- Email: [email protected]
A joint test for a unit root and common factor restrictions in the presence of a structural break
ELSEVIER
Structural Change and Economic Dynamics 8 (1997) 221-232
STRUCTURAL CHANGEAND ECONOMIC DYNAMICS
A joint test for a unit root and c o m m o n factor restrictions in the presence of a structural break J u n s o o L e e a, C h r i s t i n e A m s l e r b,. aDepartment of Economics, Vanderbilt University, Nashville, TN37235, USA b Department of Economics, Michigan State University, East Lansing, MI48824, USA
Abstract
This paper provides evidence that the common factor restrictions (CFR) play an important role in testing for a unit root in the presence of a structural break. We first show that the CFR should not be ignored in the unit root testing procedure when a structural break is allowed for. Then we provide a Wald test statistic for the joint hypothesis of a unit root and the CFR. The empirical finding suggests that the missing common factor restrictions may result in a quite different and perhaps misleading inference. © 1997 Elsevier Science B.V. Keywords: Structural break; Unit root; Common factor restrictions JEL classification: C12; C15; C22
1. Introduction
The innovative work of Perron (1989) suggests that unit root tests are biased toward accepting the false null hypothesis of a unit root if a time series exhibits stationary fluctuations around a trend containing structural breaks. He proposed an alternative testing procedure for the unit root hypothesis in the presence of a structural break. His empirical findings from applying unit root tests allowing for a break are in favor of the alternative hypothesis of stationarity. Amsler and Lee (1995) (hereafter AL) further examine this issue, and using the suitably modified version of the unit root test proposed by Schmidt and Phillips (1992) (hereafter SP), obtain theoretical and empirical evidence different from Perron (1989). One noticeable difference between Perron's test and the A L test is that the c o m m o n factor restrictions (CFR) are imposed in the A L test, but not in Perron's test. In * Corresponding author. 0954-349X/97/$15.00 © 1997 ElsevierScienceB.V. All rights reserved. PII S0954-349X (97) 00001-5
J. Lee, C Amsler / Structural Change and Economic Dynamics 8 (1997) 221-232
222
this paper, therefore, we examine the role of the C F R in testing for a unit root in the presence of a structural break. We use the Dickey and Fuller (1979) (hereafter D F ) parameterization as do Perron and others, but we add the CFR. We first show that the CFR should be included in the unit root testing procedure when a structural break is allowed for. We then provide a Wald statistic which tests the joint hypothesis of a unit root and the CFR in the presence of a structural break. To allow for autocorrelated errors, we present the Phillips and Perron (1988) type correction for the Wald test. The empirical evidence supports the finding of AL. The results in this paper may indicate that Perron's results, favoring the stationarity alternative, do not come from the presence of a break, but from the way that the D F parameterization allows for a break. The plan of the paper is as follows. In Section 2 we discuss the CFR. In Section 3 we derive a Wald test, and relevant asymptotic results. In Section 4 we provide simulation results, and apply the Wald test to the Nelson-Plosser data. Throughout this paper, ' ~ ' indicates weak convergence as T ~ Go.
2. Common factor restrictions We consider the following data generating process (DGP): yt=6'Zt+Xt,
(1)
x t = p x t _ 1 Wet,
where zt contains exogenous variables. The unit root hypothesis is p = 1, and this is the null hypothesis to be tested. This D G P is used by SP and AL. We consider zt=[1, t, Dr]' and 6=[61, 62, 63]' for the crash model, where D r = 1 for t_> TB+ 1 and zero otherwise, while TB stands for the time period when a structural change occurs. Then, Eq. (1) implies, depending upon whether p is equal to 1 or not: Null
Yt = I~o + c . Bt + y ,
1 + vt
Alternative Yt = # 1 + t " t + ( P2 - I~1)Dr + vt,
(2a)
(2b)
where vt is stationary and B t = 1 for t = Tn+ 1 and zero otherwise. This is so, since we have #o = 62, c = 63, Yo= 61 + Xo, and vt = et under the null; and /zl = 61, fl = 62, (#2 - #1) = 63 and vt = xt under the alternative. Furthermore, Eq. ( 1 ) implies: Yt = d l Y t - 1 + d 2 z t + d 3 z t - 1 + £ t ,
(3)
where d l = p , d2=6 and d3= - p 6 . Therefore, in Eq. (3) the coefficients of Y t - l , zt and zt-~ are subject to a nonlinear restriction: did 2 + d 3 = 0.
(4)
This is the CFR. In general, if zt contains k variables, there are k CFRs. The time trend variable and the intercept do not generate a CFR; given intercept and time trend, the lagged time trend variable is redundant. The variable Dt does generate a
J. Lee, C Amsler / Structural Change and Economic Dynamics 8 (1997) 221-232
223
CFR: the coefficient of Dr-1 must be equal to the negative of the coefficient of Yt-1 times the coefficient of Dr. Noticing that Bt=ADt, one can show that Eq. (3) is equivalent to the following nested equation which includes both the null and the alternative models, since Eq. (3) actually includes Dr and Bt, and this is equivalent to inclusion of Dr and Dr-1: Yt =~1 +°~2t +°~3Dt
+o~4Bt+ PYt-1 -~Et.
(5)
The relationships between Eqs. (1) and (5) are: el=(51(1-p)+52p, e2=(52(1-p), e3 = 53(1 - p ) , and e4 = 19(53. Then we have the slightly changed but equivalent form of the CFR: P~3 -- Ct4(l --p) =0.
(6)
There are two unit root restrictions in Eq. (5): p = 1 and e2 = 0. The first restriction transforms the C F R in Eq. (6) into ~3=0. This means that although the C F R is nonlinear, it becomes linear when the unit root restriction is imposed. The validity of the C F R in its general form [Eq. (6)] does not depend on the unit root restriction. However, we cannot test the C F R in Eq. (6) independently of the unit root restriction, at least based on asymptotics, since ~4, the coefficient of the regressor B, in Eq. (5), cannot be consistently estimated due to the fact that B t is asymptotically degenerate. Therefore, we settle for testing the joint restriction of the unit root and the CFR. Thus we wish to test: Ho: p = l ,
(7)
0~2=- ~3 =0.
Perron (1989) and others test only the first unit root restriction (p = 1), ignoring the two additional restrictions. The missing restrictions can potentially lead to quite different inference. It could well be that the unit root hypothesis is true, but that the other restrictions do not hold, or vice versa. It is typical of the D F regressions to include variables which are relevant under the alternative but not under the null. Here, the dummy variable (Dr) representing changed level and the time trend variable (t) are variables that belong in Eq. (2b) but not Eq. (2a), and thus are relevant in Eq. (5) under the alternative, but not under the null. The SP or AL test amounts to an LM test of p = 1, imposing the restriction ~2 = 0 and the CFR. Perron's test is a test of p = 1, not imposing 0~2 : 0 or the CFR, ~ 3 : 0.
