Directory UMM :Data Elmu:jurnal:J-a:Journal of Econometrics:Vol95.Issue1.2000:
Journal of Econometrics 95 (2000) 97}116
Detection of change in persistence of a linear
time series
Jae-Young Kim!,",*
!Department of Economics, State University of New York, Albany, NY 12222, USA
"Department of Economics, Hong Kong University of Science and Technology, Hong Kong
Received 1 January 1997; received in revised form 1 February 1999; accepted 1 April 1999
Abstract
This paper studies how to detect structural change characterized by a shift in persistence of a time series. In particular, we are interested in a process shifting from stationarity
to nonstationarity or vice versa. A general linear process is considered that includes an
ARMA process as a special one. We derive a statistic for testing the occurrence of such
a change and investigate asymptotic behavior of it. We show that our test has power
against fairly general alternatives of change in persistence. A Monte Carlo study shows
that our test has reasonably good size and power properties in "nite samples. We also
discuss how to estimate the unknown period of change. We apply our test to two
examples of time series, the series of the U.S. in#ation rate and the series of U.S. federal
government's budget de"cit in the postwar period. For these two series we have found
strong evidence of structural change from stationarity to nonstationarity. ( 2000
Elsevier Science S.A. All rights reserved.
JEL classixcation: C1; C22; C5
Keywords: Change in persistence; Unknown change period
* Department of Economics, Hong Kong University of Science and Technology, Clear Water
Bay, Kowloon, Hong Kong. Fax: #852-2358-2084.
E-mail address: [email protected] (J.-Y. Kim)
0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 3 0 4 - 4 0 7 6 ( 9 9 ) 0 0 0 3 1 - 7
98
J.-Y. Kim / Journal of Econometrics 95 (2000) 97}116
1. Introduction
Recent literature on nonstationary time series addresses the importance of
information on di!erent degrees of persistence in linear time series for
econometric inference. In this paper, we consider a stochastic process that may
undergo a shift in persistence. In particular, we are interested in a process
shifting from a lower level persistence to a higher level persistence, an example of
which is a shift from stationarity to a unit root, or vice versa. We study a formal
theory for detecting such a shift in this paper.
Examples of time series having change in persistence are found in De Long
and Summers (1988), real output series of the U.S. and the European countries.
De Long and Summers (1988) conjectured that these series shifted from stationarity to a unit root after WWII. They performed informal tests to "nd the
evidence in favor of their conjecture. Another example is in Hakkio and Rush
(1991) where they considered the U.S. federal government budget de"cit to "nd
that the de"cit process underwent a change from stationarity to a unit root.
However, Hakkio and Rush (1991)'s work is also based on informal inference.
Bai (1992) stressed the importance of change in persistence of a process but did
not discuss how to formally detect such a change.
Statistical inference on persistence change involves consideration of two
fundamentally di!erent processes connected at a point of time, especially for
change from stationarity to a unit root or vice versa. Di$culty arises in this case
because usual inferential statistics do not have standard properties. Our approach in this paper is based on residual-based tests for stationarity that are
considered in Nyblom and Makelainen (1983), Rogers (1986), Nyblom (1986),
Nabeya and Tanaka (1988), McCabe and Leybourne (1988) and Kwiatkowski et
al. (1992). We derive a test statistic by applying the analyses of these authors to
the case of persistence change with the date of change being unknown. We apply
our statistic to the following three di!erent ways for handling the problem of the
change period being unknown. First, a maximum-Chow-type test considered in
Davies (1977), Hawkins (1987), Kim and Siegmund (1989), and Andrews (1993).
Second, Hansen (1991)'s mean score test, and third, Andrews and Ploberger
(1994)'s mean-exponential test.
We investigate asymptotic behavior of our test statistics. We show that our
tests have power against fairly general alternatives of change in persistence.
Also, we discuss how to consistently estimate the unknown period of change.
A Monte Carlo study shows that our testing procedures have reasonably good
size and power properties in "nite samples. A Monte Carlo study also shows
that, among the three tests the mean score test has the smallest size distortion in
our case.
We apply our test to two examples of time series, the time series of the U.S.
in#ation rate and the series of U.S. federal government's budget de"cit in the
postwar period. For these two series we have found strong evidence of change
J.-Y. Kim / Journal of Econometrics 95 (2000) 97}116
99
from stationarity to nonstationarity. For quarterly data on the U.S. in#ation
rate the estimated period of change is 1973:3 and for the U.S. federal government's budget de"cit it is 1968:4.
