

T

(

@ tT S #11

2 yt21

t[JT1

*

T

( «t yt21 @tT #11 T

*

5 Op ~1!op ~1! 5 op ~1! by ~5d! and ~5j!+ These prove Lemma 3+

n

APPENDIX I: DERIVATION OF EQUATIONS ~16!–~18! To show that ~10T !RSST ~t! converges uniformly to a random flat line above s 2 for t [ JT2 , it is sufficient to show that ~i! ~10T !RSST ~t! converges pointwise to a random value above s 2 for all t [ JT2 and p ~ii! sup tb , ta[JT2 6~10T !RSST ~tb ! 2 ~10T !RSST ~ta !6 & 0 for any tb , ta [ JT2 +


145

To show ~i!, use Lemma 3 and ~B+1! in Appendix B+ For t [ JT2 , b1 5 1, and 6 b2 6 , 1, we have

1 T

RSST ~t! 5

T

@t0 T #

@tT #

T

(1 «t2

(1 yt21 «t

2 ~ bZ 1 ~t! 2 1!

T

(

yt21 «t

@tT #11

2 2~ bZ 2 ~t! 2 1!

T

@t0 T #

1

1

T

T

2 ( yt21 @tT #11

~T ~ bZ 2 ~t! 2 1!! 2

2

T2

2

!T

~ bZ 2 ~t! 2 b2 !

T

1 ~ bZ 2 ~t! 2 1! 2

2 ( yt21 @t T #11 0

T

( yt21 «t @t T #11 0

!T T

1 2~ bZ 2 ~t! 2 1!~1 2 b2 !

2 ( yt21 @t T #11 0

T

T

1 ~1 2 b2 ! 2

2 ( yt21 @t T #11 0

T

T

5

(1 «t2 T

n s2 1

T

2 op ~1! 2 op ~1! 1 op ~1! 2 op ~1! 1 op ~1! 1 op ~1! 1 ~1 2 b2 ! 2 ~1 2 b2 !~1 2 t0 1 B 2 ~t0 !! 1 1 b2

2 ( yt21 @t T #11 0

T

s 2 . s 2+

To show ~ii!, use ~B+1! in Appendix B, the triangle inequality, and Lemma 3+ We have

5

* T RSS ~t ! 2 T RSS ~t !* 1

sup tb , ta[JT2

1

T

sup tb , ta[JT2

*

b

T

a

@tb T #

2 ~ bZ 1 ~tb ! 2 1!

(1

@t0 T #

(

yt21 «t 2 2~ bZ 2 ~tb ! 2 1!

T

@tb T #11

T

@t0 T #

1 ~ bZ 2 ~tb ! 2 1! 2

2 ( yt21 @t T #11 b

T

yt21 «t

T

1 2~ bZ 2 ~ta ! 2 bZ 2 ~tb !!

( yt21 «t @t T #11 0

T @ta T #

T

1 ~~ bZ 2 ~tb ! 2 b2 ! 2 2 ~ bZ 2 ~ta ! 2 b2 ! 2 ! 2

2 ( yt21 @t T #11 0

T

@t0 T #

1 2~ bZ 2 ~ta ! 2 1!

( yt21 «t @t T #11 a

T

1 ~ bZ 1 ~ta ! 2 1!

@t0 T #

(

2 ~ bZ 2 ~ta ! 2 1!

