- Email: [email protected]
Diagnostic test for structural change in cointegrated regression models
~
economics letters
Ek~EVlER
Economics Letters 50 (1996) 179-187
Diagnostic test for structural change in cointegrated regression models Kang Hao, Brett Inder* Department of Econometrics, Monash University, Clayton, Victoria 3168, Australia Received 3 April 1995; accepted 19 July 1995 Abstract
We derive the asymptotic distribution of the OLS-based CUSUM test in the context of cointegrated regression models and tabulate its critical values. It is also found that the test has non-trivial local power irrespective of the particular type of structure change. Keywords: Structural change; Cointegration; Fully modified ordinary least squares; CUSUM test JEL classification: C22; C52
1. Introduction
Recently, H a n s e n (1992) examined the problem of testing for structural instability in the context of cointegrated regression models. Making use of the fully modified ( F M ) OLS estimation m e t h o d for Phillips and H a n s e n (1990), H a n s e n (1992) derived the asymptotic distribution of various test statistics against different alternatives of interest. A t the same time, Quintos and Phillips (1993) proposed a Lagrange Multiplier ( L M ) test against the r a n d o m walk alternative that corresponds to Hansen's approach. Both Quintos and Phillips' (1993) and Hansen's (1992) approaches specify precisely the form of structural break u n d e r the alternative hypothesis. H o w e v e r , it is sometimes desirable to investigate generally the stability properties of the cointegrating model without specifying the possible alternatives. In this case, testing for structural change can be m o r e or less r e g a r d e d as a diagnostic test. In this paper we consider a test designed for this purpose. It is the C U S U M test with OLS residuals that was originally suggested by Ploberger and Kriimer (1992) for stationary regressors. In addition to deriving the asymptotic distribution of the OLS-based C U S U M test, we also investigate its local power p e r f o r m a n c e . A n interesting
* Corresponding author. Tel. + 61-3-905-2303; fax: + 61-3-905-5474; e-mail: Brett. Inder@BusEco. monash.edu, au. 0165-1765/96/$12.00 © 1996 Elsevier Science S.A. All rights reserved S S D I 0165-1765(95)00750-4
K. Hao, B. Inder / Economics Letters 50 (1996) 179-187
180
finding is that the test has non-trivial local power irrespective of the particular type of structural change. Considering that the OLS-based CUSUM test can be calculated easily from the residuals, it is therefore recommended as a routine test for structural change in the context of cointegrated regression models. The paper is organized as follows. Section 2 sets up the structure of the model. Section 3 investigates the limiting distribution of the test statistic. Section 4 examines the local power performance of the test. Concluding remarks are given in Section 5.
2. The cointegrated regression model Consider the cointegrated regression model
y,=Ax,+u,,
t=l,...,T,
(1)
t
where the process x, = (xl,, xj,)' is determined by the equations Xlt
-~-
kit
, o
Xzt = Hlklt + Ilzkzt -}- X2t 0 X2t
,
0 -~- X 2 t - - 1 "[- Ot "
Let us define
= (.,, v ; ) ,
k; = (k'l,, k i t ) ,
where ~', is a k + 1 dimensional sequence of stationary disturbances with mean zero, while the elements of k, are non-negative integer powers of time of order p. That is, k[ = (1, t, 12, . . . ,
tP).
This model is fairly general and encompasses a wide range of applications (see Hansen (1992) for details). It also turns out to be especially convenient for developing the asymptotic distribution of our test statistic. Suppose ~', = [u,, v[] satisfies the multivariate invariance principle as set out by Phillips and Durlauf (1986). Let Rr(r ) =R[rrl ~',, then /2R[rr]:@W(r ) = (Wo(r), Wl(r)) , where W(r) is a k + 1 dimensional Brownian motion and is partitioned in conformity with ~',, with covariance matrix: i/t=
1//'10 qtlJ =limr_~=T - ~E
~'
,~,
where
.~ =lim T-1E A =limT__~.z-1
~'~'~
=
(TO
.a~O1
-rio
.V, J '
, [.'o .o,] t=2j=,ZE(;fl,)= LA10 A, J '
= E + a + a' ,
181
K. Hao, B. lnder / Economics Letters 50 (1996) 179-187
where ~ , Z and A are partitioned in conformity with fit. We also define
a0 a01]
+A=
A=Z
A1°
A1 j .