Section 2 discusses how to model a change in persistence of a linear process.
In Section 3 we develop test procedures for detecting change in persistence of
a linear process and investigate some asymptotic properties of the tests. Also, in
Section 3 we discuss how to estimate the date of change. Section 4 provides some
simulation results, and Section 5 applies our procedures to two time-series data.
2. Modeling change in the persistence
To study a stochastic process My N that undergoes a shift in persistence, we
t
consider the following hypotheses: The null hypothesis H that y maintains
0
t
stationarity of constant persistence throughout the sample period; An alternative hypothesis H that y maintains stationarity of constant persistence until
1
t
some period, after which it becomes a process of higher persistence such as a unit
root. Or, an alternative hypothesis H@ that y is a process of relatively high
1
t
persistence until some period, after which it becomes a process of lower persistence. The null hypothesis can be described as
H : y "r #z , for t"1,2,¹,
(2.1)
0 t
0
t
where r is a constant, and z is a stationary variable satisfying the following
0
t
(uniform mixing) regularity conditions:
Assumption 1. The process Mz N= is such that
t 0
(i) E(z)"0;
(ii) EDzDc`e(R for some c'2;
(iii) Mz N= is u-mixing with mixing coezcients u such that += u1~2@c(R;
t 0
m
m/1 m
(iv) The long-run variance p2"+= E[z z@ ] exists;
z
j/0
j`1 1
(v) lim 6 var(¹~1@2+sT z )"sp2; lim 6 var(¹~1@2+T
z )"(1!s)p2
T =
t/1 t
t/sT`1 t
z
T =
z
each s3(0, 1).
The above conditions allow for a broad class of weakly dependent time series
and have been used by Phillips (1987), Phillips and Perron (1988) and Phillips
and Solo (1992), among others, to derive limiting behavior of a stochastic
process. Alternatively, for the null hypothesis, one may consider a trend stationary process:
(2.2)
Hc : y "ct#r #z , for t"1,2,¹.
0 t
0
t
Now, consider an alternative hypothesis that y maintains stationarity of
t
constant persistence until t"[q¹] for q3(0, 1) where [q¹] is the integer part of
100
J.-Y. Kim / Journal of Econometrics 95 (2000) 97}116
q¹, but after t"[q¹] it becomes a process of higher persistence such as a unit
root:
H :
1
y "r #z , for t"1,2,[q¹],
t
0
t,0
(2.3)
y "r #z , for t"[q¹]#1,2,¹,
t
1
t,1
where z is a stationary process satisfying conditions in Assumption 1; z is
t,0
t,1
a process of higher persistence than z , and r and r are constants.
t,0
0
1
On the other hand, consider an alternative hypothesis H@ that y is a process
1
t
of relatively high persistence until t"[q¹], but after t"[q¹] it becomes
a process of lower persistence:
H@ :
1
y "r #z , for t"1,2,[q¹],
t
1
t,1
(2.3@)
y "r #z , for t"[q¹]#1,2,¹,
t
0
t,0
where z , i"0, 1, are as in (2.3).
t,i
Likewise, we can describe an alternative hypothesis for Hc in (2.2) that
0
corresponds to change in H (2.3) but with a trend:
1
Hc : y "ct#r #z , for t"1,2,[q¹],
1
t
0
t,0
(2.4)
y "ct#r #z , for t"[q¹]#1,2,¹.
t
1
t,1
Notice that y "ct#r #z
with z "z
#u is equivalent to
t
1
t,1
t,1
t~1,1
t
y "c#y #u , a unit root process with drift. Also, we can describe an
t
t~1
t
alternative hypothesis Hc@ for the null Hc that corresponds to the change in
0
1
H@ (2.3@) but with a trend:
1
Hc@: y "ct#r #z ,
for t"1,2,[q¹],
1
t
t
t,1
(2.4@)
y "ct#r #z ,
for t"[q¹]#1,2,¹.
t
0
t,0
3. Test for and estimation of structural change
In this section we discuss how to detect the occurrence of structural change
characterized by the alternative hypothesis (2.3), (2.3@), (2.4), or (2.4@). We derive
a test statistic and investigate asymptotic properties of the test statistic. In
particular, asymptotic behavior of the test statistic is studied both under the null
and under the alternative hypotheses. We "nd that our test has nontrivial power
against fairly general alternative hypotheses of change in persistence. In addition, we discuss how to estimate the unknown change point q.