2

@ta T #11

T

2 yt21

*

(1

yt21 «t T

146


*( * @tb T #

yt21 «t

# sup 6 bZ 1 ~tb ! 2 16 sup tb[JT2

1

tb[JT2

T

*(

@t0 T #

@tb T #11

1 2 sup 6 bZ 2 ~tb ! 2 16 sup tb[JT2

tb[JT2

yt21 «t

T

*

@t0 T #

2 ( yt21 @t T #11

1 sup ~ bZ 2 ~tb ! 2 1! 2 sup tb[JT2

12

b

T

tb[JT2

*

T

( yt21 «t @t T #11 0

T

*

sup 6 bZ 2 ~ta ! 2 bZ 2 ~tb !6

tb , ta[JT2

T

2 ( yt21 @t T #11

1

sup 6~ bZ 2 ~tb ! 2 b2 ! 2 2 ~ bZ 2 ~ta ! 2 b2 ! 2 6

0

T

tb , ta[JT2

1 sup 6 bZ 1 ~ta ! 2 16 sup ta[JT2

ta[JT2

*

@ta T #

(1

yt21 «t T

*

*(

@t0 T #

1 2 sup 6 bZ 2 ~ta ! 2 16 sup ta[JT2

ta[JT2

@ta T #11

T

yt21 «t

*

@t0 T #

(

1 sup ~ bZ 2 ~ta ! 2 1! ta[JT2

2

@ta T #11

sup ta[JT2

2 yt21 p

T

&

0+

This derives equation ~16!+ For t 5 t0 , 1 T

RSST ~t0 !

5

T

S( D @t0 T #

T

(1 «t2

2

T

yt21 «t

2

1

@t0 T #

T

(1

2 2~ bZ 2 ~t0 ! 2 b2 !

2 yt21

( yt21 «t @t T #11 0

T

T

1 ~ bZ 2 ~t0 ! 2 b2 ! 2

5

1 T

T

2 ( yt21 @t T #11 0

T

(1 «t2 1 op ~1! 1 op ~1! 1 op ~1!

p

&

s 2+

This derives equation ~17!+ To show that ~10T !RSST ~t! converges uniformly to a random linear function above s 2 for t [ JT1 , it is sufficient to show that


147

~i! ~10T !RSST ~t! converges pointwise to a random linear function above s 2 for all t [ JT1 and p ~ii! sup t[JT1 6~10T !RSST ~t! 2 ~10T !RSST ~t 1 ~10T !!6 & 0 for all t [ JT1 + To show ~i!, by Lemma 3 and ~B+3! in Appendix B, we have @t0 T #

T

1 T

(1 «t2

RSST ~t! 5

T

(1

2 2~ bZ 1 ~t! 2 1!

@t0 T #

(1

yt21 «t 1 ~ bZ 1 ~t! 2 1!

T

2

T

@tT #

@tT #

(

2 2~ bZ 1 ~t! 2 b2 !

2 yt21

@t0 T #11

(

yt21 «t

@t0 T #11

1 ~ bZ 1 ~t! 2 b2 ! 2

T

2 yt21

T

T

(

2 ~ bZ 2 ~t! 2 b2 !

yt21 «t

@tT #11

T @tT #

T

(1 «t2

5

T

n s2 1

(

2 op ~1! 1 op ~1! 2 op ~1! 1 ~ bZ 1 ~t! 2 b2 ! ~1 2 b2 ! 2 ~t 2 t0 1 B 2 ~t0 !!

2

2 yt21

@t0 T #11

2 op ~1!

T

s 2 $ s 2+

1 2 b22

To show ~ii!, use ~B+3! in Appendix B, the triangle inequality, and Lemma 3+ We have

* T RSS ~t! 2 T RSS St 1 T D* 1

sup t[JT1

1

1

T

T

*S S

5 sup 2 bZ 1 t 1 t[JT1

@t0 T #

1 T

D

2 bZ 1 ~t!

(1

D

yt21 «t T @t0 T #

S

S S D DD

1 ~ bZ 1 ~t! 2 1! 2 2 bZ 1 t 1

1

21

T

2

(1

2 yt21

T

@tT #

SS D

1 2 bZ 1 t 1

1

T

2 bZ 1 ~t!

D

(

yt21 «t

@t0 T #11

SS D D

1 2 bZ 1 t 1

T

1

T

2 b2

y@tT # «@tT #11

@tT #

S

S S D DD

1 ~ bZ 1 ~t! 2 b2 ! 2 bZ 1 t 1 2

1

T

2 b2

2

(

@t0 T #11

2 yt21

T T

SS D D

2 bZ 1 t 1

1

T

2 b2

2 y2 @tT #

SS D D

2 bZ 2 t 1

1

T

2 b2

T

SS D

1 bZ 2 t 1

y@tT # «@tT #11 T

*

1

T

2 bZ 2 ~t!