We denote consistent estimators of ~ and A as ~ a n d / i respectively. We partition ~ and z~ as ~ and A, and then set ('0 ^ 20.1 .
i~^20 . 1/~01 1~ . 11 XPl0 . ,
/~ 10 +
/~10
A^ 1 1/I ^ - 1 1tf10 ^ "
We define the transformed dependent variable as + ^t -1 Yt = Yt -- 1/fOllff[fl
Vt"
The cointegrated regression model (1) can be written as +
y,
+
=Ax t+u,
(2)
,
with the FM estimator of A developed by Phillips and Hansen (1990) given by A q-
=
y,
q-x tI
!
T(O,
-
XtX t
(3)
,
with the FM residuals u^ , + = y ~ + - , 3 , +X t
•
In addition to the FM estimator, we can also use other asymptotically equivalent such as the spectral estimators of Phillips (1988), the maximum likelihood estimator (1991), and the dynamic OLS estimator of Saikkonen (1991), among others. This change the asymptotic distribution of our test statistic. Before going to the next section, we modify model (1) to incorporate possible instability by allowing A to depend on time:
estimators of Phillips would not parameter
y, = Atxt + u,.
(4)
The null hypothesis can be formulated as Ho'A 1=A e .....
Ar=A
,
with the alternative being HI: at least one equality does not hold.
3. Asymptotic distribution of the test statistic
Let [.] denote 'integer part', and define the FM OLS-based CUSUM test statistic: B(r)(r) _
1
tr~l
w0.1X/T,=IZ / ~ ; ( T )
,
(5)
182
K. Hao, B. Inder / Economics Letters 50 (1996) 179-187
We reject H 0 for large values of sup0<,<, simply comprises standardized partial sums of the FM OLS residuals. This is identical to the OLS-based CUSUM test proposed by Ploberger and Kr/imer (1992), except that OLS residuals are replaced by FM OLS residuals and the estimated error variance is replaced by the long-run variance estimate ^2 0)0.1
•
For convenience, we denote
ST(r)=S
rj=
xA2-
,
and k(r)= (1, r , . . . , r P ) '. Also, we denote
X(r) = ( k ( r ) \Bl(r)] ' S(r)= fo X dB0.1 and M(r)= fo XX', where S01(r ) and Bl(r ) are independent standard Brownian motions with dimensions 1 and k, respectively. Then the asymptotic distribution of the FM OLS-based CUSUM test is found as a direct consequence of theorem 2 of Hansen (1992) in which he sets up the asymptotic distribution of Sr(r ). Observe that E ~rrl t~t+ is simply the first element of the vector Sr(r ). Therefore, by theorem 2 of Hansen (1992), we have ]
d~O.l~ ¢ / T
B(r)(z)-
[T~'¿ t~: l
d
t~ +(r)~$1(~.),
(6)
where S* (z) is the first element of the vector
S* (T) = S(r) - M(z)M(1)- ' S(1) , which is free from any nuisance parameters.
Theorem 1. The FM OLS-based CUSUM statistics (5) for parameter constancy in model (4) has an asymptotic distribution given by S* (r). A couple of special cases are worth considering. In particular, if kl, = 1 and k2, = 0, then (4) can be expressed as
y,
= ol t .gff [3tx t _~_ u t ,
t= 1,..., X t ~- X t - 1
T.
(7)
+ Vt '
In this case, x, is I(1) without drift. To get S*(z), we denote
P(r) = 1
-
;(f )l B~
0
818 1
B1,
Q(t) = B 1 -
; 0
81 .
I
V
~.~
I
..
~"
~
..