3. Wald statistic In this section, we develop a Wald statistic which considers the joint hypothesis Eq. (7). To do so, we define:
r(~)=(p-l,e2,e3)'
and
R=dr(O~)=
d0
(00i) 1 0
0
1
,
J. Lee, C. Amsler / Structural Change and Economic Dynamics 8 (1997) 221-232
224
where ~/=(p,~2,~3,0~4)' are the coefficients of the regressors wt=[y*_l, t*, D,*, ADt*]'. The asterisk (*) represents deviations from means. Then, the Wald test statistic is defined as: ~w = r(~)'[R#~ ( W W ) - ~R] - lr(~),
( 8)
where W = (wl ..... Wr)' and ~ is the OLS estimate of 0; and where ~2 is the usual error variance estimate from the regression of y, on wt. We have the following result for the asymptotics of the Wald statistic. Theorem 1. Assume that {Yt} are generated according to Eq. (1) under the null hypothesis of a unit root (p = 1) with zt=(1, t, Dr)', and the innovations e, satisfy the regularity conditions of Phillips and Perron (1988) p. 336). Suppose that there occurs a structural break at time T B, and that TB/T~2 as T~oo. Then, the asymptotic distribution of the Wald statistic in Eq. (8) follows: ~2
H
~w~--m'--
g
m,
(9)
where { H= /
(322+32-1)/12
hl+O.5hz
h, +0.5h 2
h2/(• - ,~2) - h 3
\O.5h~ +h2/(122-122 z)
-hlhz/(2-22)-O.5h3
0.5hl+h2/(122-1222) 1 -hlh2/(2-22)-O.5h3
(h~-h3/12)/(2-22) /
g=h~ +hxh2 +h~/(122-1222) + h 3 ( - 32/ + 3 2 - 1)/12 hi =
rW(r)dr-0.5
h2 =
W(r)dr-2
h3
= W(r) 2 - [
W(r) dr W(r) dr W(r) drl 2
m=[fl,f2,f31' f l =0.5[W(1) 2 _ ~ / ~ 2 ] _ W(1) f2 = 0.5 W ( 1 ) -
f2
W(r) dr
W(r) dr
f 3 ; w(,~) - x w ( 1 )
and o2=lim T l(Ze02 and a~ =lim T -l(Ze~); and where W(r) is a Brownian motion defined on [0, 1].
J. Lee, C Amsler / Structural Changeand Economic Dynamics8 (1997) 221-232
225
Table 1 Critical values 2
1%
5%
10%
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
19.55 20.16 20.03 19.72 19.58 19.08 19.99 20.16 19.61
15.17 15.98 15.92 15.51 15.50 15.17 15.68 15.80 15.30
13.13 13.72 13.65 13.41 13.42 13.30 13.66 13.63 13.25
Proofs o f theorems are given in Appendix A. T h e o r e m 1 shows that the asymptotic distribution o f the Wald statistic depends on 2 and a2/a 2. We obtain a2/a 2 = 1 under iid errors so that we can tabulate the asymptotic critical values. We obtain them t h r o u g h a simulation o f 10 000 replications with sample size T = 1000, and present them in Table 1. They do n o t vary significantly over different values o f 2. This is rather a desirable p r o p e r t y since the statistic would be less sensitive to 2 than otherwise. We used the Gauss software version 3.0 for all computations. The Gauss R N D N S procedure has been used to generate the pseudo-iid N(O, 1) r a n d o m n u m b e r s f r o m an arbitrary seed. To allow for autocorrelated innovations, we develop a 'corrected' Wald test 1 for which a Phillips and Perron (1988) type correction is used. We define s 2 and s2(k) as in Equations (24) and (25) o f SP, using the residuals ¢t f r o m an unrestricted regression on Eq. (5); these are consistent estimates o f the error variance a 2 and the long-run variance a 2 in Eq. (9), respectively. Define the corrected Wald test statistic by: s2 ~* -
s2(k)
r* ( ~ ) ' [ R d 2 ( W S V ) - ~ R l - ~r* (~),
(lO)
where
r* (~) = r ( ~ ) - - ~ D ~ [ R ( W q C ¢ ) - ~Rq r~= 0.5[s2(k) - s 2] i 3 = [ 1,0,0]'
D* = diagonal[T, T3/2,T1/2]. Theorem 2. Assume that the conditions in Theorem 1 are satisfied Then, the 1 To allow for autocorrelated innovations, one may alternatively use an augmented version of the Wald test by including the augmentation terms Ay,_j, j = 1..... k in Eq. (5). However, simulation results (not reported here) show that this augmented test has a serious size distortion problem under strongly correlated errors in finite samples.
226
J. Lee, C. Amsler / Structural Change and Economic Dynamics 8 (1997) 221-232
asymptotic distribution of the corrected WaM test statistic in Eq. (10)follows: H ~*~~m*' --m*, g
( 11 )
where m*--[/~1, f2, f3]' with J~l =0.5W(1)2-0.5 - W(1)~W(r) dr," and where f2, f3, H and g are defined in Theorem 1. The theorem indicates that the asymptotic distribution of the corrected Wald test statistic is free of the nuisance parameters a 2 and a~2, and it is the same as that of the uncorrected Wald test statistic under iid errors.
4. Monte Carlo simulation and empirical results In this section, we examine the size and power of the Wald statistic by Monte Carlo simulation, and provide empirical results of applying the Wald statistic to the Nelson and Plosser (1982) data. The D G P for the crash model is Eq. (1) with the moving average error structure: et = u, + Out_ 1. The parameter 2 for a break point is set at 0.5 for the simulation consisting of 5000 replications, and the asymptotic critical values in Table 1 are used for all simulations. In estimating the long-run error variance, the choice of the kernel is not critical and the Parzen kernel is employed for simulation. We use the optimal bandwidth parameter of Andrews (1991) as in Lee and Mossi (1996), and fixed truncation lags (k), which vary according to the sample size. We selected two values, (kl and ka); they are (3, 8), (4, 12) and (6, 15) for sample sizes T=50, 100 and 500, respectively. Table 2 contains size and power results for the Wald statistic. Experiment A considers the size of the test under the unit root null in the presence of MA errors, 0 = - 0 . 8 , - 0 . 5 , 0, 0.5, 0.8. When a structural change occurs (63 = 5), the simulation results display about the same degree of size distortions as do the usual Phillips-Perron (Phillips and Perron, 1988) unit root tests not allowing for the break; see Schwert (1989) for comparison. Experiment B considers the size of the test with different magnitudes of a structural change. The sizes of the tests do not change much, which indicates that they successfully eliminate the effect of structural change. We report this result only for sample size T = 100 and 63 = 10. Experiment C considers the power of the tests under iid errors (where virtually no significant size distortions are found) for different values of p. The truncation lag k = 0 is used in this case. The initial value Yo= Xo= 0 is chosen. The power of the Wald test is 0.138, 0.466 and 0.858 when p = 0 . 9 , 0.8 and 0.7, respectively. The Wald test is a little less powerful than the Perron test; in the same situations the power of the Perron test is simulated as 0.159, 0.529 and 0.907. This result is not different from our prior expectation that the test of the joint hypothesis is generally less powerful. Nonetheless the difference in power is not very significant. On the other hand, the power of the A L test is computed as 0.261, 0.728 and 0.998; the AL test is more powerful in the same situations than the Wald test or the Perron test. The empirical results of the corrected Wald test applied to the Nelson-Plosser
J. Lee, C. Amsler / Structural Change and Economic Dynamics 8 (1997) 221-232
227
Table 2 Size and power, 5% rejections Exp
A
T
p
63
50
1
5
100
1
5
500
1
5
B
100
1
10
C
100
0.9 0.8 0.7
5
0
- 0.8 -0.5 0.0 0.5 0.8 -0.8 - 0.5 0.0 0.5 0.8 -0.8 -0.5 0.0 0.5 0.8 - 0.8 - 0.5 0.0 0.5 0.8 0
Wald statistic Auto
kl
k2
0.995 0.635 0.082 0.062 0.074 0.999 0.666 0.062 0.046 0.043 0.999 0.486 0.052 0.045 0.045 1.00 0.660 0.067 0.044 0.046 0.153 0.484 0.878
0.991 0.592 0.088 0.066 0.068 0.999 0.639 0.069 0.044 0.055 0.998 0.490 0.050 0.047 0.048 0.999 0.673 0.069 0.039 0.043 0.138a 0.466a 0.858"
0.992 0.618 0.090 0.107 0.105 1.00 0.753 0.060 0.064 0.066 1.00 0.611 0.053 0.037 0.034 1.00 0.771 0.075 0.057 0.055
ak = 0 is used for the power of the test.