D

( yt21 «t @tT #11 T

T

148


*( * @t0 T #

yt21 «t

#2

1

T

*bZ St 1 T D 2 bZ ~t!* 1

sup t[JT1

1

1

* ( *SS @t0 T #

1

1

T

2 yt21

1

T

1 2 sup

t[JT1

1 2 sup

t[JT1

T sup 6 bZ 1 ~t! 2 16

2

t[JT1

D S 2

* S D * * * S D * * bZ 1 t 1

bZ 1 t 1

1

2 bZ 1 ~t! sup

T

1 T sup

1

(

yt21 «t

@t0 T #11

T

y@tT # «@tT #11

2 b2 sup

*

@tT #

t[JT1

T

t[JT1

T

t[JT1

S D *D D

bZ 1 t 1

1

T

21

2

*

* @tT #

1 sup

T t[J1

S S D D* SS D D * ~ bZ ~t! 2 b ! 1

1 sup bZ 1 t 1 t[JT1

1

T

2

2 bZ 1 t 1

2 b2

2

sup

1 sup

*bZ St 1 T D 2 b * sup *

t[JT1

&

1

2

t[JT1

1

2

2

sup T t[J1

2 yt21

T

T

t[JT1

*bZ St 1 T D 2 bZ ~t!* sup 2

2 b2

T

(

@t0 T #11

2 y@tT #

1 sup

t[JT1

p

2

1

2

*

T

( yt21 «t @tT #11 T

y@tT # «@tT #11

t[JT1

T

*

*

0+


APPENDIX J: DERIVATION OF EQUATIONS ~19!–~21! For b1 5 1, 6 b2 6 , 1 and for t 5 t0 2 ~m0T !, m 5 1,2,3, ++ +, and ~m0T ! r 0 as T r `, we have ~Ja!

1 T

yk202m n s 2 B 2 ~t0 ! 5 Op ~1!;


~Jb!

T

m[$1,2, + +%, ~m0T !r0

~Jc!

m[$1,2, + +%, ~m0T !r0

S

*

k0

(

k 02m11

yt21 «t 5 op ~1!;

*bZ St 2 T D 2 1* 5 o ~1!; m

sup

m

bZ 2 t0 2

*

1

sup

T

1

0

D S

5 a t0 2

p

m T

D

1

k0

11

S S

1 1 2 a t0 2

( «t yt21 k 2m11 0

k0

(

k 02m11

m T

DD

1

2 yt21

2

T

b2 1

( «t yt21 k 11 0

T

(

k 011

2 yt21

2

,

where

S

a t0 2

m T

D

k0

1 5

(

T k0

1 T

k 02m11

(

k 02m11

2 yt21

1

2 yt21

1 T

T

(

k 011

mB 2 ~t0 !

n 2 yt21

mB 2 ~t0 ! 1

1 2 t0 1 B 2 ~t0 ! 1 2 b22

Further, because k0

( «t yt21 k 2m11 0

k0

(

k 02m11

T

5 op ~1!,

( «t yt21 k 11 0

T

(

2 yt21

k 011

5 op ~1!,

2 yt21

we have

S

~Jd! bZ 2 t0 2

m T

D

n

~1 2 b22 !mB 2 ~t0 ! 1 ~1 2 t0 1 B 2 ~t0 !!b2 ~1 2 b22 !mB 2 ~t0 ! 1 1 2 t0 1 B 2 ~t0 !

Therefore

S

bZ 2 t0 2

m T

D

2 b2 n

~1 2 b22 !mB 2 ~t0 !~1 2 b2 ! ~1 2 b22 !mB 2 ~t0 ! 1 1 2 t0 1 B 2 ~t0 !


+

+

+

149

150


Using ~B+1! in Appendix B, ~5a!, ~5b!, and ~5d! in Lemma 3, and ~Ja!–~Jd!, we have 1 T

S

RSST t0 2

m T

( «t2 5

2

k0

yt21 «t

2

T

S( D k 02m

T 1

D

SS

1

2 2 bZ 2 t0 2

k 02m

(1

T

2 yt21

m T

D D 21

(

k 02m11

yt21 «t

T

k0

SS

1 bZ 2 t0 2

m T

(

D D

2 k 2m11 0

21

2 yt21

T T

SS

2 2 bZ 2 t0 2

m T

D D 2 b2

( yt21 «t k 11 0

T

T

SS

1 bZ 2 t0 2

5

T

SS

1 op ~1! 1 op ~1! 1 bZ 2 t0 2

SS

1 bZ 2 t0 2

m T

D D 21

0

T

m T

2 ( yt21 2 k 11

D D 2 b2

0

T

~1 2 t0 1 B 2 ~t0 !!~1 2 b2 ! 2 ms 2 B 2 ~t0 ! ~1 2 b22 !mB 2 ~t0 ! 1 1 2 t0 1 B 2 ~t0 !