(~
~
~
I
~
I
-.~
I
I
I
~
~ ~__~
'
~ ~
I
~-X-
~
~
'
"I
(%
~'~
K. Hao,B. Inder / EconomicsLetters50 (1996)179-187
184
We observe that
f (r,B;) M(r) =
o
;() ;() .0 0 r B1
r (r, B1) B1
1
(jP)-' ..#lv'l,lVlltJl.)-i =
-(jP)-lf
-1
-
(r,B',)[!(B1)(r,B~)]
1
(;)s(.) OO'
d
B(r)(r) ~ S ~ ' ( , ) = Bo.,(r ) - ~
-1
(/)
BI
QQ'
o
Thus,
-1"
(!)-1; ; L!()]-'()}r P
1-
(r, B ,,)
r B,
(r,B~)
0 0 ¢.r (f)--1 /( ) ;( B,r ) ; dBo.1 ] r dBol-- j (r, B'I) QQ' B, 0
=No.a0-)-1-
0
0
B1
dBo. 1
0
(!)-1 i (1 2 ; ,) (!)-1; P PdBo. 1 - 7~', B1 aa' QdBo. 1 . 0 0 0
(10)
Since the asymptotic distribution of B(r)0" ) does not depend on any nuisance parameters, we can obtain the critical values of the FM OLS-based C U S U M test by calculating suplB(r)(r)l for 0 < r < 1. In particular, we tabulate the critical values for models (7) and (9), respectively. They are found by simulation using a G A U S S program with a sample size of 1000 and 10,000 replications for one to five explanatory variables. The results are given in Tables 1 and 2, respectively. Table 1 Asymptotic critical values of the FM OLS-based CUSUM test for model (7) Number of regressors (excluding coiistant)
10%
5%
1%
1 2 3 4 5
1.0477 0.9342 0.8381 0.7687 0.7122
1.1684 1.0413 0.9336 0.8475 0.7890
1.4255 1.2561 1.1782 1.0383 0.9680
185
K. Hao, B. lnder / Economics Letters 50 (1996) 179-187
Table 2 Asymptotic critical values of the FM OLS-based CUSUM test for model (9) Number of regressors 10% 5% (excluding constant) 1 0.7678 0.8332 2 0.7214 0.7855 3 0.6796 0.7385 4 0.6421 0.6951 5 0.6173 0.6681
1% 0.9694 0.9162 0.8761 0.8270 0.7787
4. Local p o w e r p e r f o r m a n c e
For convenience, we consider the local power of the test under the condition that no constant term is included in the regression and {x,} are strictly exogenous. In this case, only OLS estimation is required to establish valid asymptotic distributions of B~r)(r). It is easy to show that under H 0 and according to theorem 3.1 of Park and Phillips (1988): B~)(~)_
[r,] 1 frail tr, l ]d 1 E a,u , - Y, ( ~ - / 3 ) x , ~B*(~) &oVT ,=1 tb0VT 1
(f)1;
= Bo.,(r ) - [. B',
BIB [
o
o B~
dBo. , ,
(11)
where a, are the OLS residuals. Suppose & r =/30 + T-lg(r), where r = tiT and g(r) can be any step function or uniform limit of step functions. Let us denote y, =/3,.rx , + u,. Under H0, /3 is estimated by ~ = Z YtXt'
XtXt'
,
1
with residuals t~t = y , - / 3 x , .
B¢T)(r ) _
1
[Trl
T h u s w e have
ff'~ [U, "~-/30Xt-~- r - ' g ( r ) x , -
&oVT 1 1 t~l[
- - -
tboVT 1
d 1[
~-'-~o
( Ut + T - l g ( r ) x g -
J u~l1/2If
W°(r) +
g(r)W1 -
o
=B*(r)+~-TZ
g(r)B 1 -
f3¢T)xt]
r x,x: ~ ,)(~ g(r) ~ - - T - + ugct
1
(! ; o
1
;
g(r)WIW'l +
g(r)BiB;
)-1 x~x;
o
/(! o
)~(i )-1 ]
W, d W ' o
] xt
W1W;
)--11 W1
o
BIB;
B1 .
(12)
K. Hao, B. lnder / Economics Letters 50 (1996) 179-187
186
When H 0 is true, /3t.r =/30, we must have g(r) = 0 for r E (0, 1). The OLS-based C U S U M test converges to its null distribution. Under H~, g(r)SO, (10) is well defined and distinct from the null distribution. The test thus has non-trivial power irrespective of the particular type of structural change. This conclusion is significant since it is in contrast to the result derived by Ploberger and Krfimer (1992) in which they found that when the regressors are stationary, the OLS-based CUSUM test has only trivial local power against structural changes that are orthogonal to the mean regressor.
5. An application Hansen (1992) investigated the stability properties of three postwar US interest rates. In particular, he fitted a cointegrated regression model between the federal funds (FF) rate and the 90-day treasury bill (TB3) rate and found some evidence of structural change using his supF, m e a n F and L c tests. Here we apply the FM OLS-based CUSUM test to test the same model. We obtain the following regression result: TB3, = 0.487 = 0.830 F F , , (0.14)
(0.02)
sup ]B(r)(z)I = 1.130,
p-value =0.063.