data are provided in Table 3. The Fejer kernel is used for estimating the l o n g - r u n error variance. We supply the results using the selected o p t i m a l b a n d w i d t h p a r a m e t e r ( c o l u m n 4) a n d fixed t r u n c a t i o n lags, k = 4 a n d 12 (columns 5 a n d 6). T h e results indicate that we c a n n o t reject the j o i n t hypothesis o f a u n i t root a n d the C F R at the 5% level for m o s t o f the series that we consider except the ' i n d u s t r i a l p r o d u c t i o n ' series. These results are fairly similar to those o f A L r e p r o d u c e d in c o l u m n 7 a n d are dissimilar to those o f P e r r o n r e p r o d u c e d in c o l u m n 8. It is n o t difficult to give the overall c o n c l u s i o n from the empirical results; namely, that m o s t series a p p e a r to c o n t a i n a u n i t root.
5. Concluding remarks I n this paper, we suggest that the c o m m o n factor restrictions ( C F R ) be included in the u n i t root test procedure in the presence o f a structural break. T h e n we develop a Wald test for the j o i n t hypothesis o f a u n i t root a n d the C F R . We observe that
228
J. Lee, C. Amsler / Structural Change and Economic Dynamics 8 (1997) 221-232
Table 3 Empirical Results Series
Real GNP Nominal GNP Real per capita GNP Industrial production Employment GNP deflator CPI Nominal wage Money stock Velocity Interest rate
T
62 62 62 111 81 82 111 71 82 102 71
Tn
21 21 21 70 40 41 70 30 41 61 30
Wald statistic Opt)
k=4
k= 12
10.3 (3) 13.3 (4) 8.55 (3) 18.3 ( 1) 10.3 (3) 13.6 (4) 2.79 (7) 12.7 (4) 10.8 (7) 6.35 ( 1) 13.7 (0)
9.99 13.3 7.97 19.0" 10.3 13.6 2.74 12.7 10.0 6.51 9.94
8.61 15.1 6.64 18.4* 9.37 13.6 2.76 12.9 10.4 7.79 10.2
ALb
Perronc
-3.15" -2.58 -2.96 - 2.64 -2.96 -2.25 -2.73 -3.04 -3.18" - 1.71 -1.12
-5.03*** -5.42*** -4.09** - 5.47*** -4.51'** -4.04** - 1.28 -5.41"** -4.39** - 1.66 -0.45
aSelected optimal bandwidth estimates are in the parentheses. bThese are obtained from Amsler and Lee (1995), p. 365. CThese are obtained from Perron (1989), p. 1383. *Significantat the 5% level. **Significantat the 2.5% level. ***Significantat the 1% level.
the empirical results from the Wald test are different f r o m those that do n o t impose the C F R . M o r e specifically, the results from the Wald test are fairly consistent with those from the L M test o f A m s l e r a n d Lee (1995), which tests the u n i t root hypothesis while i m p o s i n g the C F R , a n d are rather different from the results from P e r r o n ' s test which does n o t impose the C F R . Since the A m s l e r - L e e test is more powerful t h a n either P e r r o n ' s test or the Wald test, w h e n the C F R are true, these differences in results do n o t appear to come from a lack o f power. Rather, failing to impose the C F R m a y lead to a quite different a n d perhaps misleading inference. While we consider only a crash m o d e l in which a one time structural b r e a k occurs, the test c a n be extended to consider a c h a n g i n g growth model or a model with a s u d d e n change in the level a c c o m p a n i e d by a different growth path. We could also consider e n d o g e n i z i n g the structural break, as in Banerjee et al. (1992) a n d Z i v o t a n d A n d r e w s (1992). These extensions are n o t the focus of this paper, b u t r e m a i n as a future research.
Acknowledgements We t h a n k Peter Schmidt for helpful c o m m e n t s a n d suggestions. We are grateful for the c o m m e n t s o f two a n o n y m o u s referees.
J. Lee, C Amsler/ Structural Changeand EconomicDynamics8 (1997) 221-232
229
Appendix A Proofs o f theorems
A.1. Proof o f Theorem 1
Noticing that RD~ ~ = D * - aR, for convergence rate matrices, Dr=diagonal[T, T 3/2, T u2, 1] and D*=diagonal[T, T 3/2, TU2], we obtain: ~w=r(~)I)*[R[D~I(W'W)D~ll-IRq-IDTr(~b)/a,.* ~ A2
(A.1)
We decompose the matrix W into submatrices W~ and W2, where Wl and W 2 are T× 3 matrix and T× 1 vector with t th row (or observation) being %t=[y*_~, t*, D*]' and w2t = B*, respectively. Note that the dummy variable Bt does not affect the asymptotic distribution of the test statistic, since the matrix D~I(W~CC)D~ 1 is asymptotically block diagonal with respect to the sub-matrices, D * - I ( W ~ W l ) D *-1 and W~W2. Then, R[D~I(W~CC)D~I]-IR ' can be replaced with the sub-matrix [D*-t(W ~W1)D~-1]-1. [But it does not necessarily mean that we can exclude Bt in Eq. (5).] Also noting that under the null, r(~)l)* is same as riD*, where rl =[-Yy*_le *, St*e*, ZD*e*]', we obtain,
~w = ( D r-1 W ltr l ) [r D r*-1 ( W l/W l ) D r*-1 ] -1 (Dr*-1 Wffl)/a,. i ~2
(A.2)
Let
[1 '2 / f.~.y,2 ~"t*y~__l .~V'y*__ ID ` / " " "q:/ 10
W l W l =-
and
W,2 W13/ (W ;W')--1----'~i~ \ W13
W23
W33/
where ID[ is the determinant. Then, T[D*-~(WIW~)D~.-~] -~ = _ _
T-6IDI
7/2 W12
T-9/2 W13 ~
T-7/2 W12
T - 3 W22
T - 4W23 /"
T-9/2W13
T - 4 W2a
T - 5W33 / (A.3)
230
J. Lee, C. Amsler / Structural Change and Economic Dynamics 8 (1997) 221-232
N o w , consider the a s y m p t o t i c results: (a) T-21o = T-ZSy*_21 ~ a 2 h 3 (b) T-5/2ll = T - 5 / 2 S t *y*_ 1 ---~0-hl (C) T-3/212 = T - a / 2 S y * 1D* --r6rh2