This derives equation ~20!+ For t 5 t0 1 ~m0T !, we have ~Je!

1

sup m[$1,2, + +%, ~m0T !r0

~Jf !

1 T

k 01m

2 n ( yt21 k 11

T

*(

k 01m k 011

*

yt21 «t 5 op ~1!;

s 2 B 2 ~t0 !~1 2 b22m ! 1 2 b22

0

by ~5d! in Lemma 3; ~Jg!

sup m[$1,2, + +%, ~m0T !r0

~Jh!

1 T

T

1 T 1

*

T

( yt21 «t k 1m11 0

T

*

5 op ~1!;

2 2 # yt21 5 Op ~1! ( yt21 T k( k 1m11 11 0

2 b2

2 ( yt21 2 k 2m11

T

n s2 1

D D

0

T

k0

T

(1 «t2

m T

2 ( yt21 2 k 11

0

+

1 op ~1!

STRUCTURAL CHANGE IN AR(1) MODELS by ~5d! in Lemma 3; ~Ji! ~Jj!

*bZ St 1 T D 2 1* 5 O S T D; m *bZ St 1 T D 2 b * 5 o ~1!+ m

sup m[$1,2, + +%, ~m0T !r0

sup m[$1,2, + +%, ~m0T !r0

151

1

0

2

0

1

p

2

p

Using ~B+3! in Appendix B, by ~5a!–~5c! of Lemma 3 and ~Je! to ~Jj! given previously, we can write the residual sum of squares as follows: k0

T

1 T

S

RSST t0 1

m T

D

( «t2 5

1

SS

m

2 2 bZ 1 t0 1

T

T

D D 21

(1 yt21 «t T

k0

SS

1 bZ 1 t0 1

m T

D D 21

2

2 (1 yt21

T k 01m

SS

2 2 bZ 1 t0 1 k 01m

(

3

k 011

m T

D D 2 b2

S(

(

yt21 «t

k 011

T

T

2

D

m T

D D

2

2 b2

2

T

2 yt21

SS

1 bZ 1 t0 1

k 01m11

yt21 «t

T

T

2 ( yt21 k 1m11 0

k 01m

T

5

(1 «t2

SS

m

1 op ~1! 1 op ~1! 1 op ~1! 1 bZ 1 t0 1 T T 1 op ~1!

n s2 1

D D 2 b2

(

2 k 11 0

2 yt21

T

~1 2 b2 !s 2 B 2 ~t0 !~1 2 b22m ! 1 1 b2

+


APPENDIX K: PROOF OF THEOREM 4 It is not difficult to show that t[ T is T-consistent by using a similar proof as in Appendixes C and G+ We want to prove a stronger result that lim Tr` Pr~ kZ Þ k 0 ! 5 0+

152


Because kZ # k 0 1 Op ~1!, for any h . 0, there exists an M , ` such that Pr~6 kZ 2 k 0 6 . M ! , h+ Therefore, Pr~ kZ Þ k 0 ! 5 Pr~6 kZ 2 k 0 6 . M ! 1 Pr~6 kZ 2 k 0 6 # M # h 1 Pr~6 kZ 2 k 0 6 # M M

#h1

S S D S S D

( Pr m51

M

1

( Pr m51

and

RSST t0 2

RSST t0 1

m

T

m

T

kZ Þ k 0 !

and

kZ Þ k 0 !