0<~'<1
Our result supports Hansen's conclusion. The test rejects the null hypothesis of no structural change at the 10% level. Although the test is not significant at the 5% level, its p-value is close to 0.05. Such a result is quite similar to that of the L c test of Hansen (1992), while the FM OLS-based CUSUM test requires much less computational effort.
6. Concluding remarks In this paper we have examined the OLS-based CUSUM test in the context of the cointegrated regression model. In addition to deriving its asymptotic distribution, which is nuisance parameter free, we also show that it has non-trivial local power irrespective of the form of structural change. This significantly contrasts with its performance under stationary regression models, where it has trivial local power in certain cases. Since the associated residuals are simply a natural by-product of FM estimation of cointegrated regression models, the FM OLS-based CUSUM test can be very easily conducted. We thus recommend it as a routine test for structural change in the context of cointegrated regression models.
Acknowledgements We wish to thank Bruce Hansen for supplying his program to calculate the fully modified least squares estimator and Werner Ploberger for helpful comments on an earlier draft of this paper.
K. Hao, B. Inder / Economics Letters 50 (1996) 179-187
187
References Hansen, B., 1992, Tests for parameter instability in regressions with I(1) processes, Journal of Business and Economic Statistics 10, 321-355. Park, J.Y. and P.C.B. Phillips, 1988, Statistical influence in regressions with integrated process: Part 1, Econometric Theory 4, 468-497. Phillips, P.C.B., 1988, Spectral regression for cointegrated time series, Cowles Foundation Discussion Paper No. 872, Yale University. Phillips, P.C.B., 1991, Optimal inference in cointegrated systems, Econometrica 59, 283-306. Phillips, P.C.B. and S.N. Durlauf, 1986, Multiple time series regression with integrated processes, Review of Economic Studies 53 473-496. Phillips, P.C.B. and B. Hansen, 1990, Statistical inference in instrumental variables regression with I(1) processes, Review of Economic Studies 57, 99-125. Pioberger, W. and W. Kr~imer, 1992, The CUSUM test with OLS residuals, Econometrica 60, 271-285. Quintos, C.E. and P.C.B. Phillips, 1993, Parameter constancy in cointegrating regressions, Empirical Economics 18, 675-706. Saikkonen, P., 1991, Asymptotically efficient estimation of cointegration regressions, Econometric Theory 7, 1-20.
economics letters
Ek~EVlER
Economics Letters 50 (1996) 179-187
Diagnostic test for structural change in cointegrated regression models Kang Hao, Brett Inder* Department of Econometrics, Monash University, Clayton, Victoria 3168, Australia Received 3 April 1995; accepted 19 July 1995 Abstract
We derive the asymptotic distribution of the OLS-based CUSUM test in the context of cointegrated regression models and tabulate its critical values. It is also found that the test has non-trivial local power irrespective of the particular type of structure change. Keywords: Structural change; Cointegration; Fully modified ordinary least squares; CUSUM test JEL classification: C22; C52
1. Introduction
Recently, H a n s e n (1992) examined the problem of testing for structural instability in the context of cointegrated regression models. Making use of the fully modified ( F M ) OLS estimation m e t h o d for Phillips and H a n s e n (1990), H a n s e n (1992) derived the asymptotic distribution of various test statistics against different alternatives of interest. A t the same time, Quintos and Phillips (1993) proposed a Lagrange Multiplier ( L M ) test against the r a n d o m walk alternative that corresponds to Hansen's approach. Both Quintos and Phillips' (1993) and Hansen's (1992) approaches specify precisely the form of structural break u n d e r the alternative hypothesis. H o w e v e r , it is sometimes desirable to investigate generally the stability properties of the cointegrating model without specifying the possible alternatives. In this case, testing for structural change can be m o r e or less r e g a r d e d as a diagnostic test. In this paper we consider a test designed for this purpose. It is the C U S U M test with OLS residuals that was originally suggested by Ploberger and Kriimer (1992) for stationary regressors. In addition to deriving the asymptotic distribution of the OLS-based C U S U M test, we also investigate its local power p e r f o r m a n c e . A n interesting
* Corresponding author. Tel. + 61-3-905-2303; fax: + 61-3-905-5474; e-mail: Brett. Inder@BusEco. monash.edu, au. 0165-1765/96/$12.00 © 1996 Elsevier Science S.A. All rights reserved S S D I 0165-1765(95)00750-4
K. Hao, B. Inder / Economics Letters 50 (1996) 179-187
180
finding is that the test has non-trivial local power irrespective of the particular type of structural change. Considering that the OLS-based CUSUM test can be calculated easily from the residuals, it is therefore recommended as a routine test for structural change in the context of cointegrated regression models. The paper is organized as follows. Section 2 sets up the structure of the model. Section 3 investigates the limiting distribution of the test statistic. Section 4 examines the local power performance of the test. Concluding remarks are given in Section 5.
