Statistical Papers 36, 77-95 (1995)

Statistical Papers

~) Springer-Verlag 1995

Modified stationarity tests with improved power in small samples J. B r e i t u n g *

Received: Oct. 7, 1993; revised version: June 8, 1994

In a recent paper Kwiatkowski eta]. (1992) propose the so-called KPSS statistic for testing the null hypothesis of stationarity against the alternative of a unit root process. The statistic employs a spectra] estimator which can be shown to diverge with increasing sample size, given the alternative is true. Here, we suggest a modified spectral estimator which is shown to stabilize for moving average models. It is shown that this test statistic uniformly outperforms the KPSS statistic in an MA(1) model. Furthermore, a two-step nonparametric correction procedure is suggested, giving a test statistic with similar asymptotic properties as the original KPSS statistic. However, in small samples this correction performs better especially in detecting large random walk components.

1

Introduction

Recently Kwiatkowski, Phillips, Schmidt and Shin (1992) (hereafter KPSS) propose a simple test statistic which allows to test the null of stationarity against the alternative

of a unit root. Such tests are seen as an appealing alternative to the application of the familar Dickey-FuUer type unit root tests.

First, the test allows to control the

probability of falsely deciding for the unit root process when the process is in fact stationary. By contrast, when applying Dickey-Fuller tests, this probability is related to the power of the test against stationary alternatives which is in general unknown to *This paper was written while the author was a post-doctoral fellow at the University of Amsterdam. The author likesto thank Peter Boswijk,Inge van den Doel, Noud van Giersbergenand Jan F. Kiviet for their help during that time. Moreover, I would like to thank an anonymous referee for a number of helpful comments.

78 the investigator. Second, the new class of tests seems to give more plausible results in some instances, as has been demonstrated by KPSS (1992) and Breitung (1994) using empirical examples. In particular, for variables bounded in a fixed interval, such as the unemployment rate or the capacity utilization rate, a unit root process does not seem logically convincing. While Dickey-Fuller type tests frequently accept the unit root hypothesis for such series, the stationarity tests more clearly indicate a stationary behaviour. The idea of the test statistic is to consider the partial sum of the detrended (or demeaned) time series. Under the null hypothesis of a stationary time series the partial sum converges with a standardization factor T -1/3 to a multiple of a second-level Brownian bridge ~ ( r ) .

Under the alternative of an I(1) series, the standardization

factor is T -s/2 and the limiting distribution of the partial sum is f~ W.(r)dr, where

W,(r) stands for a detrended Wiener process. Therefore, these functionals play an important role for the asymptotic behaviour of the KPSS statistic. An important problem with this test statistic is that it has to be corrected for the short run dynamics of the time series. A standard solution to this problem is to apply a nonparametric correction due to Phillips (1987). In contrast to the test for AR unit roots, where this correction has similar properties under the null and under the alternative, it can be shown that the nonparametric correction is 01,(IT ) under the alternative, where I is the truncation lag for the nonparametric correction. As a consequence, this correction leads to a loss of power when fixed alternatives are considered. A further problem with this type of correction is that it crucially depends on an appropriately chosen truncation lag l (KPSS 1992, Leybourne and McCabe 1994). To overcome these difficulties two alternative correction factors are suggested.

79 The first correction is applicable for finite order moving average models. The idea is to compute the autocorrelation of the series given the null hypothesis of an MA unit root is true. The resulting test statistic is of the same order of magnitude no matter whether the null hypothesis holds or not. Using this type of correction, the test statistic diverges faster than the original KPSS statistic. However, it is shown that the local power is not affected, so it is expected that this correction is in particular useful in small samples. An important drawback of this modification is that it is only applicable in finite order moving average models. Although the differences of many economic time series are reasonably well approximated by MA models, it is desirable to find a similar modification for more general model classes. However, this is not as straightforward as one might expect. In this paper a modification is suggested which only partly solves this problem. The idea is to construct a series that is identical to the original one if the time series is stationary and looks like the first differences in case of an I(1) series. A two-step least-squares procedure is suggested which adopts this idea. While the first requirement that the series should be unchanged under the null is fulfilled, the second requirement that the series should be the first differences under the alternative cannot be satisfied. However the modification usually yields a series that is "more stationary" under the alternative than the original one. Thus, although a full asymptotic justification is missing, we observe that this modi/ication is able to improve the original KPSS statistic substantially. In a simulation study we found that for sample sizes of 50 the KPPS test has no power if a truncation lag is twelve. Even in these cases, however, the modified KPSS statistic still has a reasonable power. The plan of this paper is as follows. Section 2 gives a brief exposition of the KPSS test. A correction for finite MA models is proposed in Section 3. Section 4