D

2 RSST ~t0 ! , 0

D

2 RSST ~t0 ! , 0 +

Let h 1~m! and h 2 ~m! be defined as in equations ~20! and ~21!, respectively+ The preceding probability is equal to M

5h1

M

( Pr~h1~m! 1 D1T , 0! 1 m51 ( Pr~h2 ~m! 1 D2T , 0!, m51

where

S S

D 1T 5 RSST t0 2 D 2T 5 RSST t0 1

m T m T

D D

2 RSST ~t0 ! 2 h 1 ~m! 5 op ~1!, 2 RSST ~t0 ! 2 h 2 ~m! 5 op ~1!+

Because h 1~m! and h 2 ~m! are increasing with m, this implies every individual probability given earlier is declining with m+ Thus, the preceding term is bounded by h 1 M Pr~h 1~1! 1 D 1T , 0! 1 M Pr~h 2 ~1! 1 D 2T , 0!+ Further, because h 1~1! and h 2 ~1! are positive and of order Op ~1!, there exists a T large enough such that Pr~h 1~1! 1 D 1T , 0! # h and Pr~h 2 ~1! 1 D 2T , 0! # h for any h . 0+ Therefore for any h . 0, Pr~ kZ Þ k 0 ! # ~2M 1 1!h for all large T+ Because M is finite, we have lim Tr` Pr~ kZ 5 k 0 ! 5 1 and plim bZ 1~ t[ T ! 5 plim bZ 1~t0 ! 5 b1 5 1, plim bZ 2 ~ t[ T ! 5 plim bZ 2 ~t0 ! 5 b2 + These prove the consistency of bZ 1 and bZ 2 + To find the limiting distributions of bZ 1~ t[ T ! and bZ 2 ~ t[ T !, note that for any given real value x, lim Pr~T ~ bZ 1 ~ t[ T ! 2 b1 ! # x!

Tr`

5 lim Pr~T ~ bZ 1 ~t0 ! 2 b1 ! # x! lim Pr~ kZ 5 k 0 ! Tr`

Tr`

1 lim Pr~T ~ bZ 1 ~ t[ T ! 2 b1 ! # x6 kZ Þ k 0 ! lim Pr~ kZ Þ k 0 ! Tr`

Tr`

5 lim Pr~T ~ bZ 1 ~t0 ! 2 b1 ! # x!+ Tr`

Thus, bZ 1~ t[ T ! and bZ 1~t0 ! have the same asymptotic distribution+ Similarly, bZ 2 ~ t[ T ! and bZ 2 ~t0 ! have the same asymptotic distribution+ By ~5a!–~5d! in Lemma 3, we have


153

@t0 T #

(1

d

T ~ bZ 1 ~ t[ T ! 2 1! 5 T ~ bZ 1 ~t0 ! 2 1! 5

«t yt21 0T

@t0 T #

(1

B 2 ~t0 ! 2 t0

n

2

2 yt21 0T 2

E

,

t0

B2

0

T

!T ~ bZ 2 ~ t[ T ! 2 b2 ! 5d !T ~ bZ 2 ~t0 ! 2 b2 ! 5

( «t yt21 0!T @t T #11 0

T

(

@t0 T #11

n

2 yt21 0T

O !1 2 b B~1! 2 2

1 2 t0 1 B 2 ~t0 !

,

where B~{! and B~{! O are defined in Section 2+ These show the limiting distributions of bZ 1~ t[ T ! and bZ 2 ~ t[ T ! for a fixed magnitude of break+ These derive the limiting distribution of bZ 1 and bZ 2 for fixed magnitude of break+ To derive the limiting distribution of t[ T for shrinking break, we fix b1 at one and let b2T 5 1 2 @10!Tg~T !# , where g~T ! . 0, with g~T ! r ` and g~T !0!T r 0 as T r `+ Let y be a finite constant and B~{!, B1~{!, and B2 ~{! be defined as in Section 2+ For t 5 t0 1 y@ g~T !0T # and y # 0, we have ~i! L T ~t! 5 op ~1!; @t0 T #

~ii! ~ b2T 2 1!

T

2 ( yt21 ( yt21 «t @tT #11 @t T #11 0

T

52

2 ( yt21 @tT #11

Op ~Tg~T !!Op ~T !

!Tg~T !Op ~T

2

!

5 Op

S! D g~T ! T

5 op ~1!; T

~iii!