2. The cointegrated regression model Consider the cointegrated regression model
y,=Ax,+u,,
t=l,...,T,
(1)
t
where the process x, = (xl,, xj,)' is determined by the equations Xlt
-~-
kit
, o
Xzt = Hlklt + Ilzkzt -}- X2t 0 X2t
,
0 -~- X 2 t - - 1 "[- Ot "
Let us define
= (.,, v ; ) ,
k; = (k'l,, k i t ) ,
where ~', is a k + 1 dimensional sequence of stationary disturbances with mean zero, while the elements of k, are non-negative integer powers of time of order p. That is, k[ = (1, t, 12, . . . ,
tP).
This model is fairly general and encompasses a wide range of applications (see Hansen (1992) for details). It also turns out to be especially convenient for developing the asymptotic distribution of our test statistic. Suppose ~', = [u,, v[] satisfies the multivariate invariance principle as set out by Phillips and Durlauf (1986). Let Rr(r ) =R[rrl ~',, then /2R[rr]:@W(r ) = (Wo(r), Wl(r)) , where W(r) is a k + 1 dimensional Brownian motion and is partitioned in conformity with ~',, with covariance matrix: i/t=
1//'10 qtlJ =limr_~=T - ~E
~'
,~,
where
.~ =lim T-1E A =limT__~.z-1
~'~'~
=
(TO
.a~O1
-rio
.V, J '
, [.'o .o,] t=2j=,ZE(;fl,)= LA10 A, J '
= E + a + a' ,
181
K. Hao, B. lnder / Economics Letters 50 (1996) 179-187
where ~ , Z and A are partitioned in conformity with fit. We also define
a0 a01]
+A=
A=Z
A1°
A1 j .
We denote consistent estimators of ~ and A as ~ a n d / i respectively. We partition ~ and z~ as ~ and A, and then set ('0 ^ 20.1 .
i~^20 . 1/~01 1~ . 11 XPl0 . ,
/~ 10 +
/~10
A^ 1 1/I ^ - 1 1tf10 ^ "
We define the transformed dependent variable as + ^t -1 Yt = Yt -- 1/fOllff[fl
Vt"
The cointegrated regression model (1) can be written as +
y,
+
=Ax t+u,
(2)
,
with the FM estimator of A developed by Phillips and Hansen (1990) given by A q-
=
y,
q-x tI
!
T(O,
-
XtX t
(3)
,
with the FM residuals u^ , + = y ~ + - , 3 , +X t
•
In addition to the FM estimator, we can also use other asymptotically equivalent such as the spectral estimators of Phillips (1988), the maximum likelihood estimator (1991), and the dynamic OLS estimator of Saikkonen (1991), among others. This change the asymptotic distribution of our test statistic. Before going to the next section, we modify model (1) to incorporate possible instability by allowing A to depend on time:
estimators of Phillips would not parameter
y, = Atxt + u,.
(4)
The null hypothesis can be formulated as Ho'A 1=A e .....
Ar=A
,
with the alternative being HI: at least one equality does not hold.
3. Asymptotic distribution of the test statistic
Let [.] denote 'integer part', and define the FM OLS-based CUSUM test statistic: B(r)(r) _
1
tr~l
w0.1X/T,=IZ / ~ ; ( T )
,
(5)
182
K. Hao, B. Inder / Economics Letters 50 (1996) 179-187
We reject H 0 for large values of sup0<,<, simply comprises standardized partial sums of the FM OLS residuals. This is identical to the OLS-based CUSUM test proposed by Ploberger and Kr/imer (1992), except that OLS residuals are replaced by FM OLS residuals and the estimated error variance is replaced by the long-run variance estimate ^2 0)0.1
•
For convenience, we denote
ST(r)=S
rj=
xA2-
,
and k(r)= (1, r , . . . , r P ) '. Also, we denote
X(r) = ( k ( r ) \Bl(r)] ' S(r)= fo X dB0.1 and M(r)= fo XX', where S01(r ) and Bl(r ) are independent standard Brownian motions with dimensions 1 and k, respectively. Then the asymptotic distribution of the FM OLS-based CUSUM test is found as a direct consequence of theorem 2 of Hansen (1992) in which he sets up the asymptotic distribution of Sr(r ). Observe that E ~rrl t~t+ is simply the first element of the vector Sr(r ). Therefore, by theorem 2 of Hansen (1992), we have ]
d~O.l~ ¢ / T
B(r)(z)-
[T~'¿ t~: l
d
t~ +(r)~$1(~.),
(6)
where S* (z) is the first element of the vector
S* (T) = S(r) - M(z)M(1)- ' S(1) , which is free from any nuisance parameters.