8o presents a modification of the KPSS statistic which is applicable for the same class of models as considered by KPSS (1992). Section 5 concludes.

2

The KPSS statistic

To formulate the null hypothesis in a convenient way, KPSS decompose the observed time series 1tt (t = 1, 2 , . . . , T) into a sum ~f a deterministic trend, a random walk and a stationary component ~ = ~t + rt + ~t.

The random walk component is given by rt = rt-1 + ut, where ttt

(1)

are

iid(O, ~ ) and

the fixed initial value, r0, serves the role of an intercept. The stationary component is assumed to be uncorrelated with the innovations of the random walk. Moreover, the process generating et satisfies the (strong mixing) regularity conditions of phillips and Perron (1988). Let St ----~~t

e, be the partial sum process for et, the residuals from a regres-

sion of Ztt on a linear time trend and an intercept. Then, KPSS show that under the null hypothesis H0 : ~ = 0 the statistic 7- = T -~ ~ = 1 S~ is asymptotically distributed as cra f01 V2(r)2dr, where . ' is the Ulong-run variance" of e, and V2(r) is the so-called second-level Brownian bridge. If, as a special case, it is assumed that ~ = 0 the partial sum process is constructed using et -- Yt - ~, where ~ denotes the sample mean. The resuiting statistic is denoted by ~u and KPSS establish that under the null hypothesis this statistic is distributed as cr2 $~ g ( r ) ' d r , where V(r) represents the standard Brownian bridge. Usually the long-run variance, ~ , is unknown. To obtain a test statistic free

81 from nuisance parameters, an estimator of this parameter is required. KPSS suggest the estimator l

(2)

9'(l) : ~o + ~ ]~ ~,(.)5.,

where Qo denotes the sample autocovariance of et at lag 8 and zoo(s) -- 1 - s / ( i + 1). This estimate is identical to the Bartlett estimate of the power spectrum at frequency zero assuming stationary time series. Under the null hypothesis, this estimator converges weakly to a= if I -* c~ and I/T ---, 0 as T tends to infinity. This estimator is used as a nonparametric correction for the test. The respective statistics are denoted by ~, = ~7~,/82(I) and ~,. = ~,./82(I). The problem with this type of correction is that under the alternative of an integrated time series s2(l) is Op(IT). Although the test is consistent, because the numerator of the statistic is Op(T=), the test loses power by using such type of correction. Moreover, under the alternative "~o ~ ~0 and, thus, the correction factor s2(l) defined in (2) is roughly proportional to (l + 1). Accordingly, the power of the KPSS statistic crucially depends on the truncation lag, which makes it difficult to choose the appropriate truncation parameter.

3

The

modified

correction

for MA

models

As has been argued in the previous section the nonparametric correction is not well behaved under the alternative.