2 ( yt21 @t T #11

1;

&

T

(

@tT #11

~iv!

p

0

2 yt21 @6 y6 g~T !#21

1

(0

!Tg~T ! 5 5

yk02t21 «k02t @6 y6 g~T !#21

1

(0

!Tg~T ! yk 0

@6 y6 g~T !#21

(0

52

yk 0

S!

1

!T !g~T !

D

t

yk0 2 ( «k02i «k02t i50

@6 y6 g~T !#21

1

!T !g~T ! 3

S

(0

«k02t 2

g~T !

!T

t

1

( «k 2i «k 2t g~T ! i50 0

0

D

1 g~T !

@6 y6 g~T !#21

(0

~2«k02t ! 2 Op

S! D g~T ! T

n 2s 2 B~t0 !B1 ~2y!;

154


~v!

@6 y6 g~T !#21

1

(0

Tg~T ! 5

yk202t21 @6 y6 g~T !#21

1

(0

Tg~T !

5 6y6 1 5 6y6

yk20

22

!

t

yk0 2 ( «k02i i50

g~T ! yk0

T g~T !

1

T

g~T !

yk20

S

T

!T

1 Op

2

@6 y6 g~T !#21

@6 y6 g~T !#21

(0

D S!

(0

S!

( «k 2i g~T ! i50

t

1

( «k02i

g~T !

S! D S D g~T !

t

1

i50

g~T !

1 Op

T T T Thus, equation ~B+2! in Appendix B becomes

D

0

2

D

1 g~T !

n 6y6s 2 B 2 ~t0 !+

RSST ~t! 2 RSST ~t0 ! @t0 T #

52

2

@t0 T #

(

5 22~ b2T 2 1!

@tT #11

yt21 «t 1 ~ b2T 2 1! 2

(

@tT #11

@6 y6 g~T !#21

(0

!Tg~T !

~2yk02t21 «k02t ! 1

2 yt21 1 op ~1! @6 y6 g~T !#21

1 g~T !T

(0

yk202t21 1 op ~1!

n 22s 2 B~t0 !B1 ~6y6! 1 6y6s 2 B 2 ~t0 !+ Similarly, for t 5 t0 1 @yg~T !0T # and y . 0, by ~B+4! in Appendix B and the facts that ~vi! L T ~t! 5 op ~1!, @t0 T #

2 ~vii! ~ b2T 2 1!

(1

@tT #

yt21 «t

(

@t0 T #11

2 yt21

52

@tT #

2 (1 yt21

Op ~T !Op ~Tg~T !!

!Tg~T !Op ~T 2 !

5 Op

5 op ~1!, @t0 T #

~viii!

(1

2 yt21

p

@tT #

(1

&

1,

2 yt21

equation ~B+4! in Appendix B becomes RSST ~t! 2 RSST ~t0 ! @tT #

5 2~ b2T 2 1! 52

2

(

@t0 T #11

@yg~T !#21

(0

!Tg~T !

5 22

yk 0

@tT #

yt21 «t 1 ~ b2T 2 1! 2

1

!T !g~T !

yk01t «k01t11 1

Tg~T !

«k01t11 1 y

n 22s 2 B~t0 !B2 ~y! 1 ys 2 B 2 ~t0 !+

yk20 T

2 yt21 1 op ~1!

@yg~T !#21

1

@yg~T !#21

(0

(

@t0 T #11

(0

1 op ~1!

yk201t 1 op ~1!

S! D g~T ! T


155

Applying the continuous mapping theorem for argmax functionals, we have ~1 2 b2T ! 2 T 2 ~ t[ T 2 t0 ! 5 d

&

T g~T !

~ t[ T 2 t0 ! 5 y[ 5 Arg min $RSST ~t! 2 RSST ~t0 !%

H S

y[R

Arg min 22s 2 B 2 ~t0 ! y[R

5 Arg max y[R

B * ~y! B~t0 !

2

1 2

S D

B * ~y! B~t0 !

2

1 2

6y6

DJ

6y6 ,

where B * ~y! is a two-sided Brownian motion on R defined to be B * ~y! 5 B1~2y! for y # 0 and B * ~y! 5 B2 ~y! for y . 0+ n