(d) T - a d l l = T - 3 S t * 2 ~ l / 1 2 (e) T - 2da 2 = T - 2 S t * D * --, - 0.5 ( 1 - A) A (f) T - adE2 --- T - aZ'D*2 --r( 1 - 2)/l, where ha, h 2 a n d h3 are defined in T h e o r e m 1. Letting r a - [ e a , e2, e3]', we have additional a s y m p t o t i c results: (g) T - le I = T - asy*_ a e* ~ 0 - 2 f l (h) T - 3/2e2 = T - 3/2y, t * e* ~ a f 2 (i) T - 1/2e3 = T - a/2SD*e* --*0-f3, where f l , f2 a n d f3 are defined in T h e o r e m 1. Then, we o b t a i n for each t e r m in Eq. (A.3): T - 6[D[ = - ( T - 2d12)2 T - 2l0 + T - 3dll T - ad22 T - 2l0 - T - ld22 ( T - 5/2l1)2 + 2 T - 2d12 T - 5/2lI T 3/2•2 - T - 3 d l a ( T - 3/212)2 ~ - ( 1 - 2)2tr2g T-4Wll
= --(T-2da2)
2 q- T - 3 d a l
T - ad22 --~ - ( 1 - - 2 ) 2 0 " 2 [ ( - - 3 2 2
-{- 3 2 - - 1)/(120-2)]
T - 7/2 Wa 2 ~ -- ( 1 - 2) 2tr 2 [h 1/0 + h 2/(20-)] T - 9/2 W13 -~ - ( 1 - 2)2a 2 [ha/(20-) + hE/( 12( 1 -- 2) 2a)] T - 3 I4/22 ~ -- ( 1 - A)20-2 [h 2/(( 1 - 2) 2) - h3 ] T - 4 W 2 3 ~ - ( 1 - 2) 20-2 [ - h 3/2 - h lh2/(( l - - ,~)/~)] Z - 5 W33 ~ - ( 1 - ,~) ~0-2 [(h 2 _ h3 / 1 2 ) / ( ( 1 - ~) ~)].
T h e n we obtain:
[D*- 1( W i W l ) D ~ r -
l] - 1 __H* ,
(A.4)
g where H * is the s a m e as H in T h e o r e m 1 except that the first r o w o f H * is given as the first r o w o f H multiplied by I * = d i a g o n a l [ I / o - 2, 1/0-, 1/0-]. We also obtain: D * - ar I --,m # = [0-2fl ,~f2,af3]'.
(A.5)
N o w , n o t e that H * = I , . H . I~ a n d I , m # = am, where I , = diagonal [ 1/a, 1, 1 ]. T h e n
231
J. Lee, C. Amsler / Structural Change and Economic Dynamics 8 (1997) 221-232
it is directly shown from Eqs. (A.4) and (A.5) that: .
1
H*
H
0 -2
- m' - - m, Cw'a.--7, m #' - -g m # = -0-2 g
(A.6)
which proves Theorem 1.
A.2. Proof of Theorem 2 Define ~ = f l -V, where • = 0.5 (0-2_ a2)/0-2. Also define m* = [ ~ , f2, f3]' such that r e = m * q-Ti3 where i 3 =(1, 0, 0)'. Then the asymptotic distribution Eq. (A.6) of the Wald test under iid errors is expressed as: H
~w-+m*' - - m*. g
(A.7)
Here, H, g and m* do not depend on error variances, so that the above expression is free of the nuisance parameters. We want to show that the corrected Wald test follows the same asymptotic distribution in Eq. (A.7). First, we alternatively show the expression in Eq. (A.6) by: --m'--m=
0-2
g
--(m*
0-2
+7i3)'
m* +~i3
= ~-m*'
at
--m*
g
0 -2 H 0 -2 H + 2 0-2 [(0-2 _0-2)/20-2]i ~ --g m* + ~ [(0-2 _0-2)/20-212i~ __gi 3. (A.8)
Now, we rewrite the corrected Wald test in Eq. (10) after some algebra as: r* ( ~ ) ' [ R d 2 ( W'Xd¢ ) - 1R 3 - lr* ( ~ ) = [ D * r ( ~ ) I ' I D * ( W ~W1 ) D * - '1
× [D*r(~)]/# 2 - 2~i~ [D*r(~)]/# 2 t + n~l2a. t[ D r ,( W l W l)Dr
*-1
~2 ] - 1 " 13/0-~.
(A.9)
We note that the first term of the last equation is equivalent to the expression for the 'uncorrected' Wald test in Eq. (8), and its asymptotic distribution is given in Eq. (A.6). Also note that 7~-,n=~0- 2. The asymptotic distribution of the second term is given after some algebra by: H* 1 H 1 H - 2ni~ - -g m # 7 =(0-2__ 0-2~i' 2 ","3 __ [(0-2_0-2)/20-21ia. e) 3 - - m* ~ - __(0-2 __o-e)!
g
~
(A.10)
g
The third term of Eq. (A.9) follows asymptotically: 0-2 H [(0-2 __ 0-2)/20-2]2 ~2- i~ --i3. of g
(A. 11 )
Finally, after rearranging Eqs. (A.7), (A.10), (A.11), we can show that the asymptotic distribution of the whole expression in Eq. (A.9) is the same as that of the
232
J. Lee, C. Amsler / Structural Change and Economic Dynamics 8 (1997)221-232
distribution of the uncorrected Wald test statistic under lid errors [which is expressed in Eq. (A.7), and is free of nuisance parameters].
References Andrews, D.W.K., 1991. Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817-858. Amsler, C., Lee, J., 1995. An LM test for a unit root in the presence of a structural break. Econometric Theory 11,359-368. Banerjee A., Lumsdaine, R.L., Stock, J.H., 1992. Recursive and sequential tests of the unit root and trend-break hypotheses: theory and international evidence. Journal of Business and Economic Statistics 10, 271 287. Dickey, D.A., Fuller, W.A., 1979. Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427 431. Lee, J., Mossi, D., 1996. On improvements of Phillips-Perron unit root tests using optimal bandwidth estimates. Applied Economics Letters 11, 197-200. Nelson, C.R., Plosser, C.I., 1982. Trends versus random walks in macroeconomic time series: some evidence and implications. Journal of Monetary Economics 10, 139-162. Perron, P., 1989. The great crash, the oil price shock, and the unit root hypothesis. Econometrica 57, 1361-1401. Phillips, P.C.B., Perron, P., 1988. Testing for a unit root in time series regression. Biometrika 75, 335-346. Schmidt, P., Phillips, P.C.B., 1992. Testing for a unit root in the presence of deterministic trends. Oxford Bulletin of Economics and Statistics 54, 257-287. Schwert, G.W., 1989. Tests for unit roots: a Monte Carlo investigation. Journal of Business and Economic Statistics 7, 147-159. Zivot, E., Andrews, D.W.K., 1992. Further evidence on the great crash, the oil price shock and the unit root hypothesis. Journal of Business and Economic Statistics 10, 251-270.