Theorem 1. The FM OLS-based CUSUM statistics (5) for parameter constancy in model (4) has an asymptotic distribution given by S* (r). A couple of special cases are worth considering. In particular, if kl, = 1 and k2, = 0, then (4) can be expressed as
y,
= ol t .gff [3tx t _~_ u t ,
t= 1,..., X t ~- X t - 1
T.
(7)
+ Vt '
In this case, x, is I(1) without drift. To get S*(z), we denote
P(r) = 1
-
;(f )l B~
0
818 1
B1,
Q(t) = B 1 -
; 0
81 .
I
V
~.~
I
..
~"
~
..
(~
~
~
I
~
I
-.~
I
I
I
~
~ ~__~
'
~ ~
I
~-X-
~
~
'
"I
(%
~'~
K. Hao,B. Inder / EconomicsLetters50 (1996)179-187
184
We observe that
f (r,B;) M(r) =
o
;() ;() .0 0 r B1
r (r, B1) B1
1
(jP)-' ..#lv'l,lVlltJl.)-i =
-(jP)-lf
-1
-
(r,B',)[!(B1)(r,B~)]
1
(;)s(.) OO'
d
B(r)(r) ~ S ~ ' ( , ) = Bo.,(r ) - ~
-1
(/)
BI
QQ'
o
Thus,
-1"
(!)-1; ; L!()]-'()}r P
1-
(r, B ,,)
r B,
(r,B~)
0 0 ¢.r (f)--1 /( ) ;( B,r ) ; dBo.1 ] r dBol-- j (r, B'I) QQ' B, 0
=No.a0-)-1-
0
0
B1
dBo. 1
0
(!)-1 i (1 2 ; ,) (!)-1; P PdBo. 1 - 7~', B1 aa' QdBo. 1 . 0 0 0
(10)
Since the asymptotic distribution of B(r)0" ) does not depend on any nuisance parameters, we can obtain the critical values of the FM OLS-based C U S U M test by calculating suplB(r)(r)l for 0 < r < 1. In particular, we tabulate the critical values for models (7) and (9), respectively. They are found by simulation using a G A U S S program with a sample size of 1000 and 10,000 replications for one to five explanatory variables. The results are given in Tables 1 and 2, respectively. Table 1 Asymptotic critical values of the FM OLS-based CUSUM test for model (7) Number of regressors (excluding coiistant)
10%
5%
1%
1 2 3 4 5
1.0477 0.9342 0.8381 0.7687 0.7122
1.1684 1.0413 0.9336 0.8475 0.7890
1.4255 1.2561 1.1782 1.0383 0.9680
185
K. Hao, B. lnder / Economics Letters 50 (1996) 179-187
Table 2 Asymptotic critical values of the FM OLS-based CUSUM test for model (9) Number of regressors 10% 5% (excluding constant) 1 0.7678 0.8332 2 0.7214 0.7855 3 0.6796 0.7385 4 0.6421 0.6951 5 0.6173 0.6681
1% 0.9694 0.9162 0.8761 0.8270 0.7787
4. Local p o w e r p e r f o r m a n c e
For convenience, we consider the local power of the test under the condition that no constant term is included in the regression and {x,} are strictly exogenous. In this case, only OLS estimation is required to establish valid asymptotic distributions of B~r)(r). It is easy to show that under H 0 and according to theorem 3.1 of Park and Phillips (1988): B~)(~)_
[r,] 1 frail tr, l ]d 1 E a,u , - Y, ( ~ - / 3 ) x , ~B*(~) &oVT ,=1 tb0VT 1
(f)1;
= Bo.,(r ) - [. B',
BIB [
o
o B~
dBo. , ,
(11)
where a, are the OLS residuals. Suppose & r =/30 + T-lg(r), where r = tiT and g(r) can be any step function or uniform limit of step functions. Let us denote y, =/3,.rx , + u,. Under H0, /3 is estimated by ~ = Z YtXt'
XtXt'
,
1
with residuals t~t = y , - / 3 x , .