3.1.1. Test for structural change
We "rst discuss how to test H in (2.1) against H in (2.3) or H@ in (2.3@). Later
1
0
1
on we consider the procedure for testing Hc in (2.2) against Hc in (2.4) or Hc@ in
1
1
0
J.-Y. Kim / Journal of Econometrics 95 (2000) 97}116
101
(2.4@). Let z8 , t"1, 2,2,¹ be the residuals from the regression of y on intercept,
t
t
where the process y is as in H . Also, let S be the following partial sum process:
t
0
t
t
S " + z8 for t"1, 2,2,¹.
(3.1)
t
i
i/1
If a structural change is suspected to occur at time t"[q¹] for q3(0, 1), we can
de"ne the following partial sum processes separately before and after [q¹]:
t
S (q) " + z8
for t"1,2,[q¹],
0,t
i
i/1
(3.2)
t
S (q) "
+ z8
for t"[q¹]#1,2,¹.
1,t
i
i/qT`1
Now, consider the following statistic:
[(1!q)¹]~2+T
S (q)2
*qT+`1 1,t .
N (q)"
(3.3)
T
[q¹]~2+*qT+S (q)2
1 0,t
The statistic (3.3) is a key element in our testing procedure.
In the above statistic N (q) the true value of q is unknown. Under the situation
T
of the true change period being unknown three di!erent ways for testing
structural change can be derived based on N . First, a maximum-Chow-type test
T
as is considered in Davies (1977), Hawkins (1987), Kim and Siegmund (1989),
and Andrews (1993) for testing H against H with unknown break point t"q¹
0
1
is
N (q( ),max N (q),
T
T
q|T
(3.4)
where T is a compact subset of (0, 1). Second, Hansen (1991)'s mean score test is
P
N (q) dq.
(3.5)
T
q|T
Third, Andrews and Ploberger (1994)'s mean-exponential test statistic for N is
T
EN (q),
T
GP
log E exp(N (q)),log
T
q|T
H
exp(N (q)) dq .
T
(3.6)
Theorem 3.1. Suppose that Assumption 1 is true for z under the null hypothesis H .
t
0
Then, under H it is true that
0
(1!q)~2:1
Detection of change in persistence of a linear
time series
Jae-Young Kim!,",*
!Department of Economics, State University of New York, Albany, NY 12222, USA
"Department of Economics, Hong Kong University of Science and Technology, Hong Kong
Received 1 January 1997; received in revised form 1 February 1999; accepted 1 April 1999
Abstract
This paper studies how to detect structural change characterized by a shift in persistence of a time series. In particular, we are interested in a process shifting from stationarity
to nonstationarity or vice versa. A general linear process is considered that includes an
ARMA process as a special one. We derive a statistic for testing the occurrence of such
a change and investigate asymptotic behavior of it. We show that our test has power
against fairly general alternatives of change in persistence. A Monte Carlo study shows
that our test has reasonably good size and power properties in "nite samples. We also
discuss how to estimate the unknown period of change. We apply our test to two
examples of time series, the series of the U.S. in#ation rate and the series of U.S. federal
government's budget de"cit in the postwar period. For these two series we have found
strong evidence of structural change from stationarity to nonstationarity. ( 2000
Elsevier Science S.A. All rights reserved.
JEL classixcation: C1; C22; C5
Keywords: Change in persistence; Unknown change period
* Department of Economics, Hong Kong University of Science and Technology, Clear Water
Bay, Kowloon, Hong Kong. Fax: #852-2358-2084.
E-mail address: [email protected] (J.-Y. Kim)
0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 3 0 4 - 4 0 7 6 ( 9 9 ) 0 0 0 3 1 - 7
98
J.-Y. Kim / Journal of Econometrics 95 (2000) 97}116
1. Introduction
Recent literature on nonstationary time series addresses the importance of
information on di!erent degrees of persistence in linear time series for
econometric inference. In this paper, we consider a stochastic process that may
undergo a shift in persistence. In particular, we are interested in a process
shifting from a lower level persistence to a higher level persistence, an example of
which is a shift from stationarity to a unit root, or vice versa. We study a formal
theory for detecting such a shift in this paper.
Examples of time series having change in persistence are found in De Long
and Summers (1988), real output series of the U.S. and the European countries.