Structural Change and Economic Dynamics 8 (1997) 221-232
STRUCTURAL CHANGEAND ECONOMIC DYNAMICS
A joint test for a unit root and c o m m o n factor restrictions in the presence of a structural break J u n s o o L e e a, C h r i s t i n e A m s l e r b,. aDepartment of Economics, Vanderbilt University, Nashville, TN37235, USA b Department of Economics, Michigan State University, East Lansing, MI48824, USA
Abstract
This paper provides evidence that the common factor restrictions (CFR) play an important role in testing for a unit root in the presence of a structural break. We first show that the CFR should not be ignored in the unit root testing procedure when a structural break is allowed for. Then we provide a Wald test statistic for the joint hypothesis of a unit root and the CFR. The empirical finding suggests that the missing common factor restrictions may result in a quite different and perhaps misleading inference. © 1997 Elsevier Science B.V. Keywords: Structural break; Unit root; Common factor restrictions JEL classification: C12; C15; C22
1. Introduction
The innovative work of Perron (1989) suggests that unit root tests are biased toward accepting the false null hypothesis of a unit root if a time series exhibits stationary fluctuations around a trend containing structural breaks. He proposed an alternative testing procedure for the unit root hypothesis in the presence of a structural break. His empirical findings from applying unit root tests allowing for a break are in favor of the alternative hypothesis of stationarity. Amsler and Lee (1995) (hereafter AL) further examine this issue, and using the suitably modified version of the unit root test proposed by Schmidt and Phillips (1992) (hereafter SP), obtain theoretical and empirical evidence different from Perron (1989). One noticeable difference between Perron's test and the A L test is that the c o m m o n factor restrictions (CFR) are imposed in the A L test, but not in Perron's test. In * Corresponding author. 0954-349X/97/$15.00 © 1997 ElsevierScienceB.V. All rights reserved. PII S0954-349X (97) 00001-5
J. Lee, C Amsler / Structural Change and Economic Dynamics 8 (1997) 221-232
222
this paper, therefore, we examine the role of the C F R in testing for a unit root in the presence of a structural break. We use the Dickey and Fuller (1979) (hereafter D F ) parameterization as do Perron and others, but we add the CFR. We first show that the CFR should be included in the unit root testing procedure when a structural break is allowed for. We then provide a Wald statistic which tests the joint hypothesis of a unit root and the CFR in the presence of a structural break. To allow for autocorrelated errors, we present the Phillips and Perron (1988) type correction for the Wald test. The empirical evidence supports the finding of AL. The results in this paper may indicate that Perron's results, favoring the stationarity alternative, do not come from the presence of a break, but from the way that the D F parameterization allows for a break. The plan of the paper is as follows. In Section 2 we discuss the CFR. In Section 3 we derive a Wald test, and relevant asymptotic results. In Section 4 we provide simulation results, and apply the Wald test to the Nelson-Plosser data. Throughout this paper, ' ~ ' indicates weak convergence as T ~ Go.
2. Common factor restrictions We consider the following data generating process (DGP): yt=6'Zt+Xt,
(1)
x t = p x t _ 1 Wet,
where zt contains exogenous variables. The unit root hypothesis is p = 1, and this is the null hypothesis to be tested. This D G P is used by SP and AL. We consider zt=[1, t, Dr]' and 6=[61, 62, 63]' for the crash model, where D r = 1 for t_> TB+ 1 and zero otherwise, while TB stands for the time period when a structural change occurs. Then, Eq. (1) implies, depending upon whether p is equal to 1 or not: Null
Yt = I~o + c . Bt + y ,
1 + vt
Alternative Yt = # 1 + t " t + ( P2 - I~1)Dr + vt,
(2a)
(2b)
where vt is stationary and B t = 1 for t = Tn+ 1 and zero otherwise. This is so, since we have #o = 62, c = 63, Yo= 61 + Xo, and vt = et under the null; and /zl = 61, fl = 62, (#2 - #1) = 63 and vt = xt under the alternative. Furthermore, Eq. ( 1 ) implies: Yt = d l Y t - 1 + d 2 z t + d 3 z t - 1 + £ t ,
(3)
where d l = p , d2=6 and d3= - p 6 . Therefore, in Eq. (3) the coefficients of Y t - l , zt and zt-~ are subject to a nonlinear restriction: did 2 + d 3 = 0.
(4)
This is the CFR. In general, if zt contains k variables, there are k CFRs. The time trend variable and the intercept do not generate a CFR; given intercept and time trend, the lagged time trend variable is redundant. The variable Dt does generate a
J. Lee, C Amsler / Structural Change and Economic Dynamics 8 (1997) 221-232
223
CFR: the coefficient of Dr-1 must be equal to the negative of the coefficient of Yt-1 times the coefficient of Dr. Noticing that Bt=ADt, one can show that Eq. (3) is equivalent to the following nested equation which includes both the null and the alternative models, since Eq. (3) actually includes Dr and Bt, and this is equivalent to inclusion of Dr and Dr-1: Yt =~1 +°~2t +°~3Dt
+o~4Bt+ PYt-1 -~Et.
(5)
The relationships between Eqs. (1) and (5) are: el=(51(1-p)+52p, e2=(52(1-p), e3 = 53(1 - p ) , and e4 = 19(53. Then we have the slightly changed but equivalent form of the CFR: P~3 -- Ct4(l --p) =0.
(6)
There are two unit root restrictions in Eq. (5): p = 1 and e2 = 0. The first restriction transforms the C F R in Eq. (6) into ~3=0. This means that although the C F R is nonlinear, it becomes linear when the unit root restriction is imposed. The validity of the C F R in its general form [Eq. (6)] does not depend on the unit root restriction. However, we cannot test the C F R in Eq. (6) independently of the unit root restriction, at least based on asymptotics, since ~4, the coefficient of the regressor B, in Eq. (5), cannot be consistently estimated due to the fact that B t is asymptotically degenerate. Therefore, we settle for testing the joint restriction of the unit root and the CFR. Thus we wish to test: Ho: p = l ,
(7)
0~2=- ~3 =0.
Perron (1989) and others test only the first unit root restriction (p = 1), ignoring the two additional restrictions. The missing restrictions can potentially lead to quite different inference. It could well be that the unit root hypothesis is true, but that the other restrictions do not hold, or vice versa. It is typical of the D F regressions to include variables which are relevant under the alternative but not under the null. Here, the dummy variable (Dr) representing changed level and the time trend variable (t) are variables that belong in Eq. (2b) but not Eq. (2a), and thus are relevant in Eq. (5) under the alternative, but not under the null. The SP or AL test amounts to an LM test of p = 1, imposing the restriction ~2 = 0 and the CFR. Perron's test is a test of p = 1, not imposing 0~2 : 0 or the CFR, ~ 3 : 0.