To obtain a correction term which stabilizes as the

sample size approach infinity, we will assume that the stationary errors are generated by an invertible M A ( q - 1) error process et = vt - ~ =q -11 ~vt-~, where vt is a white noise sequence and the lag order q is known. Thus, the difference of Yt is an MA(q) process. As in the previous section let e~ denote the residuals from the regression of Yt on a trend

82

and an intercept. First we note that under the hypothesis H0 : a~ = 0 the differenced series z= = Ay~ has a unit root in its MA representation. This fact will be used when computing the autocovariance function (ACF) of e=. It can easily be verified that the autocovariance function "r~(./) = E(y,

- ,'o - ~t)(~,_~

- ,'o - ~(t - j))

(3)

is related to the ACF of the differenced series z= = Y=- yt-1

-y, Cj)

(4)

E(=, - ~)(~,_j - ~)

=

by the relationship

~, = F ~ ,

(5)

where F is a (q + 2) • (q + 2) matrix given by 2

-2

0

0

0

0

0

-1

2

-1

0

0

0

0

0

0

--I

2

-I

0

0

0

-I

2

F=

and % is a (q+2) • 1 vector stacking the covariances of zt = Y~-Y,-I for j = 0, I , . . . , q + l . The covariance vector of Yt is defined similarly. A consistent estimate of the autocovariances % ( j ) is obtained from the sample

83 autocovariance of the differenced residuals, -~,(j) = T -~ ~t AetAet_j ( j = 0, 1 , . . . , q + 1). Inserting these estimates we obtain estimates of % ( j ) from

+,=F-'+..

(6)

This estimator imposes the MA unit root constraint and, thus, is valid only under the null hypothesis. In other words our modified test uses estimates of the nuisance parameters derived under the null hypothesis, t Since Aet is stationary under the null as well as under the alternative the covariance estimates "~z(j) converge to E(AetAet_j). As a result "Yv converges in probability to F - 1 % even if ~t is generated by an I(1) process. By determining the inverse F - I we find

q+l ~'(3) =

q+2-Jh(0)+~'~ 2

.=,

min[q+2-j,q+2-rl~,(r)

(7)

(cf. Breitung 1994). The modified correction term results as

q

~'(q)=~(o)+2~(s).

(8)

Since we assume the order of the MA model to be finite and known, it is not necessary to introduce a weight function or a data dependent truncation lag to obtain a consistent spectral estimator. tThis approach can be rdated to the more famili~ situation of testing the coei~cieuts in a tineaz regression model. While the Wald p~cip]e amounts to a test using a variance estimate under the alternative, the score principle employs an estimate of the error variance given the null hypothesis is

true (cf. Eng]e 1984).

84

By construction the correction term ~:(q) is Op(1) under the null as well as under the alternative implying that the test statistic 4. = sl./~2(q) diverges at a rate of O ( T 2) under a fixed alternative. One might expect that the test gains asymptotic power

from such a modification, in the sense that it is able to detect alternatives converging with a higher rate to the null hypothesis than is detected by using the KPSS correction. However, as the following proposition demonstrates for the case that A!/r is an MA(1) process (i.e. et is white noise) this is not the case. PROPOSITION 1: Let the sequence of alternatives in (1) be c,,, = c~c/T, where c > 0 and et is a white noise series with E(et) = 0 and Var(et) = cr2, . Then, the local power of the tests is given by

L where W.(,') denotes a d~t~nd~a Wiener p , ' o ~

,,rid V2(,') is a ~o,~d-l~,,el Brownian

bridge. PROOF: See Appendix. Next we compare the small sample properties of both statisticsfor an MA(1)

model which is also used by Tanaka (1990). The relationship to the framework used by KPSS is as follows (cf. Hwang and Schmidt 1993). Consider the model (I) letting = 0 and assuming that e, follows a white noise process. Then,

A~h = zt

=

et--et-i

Jrt~t

=

v t -- ~ v t - 1

85 T A B L E 1: Size and Power in an MA( ) model T=50

T=100

T=150

Level: 0.05 0.05 0.05 0.05 0.05 0.30 0.31 0.49 0.50 0.60 0.63 0.76 0.78 0.84 0.87 0.94 0.96 0.93 0.96 0.99 0.99 0.98 1.00 1.00 1.00 Level: 0.01 = 1.00 0.01 0.01 0.00 0.01 0.01 0.01 0.95 0.03 0.04 0.17 0.19 0.66 0.69 0.90 0.15 0.19 0.44 0.47 0.66 0.69 0.80 0.40 0.48 0.75 0.81 0.88 0.91 0.70 0.59 0.69 0.81 0.87 0.95 0.97 0.50 0.77 0.88 0.93 0.98 0.99 1.00 Notes: Entries are the rejection frequencies for the null of an MA unit root in an Gaussian MA(1) model zt = Ayt = ~'t - ~ ' t - 1 using 2000 Monte Carlo replications. The significance level is 0.05 and 0.01, respectively. = 1.00 0.95 0.90 0.80 0.70 0.50

