nonparametric tests for unit roots and cointegration

Journal of Econometrics 108 (2002) 343–363www.elsevier.com/locate/econbase

Nonparametric tests for unit roots andcointegrationJ#org Breitung ∗

Institute of Statistics and Econometrics, Humboldt University Berlin, Spandauer Strasse 1,D-10178 Berlin, Germany

Received 19 January 2000; received in revised form 29 October 2001; accepted 13 November 2001

Abstract

It is possible to construct unit root tests without speci,cation of the short-run dynamics. Thesetests are robust against misspeci,cation and structural breaks in the short-run components andcan be used to test a wide range of nonlinear models. The variance ratio statistic is similarto the test statistic suggested by Kwiatkowski et al. (J. Econom. 15 (1992) 159) but assumesnonstationarity under the null hypothesis. A straightforward generalization of the variance ratiostatistic is suggested, which can be used to test the cointegration rank in the spirit of Johansen (J.Econ. Dyn. Control 12 (1988) 231). Monte Carlo simulations suggest that the tests perform wellin linear and nonlinear models with a su8ciently large sample size. c© 2002 Elsevier ScienceB.V. All rights reserved.

JEL classi#cation: C22; C32

Keywords: Unit roots; Cointegration; Nonlinear processes

1. Introduction

Many tests for unit roots and cointegration are based on a parametric model. Forexample, the Dickey–Fuller test (Dickey and Fuller, 1979) and the cointegration testsproposed by Johansen (1988, 1991) employ an autoregressive representation of the timeseries. Other tests (e.g. Phillips and Perron, 1988; Quintos, 1998) use kernel estimatorsfor the nuisance parameters implied by the short-run dynamics of the process. It is,however, possible to construct test statistics that do not require the speci,cation ofthe short-run dynamics or the estimation of nuisance parameters. Such an approach is

∗ Tel.: +49-30-2093-5711; fax: +49-30-2093-5712.E-mail address: [email protected] (J. Breitung).

0304-4076/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S 0304 -4076(01)00139 -7

344 J. Breitung / Journal of Econometrics 108 (2002) 343–363

called “model free” in Bierens (1997a) and “nonparametric” in Bierens (1997b). Albeitboth terms may be somewhat misleading, we follow Bierens (1997b) and use the term“nonparametric”. In fact, it is di8cult to think of any test, which is “less parametric”.The idea behind this approach is the following. Under the null hypothesis it is

assumed that T−1=2y[aT ] converges weakly to �W (a), where W (a) denotes a stan-dard Brownian motion and �2 is the so-called “long-run variance” which is equal tothe limit of E(T−1y2

T ). To obtain a test statistic that is asymptotically free of un-known parameters, � may be replaced by a consistent estimate. It is, however, possibleto get rid of the nuisance parameter without using an estimator of �. For example,since the rank transformation RT (·) is invariant to a scale transformation of the serieswe have

RT (y[rT ]) = RT

(1

�√T

y[rT ]

)

and, thus, the asymptotic theory for a unit root test based on ranks does not involvethe parameters involved by the short-run dynamics of the process (see Breitung andGouriLeroux, 1997; Breitung, 2001).Park and Choi (1988) and Park (1990) were the ,rst who proposed test statistics that

do not require corrections for short-run dynamics. They observed that the normalizedF-statistic for superMuous regressors tends to a (nonstandard) limiting distribution notdepending on nuisance parameters. An important problem with this approach is thechoice of superMuous variables. If arti,cial stochastic variables are used, then the testdecision depends on a random draw so that another set of generated variables mayyield a conMicting result. Moreover, superMuous stochastic regressors are an additionalsource of randomness that can result in a serious loss of power.In this paper, we adopt the approach of Bierens (1997a, b) and Vogelsang (1998a, b)

to eliminate the nuisance parameter �. For example, consider the following test statistic:

T =T−1(

∑Tt=1 Nyt)2

T−2∑T

t=1 y2t

=(yT − y0)2

T∑T

t=1 y2t

: (1)

Under the null hypothesis of a unit root process and suitable assumptions on the initialvalue, we have

T ⇒ �2W (1)2

�2∫ 10 W (a)2 da

and, thus, the parameter �2 cancels from the limiting distribution.Unfortunately, a test based on T is inconsistent. The reason is that under the alter-

native of a stationary process, the numerator and denominator are of the same order ofmagnitude so that T is Op(1) under the alternative as well. Bierens (1997a) resolvesthis problem by using the squares of the weighted sum