De Long and Summers (1988) conjectured that these series shifted from stationarity to a unit root after WWII. They performed informal tests to "nd the
evidence in favor of their conjecture. Another example is in Hakkio and Rush
(1991) where they considered the U.S. federal government budget de"cit to "nd
that the de"cit process underwent a change from stationarity to a unit root.
However, Hakkio and Rush (1991)'s work is also based on informal inference.
Bai (1992) stressed the importance of change in persistence of a process but did
not discuss how to formally detect such a change.
Statistical inference on persistence change involves consideration of two
fundamentally di!erent processes connected at a point of time, especially for
change from stationarity to a unit root or vice versa. Di$culty arises in this case
because usual inferential statistics do not have standard properties. Our approach in this paper is based on residual-based tests for stationarity that are
considered in Nyblom and Makelainen (1983), Rogers (1986), Nyblom (1986),
Nabeya and Tanaka (1988), McCabe and Leybourne (1988) and Kwiatkowski et
al. (1992). We derive a test statistic by applying the analyses of these authors to
the case of persistence change with the date of change being unknown. We apply
our statistic to the following three di!erent ways for handling the problem of the
change period being unknown. First, a maximum-Chow-type test considered in
Davies (1977), Hawkins (1987), Kim and Siegmund (1989), and Andrews (1993).
Second, Hansen (1991)'s mean score test, and third, Andrews and Ploberger
(1994)'s mean-exponential test.
We investigate asymptotic behavior of our test statistics. We show that our
tests have power against fairly general alternatives of change in persistence.
Also, we discuss how to consistently estimate the unknown period of change.
A Monte Carlo study shows that our testing procedures have reasonably good
size and power properties in "nite samples. A Monte Carlo study also shows
that, among the three tests the mean score test has the smallest size distortion in
our case.
We apply our test to two examples of time series, the time series of the U.S.
in#ation rate and the series of U.S. federal government's budget de"cit in the
postwar period. For these two series we have found strong evidence of change
J.-Y. Kim / Journal of Econometrics 95 (2000) 97}116
99
from stationarity to nonstationarity. For quarterly data on the U.S. in#ation
rate the estimated period of change is 1973:3 and for the U.S. federal government's budget de"cit it is 1968:4.
Section 2 discusses how to model a change in persistence of a linear process.
In Section 3 we develop test procedures for detecting change in persistence of
a linear process and investigate some asymptotic properties of the tests. Also, in
Section 3 we discuss how to estimate the date of change. Section 4 provides some
simulation results, and Section 5 applies our procedures to two time-series data.
2. Modeling change in the persistence
To study a stochastic process My N that undergoes a shift in persistence, we
t
consider the following hypotheses: The null hypothesis H that y maintains
0
t
stationarity of constant persistence throughout the sample period; An alternative hypothesis H that y maintains stationarity of constant persistence until
1
t
some period, after which it becomes a process of higher persistence such as a unit
root. Or, an alternative hypothesis H@ that y is a process of relatively high
1
t
persistence until some period, after which it becomes a process of lower persistence. The null hypothesis can be described as
H : y "r #z , for t"1,2,¹,
(2.1)
0 t
0
t
where r is a constant, and z is a stationary variable satisfying the following
0
t
(uniform mixing) regularity conditions:
Assumption 1. The process Mz N= is such that
t 0
(i) E(z)"0;
(ii) EDzDc`e(R for some c'2;
(iii) Mz N= is u-mixing with mixing coezcients u such that += u1~2@c(R;
t 0
m
m/1 m
(iv) The long-run variance p2"+= E[z z@ ] exists;
z
j/0
j`1 1
(v) lim 6 var(¹~1@2+sT z )"sp2; lim 6 var(¹~1@2+T
z )"(1!s)p2
T =
t/1 t
t/sT`1 t
z
T =
z
each s3(0, 1).
The above conditions allow for a broad class of weakly dependent time series
and have been used by Phillips (1987), Phillips and Perron (1988) and Phillips
and Solo (1992), among others, to derive limiting behavior of a stochastic
process. Alternatively, for the null hypothesis, one may consider a trend stationary process:
(2.2)
Hc : y "ct#r #z , for t"1,2,¹.