3. Wald statistic In this section, we develop a Wald statistic which considers the joint hypothesis Eq. (7). To do so, we define:
r(~)=(p-l,e2,e3)'
and
R=dr(O~)=
d0
(00i) 1 0
0
1
,
J. Lee, C. Amsler / Structural Change and Economic Dynamics 8 (1997) 221-232
224
where ~/=(p,~2,~3,0~4)' are the coefficients of the regressors wt=[y*_l, t*, D,*, ADt*]'. The asterisk (*) represents deviations from means. Then, the Wald test statistic is defined as: ~w = r(~)'[R#~ ( W W ) - ~R] - lr(~),
( 8)
where W = (wl ..... Wr)' and ~ is the OLS estimate of 0; and where ~2 is the usual error variance estimate from the regression of y, on wt. We have the following result for the asymptotics of the Wald statistic. Theorem 1. Assume that {Yt} are generated according to Eq. (1) under the null hypothesis of a unit root (p = 1) with zt=(1, t, Dr)', and the innovations e, satisfy the regularity conditions of Phillips and Perron (1988) p. 336). Suppose that there occurs a structural break at time T B, and that TB/T~2 as T~oo. Then, the asymptotic distribution of the Wald statistic in Eq. (8) follows: ~2
H
~w~--m'--
g
m,
(9)
where { H= /
(322+32-1)/12
hl+O.5hz
h, +0.5h 2
h2/(• - ,~2) - h 3
\O.5h~ +h2/(122-122 z)
-hlhz/(2-22)-O.5h3
0.5hl+h2/(122-1222) 1 -hlh2/(2-22)-O.5h3
(h~-h3/12)/(2-22) /
g=h~ +hxh2 +h~/(122-1222) + h 3 ( - 32/ + 3 2 - 1)/12 hi =
rW(r)dr-0.5
h2 =
W(r)dr-2
h3
= W(r) 2 - [
W(r) dr W(r) dr W(r) drl 2
m=[fl,f2,f31' f l =0.5[W(1) 2 _ ~ / ~ 2 ] _ W(1) f2 = 0.5 W ( 1 ) -
f2
W(r) dr
W(r) dr
f 3 ; w(,~) - x w ( 1 )
and o2=lim T l(Ze02 and a~ =lim T -l(Ze~); and where W(r) is a Brownian motion defined on [0, 1].
J. Lee, C Amsler / Structural Changeand Economic Dynamics8 (1997) 221-232
225
Table 1 Critical values 2
1%
5%
10%
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
19.55 20.16 20.03 19.72 19.58 19.08 19.99 20.16 19.61
15.17 15.98 15.92 15.51 15.50 15.17 15.68 15.80 15.30
13.13 13.72 13.65 13.41 13.42 13.30 13.66 13.63 13.25
Proofs o f theorems are given in Appendix A. T h e o r e m 1 shows that the asymptotic distribution o f the Wald statistic depends on 2 and a2/a 2. We obtain a2/a 2 = 1 under iid errors so that we can tabulate the asymptotic critical values. We obtain them t h r o u g h a simulation o f 10 000 replications with sample size T = 1000, and present them in Table 1. They do n o t vary significantly over different values o f 2. This is rather a desirable p r o p e r t y since the statistic would be less sensitive to 2 than otherwise. We used the Gauss software version 3.0 for all computations. The Gauss R N D N S procedure has been used to generate the pseudo-iid N(O, 1) r a n d o m n u m b e r s f r o m an arbitrary seed. To allow for autocorrelated innovations, we develop a 'corrected' Wald test 1 for which a Phillips and Perron (1988) type correction is used. We define s 2 and s2(k) as in Equations (24) and (25) o f SP, using the residuals ¢t f r o m an unrestricted regression on Eq. (5); these are consistent estimates o f the error variance a 2 and the long-run variance a 2 in Eq. (9), respectively. Define the corrected Wald test statistic by: s2 ~* -
s2(k)
r* ( ~ ) ' [ R d 2 ( W S V ) - ~ R l - ~r* (~),
(lO)
where
r* (~) = r ( ~ ) - - ~ D ~ [ R ( W q C ¢ ) - ~Rq r~= 0.5[s2(k) - s 2] i 3 = [ 1,0,0]'
D* = diagonal[T, T3/2,T1/2]. Theorem 2. Assume that the conditions in Theorem 1 are satisfied Then, the 1 To allow for autocorrelated innovations, one may alternatively use an augmented version of the Wald test by including the augmentation terms Ay,_j, j = 1..... k in Eq. (5). However, simulation results (not reported here) show that this augmented test has a serious size distortion problem under strongly correlated errors in finite samples.
226
J. Lee, C. Amsler / Structural Change and Economic Dynamics 8 (1997) 221-232
asymptotic distribution of the corrected WaM test statistic in Eq. (10)follows: H ~*~~m*' --m*, g
( 11 )
where m*--[/~1, f2, f3]' with J~l =0.5W(1)2-0.5 - W(1)~W(r) dr," and where f2, f3, H and g are defined in Theorem 1. The theorem indicates that the asymptotic distribution of the corrected Wald test statistic is free of the nuisance parameters a 2 and a~2, and it is the same as that of the uncorrected Wald test statistic under iid errors.
4. Monte Carlo simulation and empirical results In this section, we examine the size and power of the Wald statistic by Monte Carlo simulation, and provide empirical results of applying the Wald statistic to the Nelson and Plosser (1982) data. The D G P for the crash model is Eq. (1) with the moving average error structure: et = u, + Out_ 1. The parameter 2 for a break point is set at 0.5 for the simulation consisting of 5000 replications, and the asymptotic critical values in Table 1 are used for all simulations. In estimating the long-run error variance, the choice of the kernel is not critical and the Parzen kernel is employed for simulation. We use the optimal bandwidth parameter of Andrews (1991) as in Lee and Mossi (1996), and fixed truncation lags (k), which vary according to the sample size. We selected two values, (kl and ka); they are (3, 8), (4, 12) and (6, 15) for sample sizes T=50, 100 and 500, respectively. Table 2 contains size and power results for the Wald statistic. Experiment A considers the size of the test under the unit root null in the presence of MA errors, 0 = - 0 . 8 , - 0 . 5 , 0, 0.5, 0.8. When a structural change occurs (63 = 5), the simulation results display about the same degree of size distortions as do the usual Phillips-Perron (Phillips and Perron, 1988) unit root tests not allowing for the break; see Schwert (1989) for comparison. Experiment B considers the size of the test with different magnitudes of a structural change. The sizes of the tests do not change much, which indicates that they successfully eliminate the effect of structural change. We report this result only for sample size T = 100 and 63 = 10. Experiment C considers the power of the tests under iid errors (where virtually no significant size distortions are found) for different values of p. The truncation lag k = 0 is used in this case. The initial value Yo= Xo= 0 is chosen. The power of the Wald test is 0.138, 0.466 and 0.858 when p = 0 . 9 , 0.8 and 0.7, respectively. The Wald test is a little less powerful than the Perron test; in the same situations the power of the Perron test is simulated as 0.159, 0.529 and 0.907. This result is not different from our prior expectation that the test of the joint hypothesis is generally less powerful. Nonetheless the difference in power is not very significant. On the other hand, the power of the A L test is computed as 0.261, 0.728 and 0.998; the AL test is more powerful in the same situations than the Wald test or the Perron test. The empirical results of the corrected Wald test applied to the Nelson-Plosser
J. Lee, C. Amsler / Structural Change and Economic Dynamics 8 (1997) 221-232
227
Table 2 Size and power, 5% rejections Exp
A
T
p
63
50
1
5
100
1
5
500
1
5
B
100
1
10
C
100
0.9 0.8 0.7
5
0
- 0.8 -0.5 0.0 0.5 0.8 -0.8 - 0.5 0.0 0.5 0.8 -0.8 -0.5 0.0 0.5 0.8 - 0.8 - 0.5 0.0 0.5 0.8 0
Wald statistic Auto
kl
k2
0.995 0.635 0.082 0.062 0.074 0.999 0.666 0.062 0.046 0.043 0.999 0.486 0.052 0.045 0.045 1.00 0.660 0.067 0.044 0.046 0.153 0.484 0.878
0.991 0.592 0.088 0.066 0.068 0.999 0.639 0.069 0.044 0.055 0.998 0.490 0.050 0.047 0.048 0.999 0.673 0.069 0.039 0.043 0.138a 0.466a 0.858"
0.992 0.618 0.090 0.107 0.105 1.00 0.753 0.060 0.064 0.066 1.00 0.611 0.053 0.037 0.034 1.00 0.771 0.075 0.057 0.055
ak = 0 is used for the power of the test.