0.04 0.12 0.27 0.57 0.73 0.87

0.06 0.13 0.30 0.62 0.79 0.93

where E ( e , u , ) = O, a~ = ~ V a r ( v t ) a n d a~2 = (1 - f l ) Z V a r ( v t ) .

Thus we have a ~2/ a ; 2 =

(1 - ~)2/fl which is the crucial parameter for the power of the test. Formin 8 partial sums gives

t

Yt = ~_~ z . = eo § et -~ rt, a----1

where rt is the random walk process with ut as increments,

e0 can be seen as an

intercept which cancels by subtracting the mean. Thus, testing for a unit MA root = 1 is identical to test for a~ = 0. In Table 1 we present the empirical frequencies using the statistics ~ and ~]~,from 2000 Monte Carlo replications of the MA(1) model. In this case the nonparametric correction t e r m is simply 82(0) = "Yo (cf KPSS, p. 163), which is employed in our simulations, t tChoosing higher truncation lags for the KPSS statistic leads to a loss of power so that the peffor-

86 It turns out that the modified test is slightly more powerful than the original KPSS statistic. The difference in power is more substantial in small samples and by using lower significance levels.

4

A modified nonparametric

correction

In the previous section we have assumed that the difference of the time series follows an MA process with known lag order q. In this section a nonparametric correction according to (2) is proposed which is applicable for the class of models considered by KPSS. We will show that this correction behaves asymptotically identical to the one proposed by KPSS. However, there are reasons to expect that it has a better small sample behaviour. The idea behind the modification is simple. Consider the first order autoregression of the detrended series

et =

&et-1 + ~,,

(9)

where & denotes the OLS estimator. As is well known, & will converge to one with a rate of T whenever the original series ~t is an I(1) series. Now construct the partial sum S~' = Eto=l ~o and consider the regression

e, = ~,.,e~'_I + e;.

(10)

If et is stationary, the estimator ~ should be close to zero, because the partial sum mance of such tests would be worse than those reported here.

87 process S~'_a caamot exp!dn a stationaxy variable. If et is an I(1) series, then & is close to one giving S~_I ~ et-x and we expect "~ to be close to one. Hence, e~ ~ et for s stationary series and e~ ~, Act for an integrated series. This reasoning suggests that the autocovariances of e~ are bounded under the null as well as under the alternative and it is expected that computing the nonparametric correction using e~ leads to a more powerful test procedure. Thus far our reasoning is somewhat heuristic. In fe,ct it will be shown in the following proposition that our correction is valid under the null hypothesis but will not stabilize under the alternative. The reason is that the OLS estimator for 3' from (10) will converge to zero in the stationary case but does not eLpproach one if the series is I(1).

PROPOSITION 2: Let Yt be generated by (1) where the assumptions are the same

as in section ~. Then, let S~' = ~t.= 1 f/, be the partial sum process constructed from the residuals of (9). The limiting distribution of the regression coefficient from (10) is for (i) ~'. = o

v~,~ =~

[1 --"/(1)/"/(0)][o"

f~ 'V2d~

-

(o"-- o'~)/2]+ "/(1)2/3'(0) V2(r)2dr

o,~[1 - ~,(1)/3,(0)] z f

where 7(J) = plim T - I ( E e,e,_j), and for (ii) r > 0

f w.Cr)'dr - 2Z, f~ (f: w.Cr)dr) w.Ca)da + Z~ Sd (~

w.Cr)dr)"d,,'

where W,(r) denotes the detrended Wiener process, V2(r) represents a second-level Brownian bridge and Z, is a random variable distributed with the asymptotic distri.