T−1=2T∑

t=1

g(t=T ) t ⇒ �g(1)W (1)− �∫ 1

0∇g(r) dW (r) (2)

J. Breitung / Journal of Econometrics 108 (2002) 343–363 345

as the numerator in (1), where ∇g(r) denotes the derivative of g(r). 1 For someappropriate weight function g(r), it can be shown that the test is consistent againststationary alternatives.In the present paper a similar idea is adopted. However, instead of using weighted

sums in the numerator of the test statistic, we follow Vogelsang (1998a, b) and usefunctionals on the partial sum Yt = y1 + · · · + yt . The advantage of this approach isthat no weights are needed to make the test consistent. Furthermore, it turns out thatour tests are more powerful than (the stylized version of) Bierens’ (1997a) test andmay even outperform the augmented Dickey–Fuller test.The plan of the paper is as follows. In Section 2, a general framework is suggested

which allows for a wide range of nonlinear processes generating the transitory com-ponent. The power of Bierens’ (1997a) test is considered in Section 3. In Section 4,a variance ratio statistic based on partial sums is proposed. Section 5 generalizes thevariance ratio statistic to cointegrated systems and restrictions on the trend function areconsidered in Section 6. Section 7 presents the results of a Monte Carlo comparisonof alternative test statistics and Section 8 concludes.

2. The null hypothesis

Let {yt}T1 be an observed time series that can be decomposed as yt =�t + xt , where�t = E(yt) = �′dt is the deterministic component modeled as a linear combination ofa vector of nonrandom regressors dt . Typical components of dt are a constant, a timetrend or dummy variables.Following by Davidson (2002), we employ a de,nition of integration that is not

based on a particular time series model.

Assumption 1. A time series xt is integrated of order one; or xt ∼ I(1); if as T →∞T−1=2x[aT ] ⇒ �W (a);

where �¿ 0 is a constant; [ · ] represents the integer part; and W (a) is a Brownianmotion de,ned on C[0; 1].

DiRerent sets of su8cient conditions for the functional central limit theorem (FCLT)involved can be found in Herrndorf (1984), Gallant and White (1988), and Phillipsand Solo (1992).It is useful to decompose the stochastic part of the time series as xt = �t + vt ,

where �t is a random walk component with uncorrelated increments and T−1=2�[aT ] ⇒�W (a). The transitory component vt is Op(1) and represents the short-run dynam-ics of the process. For a linear process Nyt =

∑∞j=0 �j t−j with �0 = 1, E( t) = 0

and E( 2t ) = �2 ¡∞, the decomposition is valid whenever the process admits a

1 In fact, Bierens’ (1997a) test is more complicated because he uses vector weights and extra terms toaccommodate a nonlinear mean function. However, for our purpose it is su8cient to consider a simpli,edversion of his test given by (2).


Beveridge–Nelson decomposition, i.e., if∑∞

j=0 j2�2j ¡∞ (cf. Phillips and Solo, 1992).

Similarly, a Beveridge–Nelson type of decomposition is available for nonlinear pro-cesses(cf. Clarida and Taylor, 1999).Since the asymptotic properties of our tests do not depend on the transitory compo-

nent, the tests are robust against a possible misspeci,cation of the short-run dynamicsrepresented by vt . Furthermore, vt may be fractionally integrated with (1− L)dvt = t ,where L is the lag operator, d is a real number and t is white noise. From Sowell(1990, Theorem 1) it follows that Assumption 1 is satis,ed for d¡ 1. In such situ-ations, the augmented Dickey–Fuller test is expected to have poor power, because ahigh augmentation lag is needed to account for the long memory of the errors.

3. Bierens’ approach

In this section we consider Bierens’ (1997a) nonparametric test statistic. For conve-nience, we will neglect the deterministic part of series, dt so that yt = xt . As alreadymentioned, the statistic we consider is a “stylized version” of the test suggested byBierens. Since we neglect a (possibly nonlinear) time trend and consider a scalarweight function, the test statistic simpli,es to

�T =T (

∑Tt=1 g(t=T )Nyt)2∑T

t=1 y2t

: (3)

Bierens (1997a) construct the weights using Chebishev time polynomials but any otherdiRerentiable weight function may be used as well. It is interesting to consider theeRects of the weight function on the power of the test. The following propositioncharacterizes the asymptotic behavior of the test statistic under the alternative of astationary process.