data are provided in Table 3. The Fejer kernel is used for estimating the l o n g - r u n error variance. We supply the results using the selected o p t i m a l b a n d w i d t h p a r a m e t e r ( c o l u m n 4) a n d fixed t r u n c a t i o n lags, k = 4 a n d 12 (columns 5 a n d 6). T h e results indicate that we c a n n o t reject the j o i n t hypothesis o f a u n i t root a n d the C F R at the 5% level for m o s t o f the series that we consider except the ' i n d u s t r i a l p r o d u c t i o n ' series. These results are fairly similar to those o f A L r e p r o d u c e d in c o l u m n 7 a n d are dissimilar to those o f P e r r o n r e p r o d u c e d in c o l u m n 8. It is n o t difficult to give the overall c o n c l u s i o n from the empirical results; namely, that m o s t series a p p e a r to c o n t a i n a u n i t root.
5. Concluding remarks I n this paper, we suggest that the c o m m o n factor restrictions ( C F R ) be included in the u n i t root test procedure in the presence o f a structural break. T h e n we develop a Wald test for the j o i n t hypothesis o f a u n i t root a n d the C F R . We observe that
228
J. Lee, C. Amsler / Structural Change and Economic Dynamics 8 (1997) 221-232
Table 3 Empirical Results Series
Real GNP Nominal GNP Real per capita GNP Industrial production Employment GNP deflator CPI Nominal wage Money stock Velocity Interest rate
T
62 62 62 111 81 82 111 71 82 102 71
Tn
21 21 21 70 40 41 70 30 41 61 30
Wald statistic Opt)
k=4
k= 12
10.3 (3) 13.3 (4) 8.55 (3) 18.3 ( 1) 10.3 (3) 13.6 (4) 2.79 (7) 12.7 (4) 10.8 (7) 6.35 ( 1) 13.7 (0)
9.99 13.3 7.97 19.0" 10.3 13.6 2.74 12.7 10.0 6.51 9.94
8.61 15.1 6.64 18.4* 9.37 13.6 2.76 12.9 10.4 7.79 10.2
ALb
Perronc
-3.15" -2.58 -2.96 - 2.64 -2.96 -2.25 -2.73 -3.04 -3.18" - 1.71 -1.12
-5.03*** -5.42*** -4.09** - 5.47*** -4.51'** -4.04** - 1.28 -5.41"** -4.39** - 1.66 -0.45
aSelected optimal bandwidth estimates are in the parentheses. bThese are obtained from Amsler and Lee (1995), p. 365. CThese are obtained from Perron (1989), p. 1383. *Significantat the 5% level. **Significantat the 2.5% level. ***Significantat the 1% level.
the empirical results from the Wald test are different f r o m those that do n o t impose the C F R . M o r e specifically, the results from the Wald test are fairly consistent with those from the L M test o f A m s l e r a n d Lee (1995), which tests the u n i t root hypothesis while i m p o s i n g the C F R , a n d are rather different from the results from P e r r o n ' s test which does n o t impose the C F R . Since the A m s l e r - L e e test is more powerful t h a n either P e r r o n ' s test or the Wald test, w h e n the C F R are true, these differences in results do n o t appear to come from a lack o f power. Rather, failing to impose the C F R m a y lead to a quite different a n d perhaps misleading inference. While we consider only a crash m o d e l in which a one time structural b r e a k occurs, the test c a n be extended to consider a c h a n g i n g growth model or a model with a s u d d e n change in the level a c c o m p a n i e d by a different growth path. We could also consider e n d o g e n i z i n g the structural break, as in Banerjee et al. (1992) a n d Z i v o t a n d A n d r e w s (1992). These extensions are n o t the focus of this paper, b u t r e m a i n as a future research.
Acknowledgements We t h a n k Peter Schmidt for helpful c o m m e n t s a n d suggestions. We are grateful for the c o m m e n t s o f two a n o n y m o u s referees.
J. Lee, C Amsler/ Structural Changeand EconomicDynamics8 (1997) 221-232
229
Appendix A Proofs o f theorems
A.1. Proof o f Theorem 1
Noticing that RD~ ~ = D * - aR, for convergence rate matrices, Dr=diagonal[T, T 3/2, T u2, 1] and D*=diagonal[T, T 3/2, TU2], we obtain: ~w=r(~)I)*[R[D~I(W'W)D~ll-IRq-IDTr(~b)/a,.* ~ A2
(A.1)
We decompose the matrix W into submatrices W~ and W2, where Wl and W 2 are T× 3 matrix and T× 1 vector with t th row (or observation) being %t=[y*_~, t*, D*]' and w2t = B*, respectively. Note that the dummy variable Bt does not affect the asymptotic distribution of the test statistic, since the matrix D~I(W~CC)D~ 1 is asymptotically block diagonal with respect to the sub-matrices, D * - I ( W ~ W l ) D *-1 and W~W2. Then, R[D~I(W~CC)D~I]-IR ' can be replaced with the sub-matrix [D*-t(W ~W1)D~-1]-1. [But it does not necessarily mean that we can exclude Bt in Eq. (5).] Also noting that under the null, r(~)l)* is same as riD*, where rl =[-Yy*_le *, St*e*, ZD*e*]', we obtain,
~w = ( D r-1 W ltr l ) [r D r*-1 ( W l/W l ) D r*-1 ] -1 (Dr*-1 Wffl)/a,. i ~2
(A.2)
Let
[1 '2 / f.~.y,2 ~"t*y~__l .~V'y*__ ID ` / " " "q:/ 10
W l W l =-
and
W,2 W13/ (W ;W')--1----'~i~ \ W13
W23
W33/
where ID[ is the determinant. Then, T[D*-~(WIW~)D~.-~] -~ = _ _
T-6IDI
7/2 W12
T-9/2 W13 ~
T-7/2 W12
T - 3 W22
T - 4W23 /"
T-9/2W13
T - 4 W2a
T - 5W33 / (A.3)
230
J. Lee, C. Amsler / Structural Change and Economic Dynamics 8 (1997) 221-232
N o w , consider the a s y m p t o t i c results: (a) T-21o = T-ZSy*_21 ~ a 2 h 3 (b) T-5/2ll = T - 5 / 2 S t *y*_ 1 ---~0-hl (C) T-3/212 = T - a / 2 S y * 1D* --r6rh2