B¢T)(r ) _
1
[Trl
T h u s w e have
ff'~ [U, "~-/30Xt-~- r - ' g ( r ) x , -
&oVT 1 1 t~l[
- - -
tboVT 1
d 1[
~-'-~o
( Ut + T - l g ( r ) x g -
J u~l1/2If
W°(r) +
g(r)W1 -
o
=B*(r)+~-TZ
g(r)B 1 -
f3¢T)xt]
r x,x: ~ ,)(~ g(r) ~ - - T - + ugct
1
(! ; o
1
;
g(r)WIW'l +
g(r)BiB;
)-1 x~x;
o
/(! o
)~(i )-1 ]
W, d W ' o
] xt
W1W;
)--11 W1
o
BIB;
B1 .
(12)
K. Hao, B. lnder / Economics Letters 50 (1996) 179-187
186
When H 0 is true, /3t.r =/30, we must have g(r) = 0 for r E (0, 1). The OLS-based C U S U M test converges to its null distribution. Under H~, g(r)SO, (10) is well defined and distinct from the null distribution. The test thus has non-trivial power irrespective of the particular type of structural change. This conclusion is significant since it is in contrast to the result derived by Ploberger and Krfimer (1992) in which they found that when the regressors are stationary, the OLS-based CUSUM test has only trivial local power against structural changes that are orthogonal to the mean regressor.
5. An application Hansen (1992) investigated the stability properties of three postwar US interest rates. In particular, he fitted a cointegrated regression model between the federal funds (FF) rate and the 90-day treasury bill (TB3) rate and found some evidence of structural change using his supF, m e a n F and L c tests. Here we apply the FM OLS-based CUSUM test to test the same model. We obtain the following regression result: TB3, = 0.487 = 0.830 F F , , (0.14)
(0.02)
sup ]B(r)(z)I = 1.130,
p-value =0.063.
0<~'<1
Our result supports Hansen's conclusion. The test rejects the null hypothesis of no structural change at the 10% level. Although the test is not significant at the 5% level, its p-value is close to 0.05. Such a result is quite similar to that of the L c test of Hansen (1992), while the FM OLS-based CUSUM test requires much less computational effort.
6. Concluding remarks In this paper we have examined the OLS-based CUSUM test in the context of the cointegrated regression model. In addition to deriving its asymptotic distribution, which is nuisance parameter free, we also show that it has non-trivial local power irrespective of the form of structural change. This significantly contrasts with its performance under stationary regression models, where it has trivial local power in certain cases. Since the associated residuals are simply a natural by-product of FM estimation of cointegrated regression models, the FM OLS-based CUSUM test can be very easily conducted. We thus recommend it as a routine test for structural change in the context of cointegrated regression models.
Acknowledgements We wish to thank Bruce Hansen for supplying his program to calculate the fully modified least squares estimator and Werner Ploberger for helpful comments on an earlier draft of this paper.
K. Hao, B. Inder / Economics Letters 50 (1996) 179-187
187
References Hansen, B., 1992, Tests for parameter instability in regressions with I(1) processes, Journal of Business and Economic Statistics 10, 321-355. Park, J.Y. and P.C.B. Phillips, 1988, Statistical influence in regressions with integrated process: Part 1, Econometric Theory 4, 468-497. Phillips, P.C.B., 1988, Spectral regression for cointegrated time series, Cowles Foundation Discussion Paper No. 872, Yale University. Phillips, P.C.B., 1991, Optimal inference in cointegrated systems, Econometrica 59, 283-306. Phillips, P.C.B. and S.N. Durlauf, 1986, Multiple time series regression with integrated processes, Review of Economic Studies 53 473-496. Phillips, P.C.B. and B. Hansen, 1990, Statistical inference in instrumental variables regression with I(1) processes, Review of Economic Studies 57, 99-125. Pioberger, W. and W. Kr~imer, 1992, The CUSUM test with OLS residuals, Econometrica 60, 271-285. Quintos, C.E. and P.C.B. Phillips, 1993, Parameter constancy in cointegrating regressions, Empirical Economics 18, 675-706. Saikkonen, P., 1991, Asymptotically efficient estimation of cointegration regressions, Econometric Theory 7, 1-20.