88 bution of the antoregressive parameter in a Dickey-Fuller regression with linear time trend.

PROOF: See Appendix.

Under the null the residuals e~ behave like ordinary residuals from a regression with stationary variables while under the alternative the residuals have a nonstationary component. Nevertheless it is legitimate to hope that our intuitive reasoning is not a poor guide in small samples.

Since the regression minimize the variances of the

residuals, the regression will render a residual series that resembles a stationary series as much as possible and thus produces a correction term which is usually smaller than the one computed by the original KPSS correction. The modified test, denoted by Q~ and ~): results from replacing the correction term s~(1) by !

8~.(I) = ~ + 2 y~ wl(j)~,

(11)

j=l

where "~ = T -1 ~ t e;e~_j. In order to study the relative performance of the alternative corrections we perform a number of Monte Carlo experiments. We simulate data from model (1) generating et according to the stationary AR(1) process et = pet-a § vt, where vt " N(0,1 - p2) such that E(e~) = 1. Table 2 and Table 3 report the rejection frequencies of the test statistics ~, and ~ for p = 0.2, p = 0.5 and p -- 0.8, where the sample size is T = 50 and 100, respectively. We confine ourselves to a model without time trend, because the conclusions are very similar for the model with trend. From Table 2 and Table 3 it

89 TABLE 2: Size and Power in a model with AR(1) errors (T=50) l :

4

8

12

p=0.2 0.03 0.02 0.03 0.01 0.22 0.28 0.03 0.09 0.29 0.50 0.01 0.41 0.30 0.62 0.01 0.57 p=O.5 ~ = 0.00 0.07 0.07 0.03 0.03 0.02 0.02 0.41 0.10 0.45 0.19 0.29 0.02 0.15 0.56 0.67 0.29 0.55 0.01 0.47 1.00 10.0 0.59 0.75 0.31 0.63 0.01 0.58 p=O.8 2 or= = 0.00 0.20 0.27 0.05 0.13 0.01 0.08 0.42 0.54 0.17 0.40 0.01 0.08 0.10 0.57 1.00 0.71 0.28 0.59 0.01 0.54 I0.0 0.59 0.75 0.31 0.64 0.01 0.58 Notes: Entries report the rejection frequenciesfrom 2000 Monte Carlo runs of the model Yt = rt + e=, where rr = rr + ur and er = pet-1 + ~ using ut N(0,~.=) and vr ~ N(0,1 -p2). For all simulations the nominal significance ~ = 0.00 0.10 1.00 10.0

0.05 0.45 0.57 0.59

0.04 0.47 0.66 0.73

level is 0.05.

emerges that for p = 0.2 and p = 0.5 both test statistics perform similar under the null hypothesis. For p = 0.8 and T = 50 the modified KPSS statistic tends to reject the null hypothesis more often, so that tfigher truncation lags are required to actL~eve a significance level of 0.05. With respect to the power of the tests it turns out that for sample sizes of 50 and 100 the modified KPSS statistic outperforms the original KPSS statistic. In particular if there is an important random walk component, the modified statistic will detect it with a significantly higher probability than the original KPSS test. If the random walk component is small, both tests perform similar. In practical situations it is usually more important to detect nonstationarity when there is in fact a large random walk component. In this case the dynamic behaviour of the time series is dominated