(d) T - a d l l = T - 3 S t * 2 ~ l / 1 2 (e) T - 2da 2 = T - 2 S t * D * --, - 0.5 ( 1 - A) A (f) T - adE2 --- T - aZ'D*2 --r( 1 - 2)/l, where ha, h 2 a n d h3 are defined in T h e o r e m 1. Letting r a - [ e a , e2, e3]', we have additional a s y m p t o t i c results: (g) T - le I = T - asy*_ a e* ~ 0 - 2 f l (h) T - 3/2e2 = T - 3/2y, t * e* ~ a f 2 (i) T - 1/2e3 = T - a/2SD*e* --*0-f3, where f l , f2 a n d f3 are defined in T h e o r e m 1. Then, we o b t a i n for each t e r m in Eq. (A.3): T - 6[D[ = - ( T - 2d12)2 T - 2l0 + T - 3dll T - ad22 T - 2l0 - T - ld22 ( T - 5/2l1)2 + 2 T - 2d12 T - 5/2lI T 3/2•2 - T - 3 d l a ( T - 3/212)2 ~ - ( 1 - 2)2tr2g T-4Wll
= --(T-2da2)
2 q- T - 3 d a l
T - ad22 --~ - ( 1 - - 2 ) 2 0 " 2 [ ( - - 3 2 2
-{- 3 2 - - 1)/(120-2)]
T - 7/2 Wa 2 ~ -- ( 1 - 2) 2tr 2 [h 1/0 + h 2/(20-)] T - 9/2 W13 -~ - ( 1 - 2)2a 2 [ha/(20-) + hE/( 12( 1 -- 2) 2a)] T - 3 I4/22 ~ -- ( 1 - A)20-2 [h 2/(( 1 - 2) 2) - h3 ] T - 4 W 2 3 ~ - ( 1 - 2) 20-2 [ - h 3/2 - h lh2/(( l - - ,~)/~)] Z - 5 W33 ~ - ( 1 - ,~) ~0-2 [(h 2 _ h3 / 1 2 ) / ( ( 1 - ~) ~)].
T h e n we obtain:
[D*- 1( W i W l ) D ~ r -
l] - 1 __H* ,
(A.4)
g where H * is the s a m e as H in T h e o r e m 1 except that the first r o w o f H * is given as the first r o w o f H multiplied by I * = d i a g o n a l [ I / o - 2, 1/0-, 1/0-]. We also obtain: D * - ar I --,m # = [0-2fl ,~f2,af3]'.
(A.5)
N o w , n o t e that H * = I , . H . I~ a n d I , m # = am, where I , = diagonal [ 1/a, 1, 1 ]. T h e n
231
J. Lee, C. Amsler / Structural Change and Economic Dynamics 8 (1997) 221-232
it is directly shown from Eqs. (A.4) and (A.5) that: .
1
H*
H
0 -2
- m' - - m, Cw'a.--7, m #' - -g m # = -0-2 g
(A.6)
which proves Theorem 1.
A.2. Proof of Theorem 2 Define ~ = f l -V, where • = 0.5 (0-2_ a2)/0-2. Also define m* = [ ~ , f2, f3]' such that r e = m * q-Ti3 where i 3 =(1, 0, 0)'. Then the asymptotic distribution Eq. (A.6) of the Wald test under iid errors is expressed as: H
~w-+m*' - - m*. g
(A.7)
Here, H, g and m* do not depend on error variances, so that the above expression is free of the nuisance parameters. We want to show that the corrected Wald test follows the same asymptotic distribution in Eq. (A.7). First, we alternatively show the expression in Eq. (A.6) by: --m'--m=
0-2
g
--(m*
0-2
+7i3)'
m* +~i3
= ~-m*'
at
--m*
g
0 -2 H 0 -2 H + 2 0-2 [(0-2 _0-2)/20-2]i ~ --g m* + ~ [(0-2 _0-2)/20-212i~ __gi 3. (A.8)
Now, we rewrite the corrected Wald test in Eq. (10) after some algebra as: r* ( ~ ) ' [ R d 2 ( W'Xd¢ ) - 1R 3 - lr* ( ~ ) = [ D * r ( ~ ) I ' I D * ( W ~W1 ) D * - '1
× [D*r(~)]/# 2 - 2~i~ [D*r(~)]/# 2 t + n~l2a. t[ D r ,( W l W l)Dr
*-1
~2 ] - 1 " 13/0-~.
(A.9)
We note that the first term of the last equation is equivalent to the expression for the 'uncorrected' Wald test in Eq. (8), and its asymptotic distribution is given in Eq. (A.6). Also note that 7~-,n=~0- 2. The asymptotic distribution of the second term is given after some algebra by: H* 1 H 1 H - 2ni~ - -g m # 7 =(0-2__ 0-2~i' 2 ","3 __ [(0-2_0-2)/20-21ia. e) 3 - - m* ~ - __(0-2 __o-e)!
g
~
(A.10)
g
The third term of Eq. (A.9) follows asymptotically: 0-2 H [(0-2 __ 0-2)/20-2]2 ~2- i~ --i3. of g
(A. 11 )
Finally, after rearranging Eqs. (A.7), (A.10), (A.11), we can show that the asymptotic distribution of the whole expression in Eq. (A.9) is the same as that of the
232
J. Lee, C. Amsler / Structural Change and Economic Dynamics 8 (1997)221-232
distribution of the uncorrected Wald test statistic under lid errors [which is expressed in Eq. (A.7), and is free of nuisance parameters].
References Andrews, D.W.K., 1991. Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817-858. Amsler, C., Lee, J., 1995. An LM test for a unit root in the presence of a structural break. Econometric Theory 11,359-368. Banerjee A., Lumsdaine, R.L., Stock, J.H., 1992. Recursive and sequential tests of the unit root and trend-break hypotheses: theory and international evidence. Journal of Business and Economic Statistics 10, 271 287. Dickey, D.A., Fuller, W.A., 1979. Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427 431. Lee, J., Mossi, D., 1996. On improvements of Phillips-Perron unit root tests using optimal bandwidth estimates. Applied Economics Letters 11, 197-200. Nelson, C.R., Plosser, C.I., 1982. Trends versus random walks in macroeconomic time series: some evidence and implications. Journal of Monetary Economics 10, 139-162. Perron, P., 1989. The great crash, the oil price shock, and the unit root hypothesis. Econometrica 57, 1361-1401. Phillips, P.C.B., Perron, P., 1988. Testing for a unit root in time series regression. Biometrika 75, 335-346. Schmidt, P., Phillips, P.C.B., 1992. Testing for a unit root in the presence of deterministic trends. Oxford Bulletin of Economics and Statistics 54, 257-287. Schwert, G.W., 1989. Tests for unit roots: a Monte Carlo investigation. Journal of Business and Economic Statistics 7, 147-159. Zivot, E., Andrews, D.W.K., 1992. Further evidence on the great crash, the oil price shock and the unit root hypothesis. Journal of Business and Economic Statistics 10, 251-270.