Proposition 1. Let yt =∑∞

j=0 �j t be stationary and ergodic; E( 2t )=�2

and the rootsof the polynomial �(z) = �0 + �1z + �2z2 + · · · are all outside the complex unit circle.For T →∞; we have

T−1�T =v� ⇒ �2;

where

v� = 1 +∞∑j=1

2 Tcj�j; �j = E(NytNyt+j)=E(y2t );

cj;T =T−1 ∑Tt=1 g(t=T )g[(t+j)=T ]; Tcj=limT →∞ cj;T and �2 represents a �2-distributed

random variable with one degree of freedom.

Proof. From the central limit theorem for stationary processes (Hall and Heyde; 1980)it follows that

T−1=2T∑

t=1

g(t=T )Nyt ⇒ N (0; �2g);


where

�2g =

∞∑j=−∞

TcjE(NytNyt+j):

Furthermore; T−1 ∑y2t converges in probability to E(y2

t ); so that T−1�T =v� ⇒ �2;where v� = �2

g=E(y2t ).

This proposition shows that the power of the test crucially depends on the weightfunction. Thus, it is important to specify the weight function carefully. For illustration,consider the trigonometric weights

gk(t=T ) = cos(!kt=T );

where !k = k · 2$, k = 1; 2; : : : : A similar weight function is considered in Bierens(1997b). The main diRerence between gk(t=T ) and the Chebychev polynomial used inBierens (1997a) is that the Chebychev polynomial introduces a phase shift. However,this does not have any eRect on our discussion.A second order Taylor expansion gives

cos(!k

t + jT

)� cos(!kt=T ) + sin(!kt=T )

j!k

T− 1

2cos(!kt=T )

(j!k

T

)2

and, thus,

2cj;T � 1− ( j!k)2

2T 2 :

If yt is white noise, we have �1 = −1 and �j = 0 for j¿ 2. Therefore, to achieve agood power of the test, c1;T should be as small as possible, that is, a high frequencyshould be used for the trigonometric weights. On the other hand, if Nyt is positivelycorrelated, a low frequency is more appropriate. This example demonstrates, that thereis no uniformly optimal weight function for the test and it is di8cult to specify theweight function without an idea about the autocorrelation function of the series (seealso Tschernig, 1997).Another problem is that the frequency of the trigonometric weight function must be

low relative to the sample size. Assume that the frequency grows with the sample sizesuch that k = T=(2q) and, therefore, gq(t=T ) = cos($t=q), where q = 1; 2; : : : : For themaximal frequency q=1, the weight function Mips between the values 1 and −1. Notethat in this case the weight function is not diRerentiable. From the above reasoning, weexpect that setting q= 1 yields a test with optimal power against a white-noise series.However, as shown by the following proposition, the asymptotic theory for such a testis diRerent.

Proposition 2. Let gq(t=T ) = cos($t=q); where q¡∞ and assume that yt obeysAssumption 1. Then; as T →∞ we have

T−1=2T∑

t=2

cos(t$=q)Nyt ⇒ $√2fNy($=q)W (1); (4)

where fNy($=q) denotes the spectral density of Nyt at frequency $=q.


Proof. From Eq. (32) of Phillips and Solo (1992) we haveT∑

t=2

cos(t$=q)Nyt =Re

[T∑

t=2

eit$=qNyt

]

=Re

[�(ei$=q)

T∑t=2

ei$=q t +Op(1)

]

=T∑

t=1

{Re[�(ei$=q)] cos(t$=q)− Im[�(ei$=q)] sin(t$=q)}+Op(1);

where Re(a) and Im(a) denote the real and imaginary part of the complex number a.The phase of the ,lter �(L) is de,ned as

'�(!) = tan−1{−Im[�(e−i!)]=Re[�(e−i!)]}:Furthermore,

a cos j!+ b sin j!= cos[j!+ tan−1(−b=a)]:

This givesT∑

t=1

cos(t$=q)Nyt =T∑

t=1

cos[t$=q+ '�($=q)] t +Op(1):

Using the results of Chan and Wei (1988), we have

T−1=2T∑

t=1

cos[t$=q+ '�($=q)] t ⇒ 2$fNy($=q)√2

W (1)

which yields the desired result.