0 t
0
t
Now, consider an alternative hypothesis that y maintains stationarity of
t
constant persistence until t"[q¹] for q3(0, 1) where [q¹] is the integer part of
100
J.-Y. Kim / Journal of Econometrics 95 (2000) 97}116
q¹, but after t"[q¹] it becomes a process of higher persistence such as a unit
root:
H :
1
y "r #z , for t"1,2,[q¹],
t
0
t,0
(2.3)
y "r #z , for t"[q¹]#1,2,¹,
t
1
t,1
where z is a stationary process satisfying conditions in Assumption 1; z is
t,0
t,1
a process of higher persistence than z , and r and r are constants.
t,0
0
1
On the other hand, consider an alternative hypothesis H@ that y is a process
1
t
of relatively high persistence until t"[q¹], but after t"[q¹] it becomes
a process of lower persistence:
H@ :
1
y "r #z , for t"1,2,[q¹],
t
1
t,1
(2.3@)
y "r #z , for t"[q¹]#1,2,¹,
t
0
t,0
where z , i"0, 1, are as in (2.3).
t,i
Likewise, we can describe an alternative hypothesis for Hc in (2.2) that
0
corresponds to change in H (2.3) but with a trend:
1
Hc : y "ct#r #z , for t"1,2,[q¹],
1
t
0
t,0
(2.4)
y "ct#r #z , for t"[q¹]#1,2,¹.
t
1
t,1
Notice that y "ct#r #z
with z "z
#u is equivalent to
t
1
t,1
t,1
t~1,1
t
y "c#y #u , a unit root process with drift. Also, we can describe an
t
t~1
t
alternative hypothesis Hc@ for the null Hc that corresponds to the change in
0
1
H@ (2.3@) but with a trend:
1
Hc@: y "ct#r #z ,
for t"1,2,[q¹],
1
t
t
t,1
(2.4@)
y "ct#r #z ,
for t"[q¹]#1,2,¹.
t
0
t,0
3. Test for and estimation of structural change
In this section we discuss how to detect the occurrence of structural change
characterized by the alternative hypothesis (2.3), (2.3@), (2.4), or (2.4@). We derive
a test statistic and investigate asymptotic properties of the test statistic. In
particular, asymptotic behavior of the test statistic is studied both under the null
and under the alternative hypotheses. We "nd that our test has nontrivial power
against fairly general alternative hypotheses of change in persistence. In addition, we discuss how to estimate the unknown change point q.
3.1.1. Test for structural change
We "rst discuss how to test H in (2.1) against H in (2.3) or H@ in (2.3@). Later
1
0
1
on we consider the procedure for testing Hc in (2.2) against Hc in (2.4) or Hc@ in
1
1
0
J.-Y. Kim / Journal of Econometrics 95 (2000) 97}116
101
(2.4@). Let z8 , t"1, 2,2,¹ be the residuals from the regression of y on intercept,
t
t
where the process y is as in H . Also, let S be the following partial sum process:
t
0
t
t
S " + z8 for t"1, 2,2,¹.
(3.1)
t
i
i/1
If a structural change is suspected to occur at time t"[q¹] for q3(0, 1), we can
de"ne the following partial sum processes separately before and after [q¹]:
t
S (q) " + z8
for t"1,2,[q¹],
0,t
i
i/1
(3.2)
t
S (q) "
+ z8
for t"[q¹]#1,2,¹.
1,t
i
i/qT`1
Now, consider the following statistic:
[(1!q)¹]~2+T
S (q)2
*qT+`1 1,t .
N (q)"
(3.3)
T
[q¹]~2+*qT+S (q)2
1 0,t
The statistic (3.3) is a key element in our testing procedure.
In the above statistic N (q) the true value of q is unknown. Under the situation
T
of the true change period being unknown three di!erent ways for testing
structural change can be derived based on N . First, a maximum-Chow-type test
T
as is considered in Davies (1977), Hawkins (1987), Kim and Siegmund (1989),
and Andrews (1993) for testing H against H with unknown break point t"q¹
0
1
is
N (q( ),max N (q),
T
T
q|T
(3.4)
where T is a compact subset of (0, 1). Second, Hansen (1991)'s mean score test is
P
N (q) dq.
(3.5)
T
q|T
Third, Andrews and Ploberger (1994)'s mean-exponential test statistic for N is
T
EN (q),
T
GP
log E exp(N (q)),log
T
q|T
H
exp(N (q)) dq .
T
(3.6)
Theorem 3.1. Suppose that Assumption 1 is true for z under the null hypothesis H .
t
0
Then, under H it is true that
0
(1!q)~2:1