90 TABLE 3: Size and Power in a model with AR(1) errors (T=100) l:

4

8

12

p=0.2 0.04 0.04 0.04 0.03 0.60 0.61 0.50 0.51 0.65 0.72 0.54 0.63 0.66 0.78 0.54 0.72 p=0.5 cr~ = 0.00 0.09 0.09 0.05 0.05 0.04 0.04 0.58 0.59 0.47 0.50 0.10 0.70 0.72 1.00 0.79 0.85 0.65 0.74 0.53 0.65 10.0 0.81 0.90 0.66 0.79 0.55 0.73 p=0.8 ~ = 0.00 0.24 0.27 0.13 0.14 0.08 0.09 0.10 0.69 0.75 0.54 0.62 0.43 0.53 0.64 0.77 0.52 ,0.69 1.00 0.79 0.88 0.66 0.79 0.55 0.72 10.0 0.81 0.91 Notes: Entries report the rejection frequenciesfrom 2000 Monte Carlo vans of the model la = rt -t-~=, where rt = rt-1 -t- ut and et = pej-I + ~t using u= ,,, N(0,o~) and v, ~ N(0, 1 -p2). For all simulations the nominal significance level is 0.05.

o~ = 0.00 0.10 1.00 10.0

0.06 0.72 0.80 0.82

0.06 0.73 0.85 0.90

by the random walk component, so that ignoring the nonstationarity of the data would give misleading results. Therefore, it is highly desirable to have a high power against alternatives with a large nonstationary component. Furthermore it turns out that the KPSS statistic fails to give a reasonable test if the truncation lag is large in comparison with the sample size. For a truncation lag of 12 the original KPSS statistic does not give a useful test for a sample size of 50. In contrast, the modified statistic ~ still has a reasonable power against alternatives with large random walk components.

9]

5

Conclusions

In this paper, two modified correction terms for the KPSS statistic are proposed. While the original correction for the short run dynamics of the series increases with Op(IT) if the alternative is true, the modified corrections are shown to be more stable. For finite order MA models one can easily incorporate the MA unit root constraint in order to get a correction which is Op(1) no matter whether the null hypothesis or the alternative is true. For more general models, however, it is much more difficult to construct a correction term with similar properties. Here we suggest a two-step approach which makes a step toward this end, but fails to stabilize asymptotically. Nevertheless, there are good reasons to expect that the modification has some merits in small samples. In fact, as turns out from a couple of Monte Carlo experiments, both modifications have the potential to improve the original KPSS procedure in small samples. In general, the gain increases with the size of the random walk component and decreases with growing sample size. For the two-step nonparametric correction we found that the power of the test is not as much affected by the choice of the truncation lag as is observed for the KPSS statistic. This is an important advantage for empirical applications. Since the test may have a serious size bias if the truncation lag is chosen too short, it is desirable to choose the lag length liberally. While in this case the original KPSS test suffers from a serious loss of power, the modified statistic is shown to be more robust.

92

Appendix P r o o f of P r o p o s i t i o n 1. Under the sequence of alternatives we have

C

where the increments of the random walk rt have unit variance. This gives

=~

..o. r (/',..~.~.)' ~. + ~4oZ' "

,/o

(/o','.~,~,) ~.~,~, +-.' Z'".~,~ '~,,

where ~ and e~ are the detrended analogs of rt and er W.(r) is the detrended Wiener process as given in Park and Phillips (1088) and V2(r) is a second-level Brownian bridge. Moreover we have

2c ~ . ,

,

1~

e: 2

' + o,,(T-'/')

O"G

and similarly

~(e,-

e,_,) ~ + 2(e, - e,-O(e,-~ -- e,_,) = ~.~ +

O~(T -~/~)

so that the limiting distributionsof ~, and ~, are as stated in Proposition I.

P r o o f o f P r o p o s i t i o n 2. Under the null hypothesis er is strong mixing with

93

7(J) = plim T - : E,(ete,-~) < oo. We then have

j, =

c,(c'_-s,.-,-,.,.-.),, v., (v.:__, e.'_,- 2ae._,,._, + ,~',:_,)"

Using standard asymptotic theory we find

T-'~(.=~3.~_.2&e._.eo_. + &'.,_.) ~ .:(1--~,7(1)~: : ~(r):dr For the numerator we have

=

O. Letting

z." =

T(~ - 1)

WC h a v e

and for the partial sum S [ = ~, - Z ~.

e._x

)

/T.

Thus, we find

:c, [,,_, - zr (V.:=,e._,)/T] ,,

j, =

r.,e:_,-

( = -)e,-,/~+