Note that for q¿ 1, we have limT →∞ T−1 ∑Tt=1 cos(t$=q)

2 = 12 so that if Nyt is

white noise, expression (4) is normally distributed with variance �2 =2. Furthermore, it

follows from Proposition 2 that, in general, the limiting distribution of the test statistic�T depends on fNy($=q).To summarize, Bierens’ (1997a) asymptotic theory is valid for ,xed k in the weight

function g(t=T )=cos(k2$t=T ). However, if k →∞ at the rate T , then a diRerent asymp-totic theory applies and using the limiting �2 distribution results in severe size distor-tions.

4. The variance ratio statistic

To test the null hypothesis that yt is I(0) against the alternative yi ∼ I(1), Tanaka(1990) and Kwiatkowski et al. (1992) suggest an LM-type test statistic. If it is assumedthat under the null hypothesis yt is white noise with zero mean, the test statistic is

%T =T−2 ∑T

t=1 Y2t

T−1∑T

t=1 y2t

; (5)


where Yt = y1 + · · · + yt denotes the partial sum process. If yt is serially correlated,the denominator is replaced by the estimated long-run variance (cf. Kwiatkowski etal., 1992). Note that %T is the (normalized) variance ratio of the partial sums andthe original series. Such statistics have a long tradition in time series analysis. Forexample, the Durbin–Watson statistic is the ratio of sample variances computed fromthe original and the diRerenced series (e.g. Anderson, 1971, Section 3.4.5).In contrast to Kwiatkowski et al. (1992), the variance ratio statistic is employed to

test the null hypothesis that yt is I(1) against the alternative yt ∼ I(0). Thus, our testMips the null and alternative hypothesis of the test suggested by Kwiatkowski et al.(1992). To adjust for a nonzero mean of the form dt = �′zt , the time series yt is re-gressed on zt and the residuals u t=yt−�

′zt are used to form the variance ratio statistic:

%T =T−1 ∑T

t=1 U2t∑T

t=1 u2t; (6)

where U t = u 1 + · · ·+ u t . For d= 0 we let u t = yt .As suggested by a referee, the power of the test statistic can be improved by using

the local-to-unity GLS procedure of Elliott et al. (1996). The details of such a testprocedure are considered in Appendix B. It should be noted, however, that such alocal-to-unity GLS procedure is based on a particular sequence of parametric alterna-tives, so that it does not ,t well to the nonparametric Mavor of our test. Nevertheless,the simulations reported in Appendix B suggest that such a modi,cation can improvethe power of the test against linear alternatives.In contrast to the stationarity test (5), the variance ratio statistic is a left tailed test

that rejects for small values of the test statistic. For the usual mean functions, thefollowing proposition presents the limiting null distribution of the test statistics.

Proposition 3. Under Assumption 1 we have

T−1%T ⇒∫ 10

[∫ a0 W j(s) ds

]2da∫ 1

0 W j(a)2 da;

where

W0(s) ≡ W (s) for dt = 0;

W1(s) ≡ W (s)−∫ 1

0W (a) da for dt = 1;

W2(s) ≡ W (s)− (4− 6s)∫ 1

0W (a) da− (12s− 6)

∫ 1

0aW (a) da for dt = [1; t]′:

Proof. From Assumption 1 it follows that

T−1=2u [aT ] ⇒ Wj(a);

T−3=2U [sT ] ⇒∫ s

0Wj(a) da


(e.g. Park and Phillips; 1988). Thus; we obtain

T−1%T =T−4 ∑T

t=1 U2t

T−2∑T

t=1 u2t

⇒∫ 10

[∫ a0 Wj(s) ds

]2da∫ 1

0 Wj(a)2 da:

It is important to note that the null distributions do not depend on nuisance parame-ters. This is due to the fact that the parameter �2 cancels from the variance ratio. Sim-ulated critical values of the asymptotic null distributions are provided in Appendix A.The following proposition shows that the test is consistent against stationary alternativesand considers the usual class of local alternatives (e.g. Phillips, 1987).

Proposition 4. Let yt be stationary with Wold representation yt =∑∞

j=0 �j t−j; where�0 = 1;

∑∞j=0 �

2j ¡∞; and t is white noise with E( t) = 0 and E( 2t ) = �2

. Under thisalternative; we have T−1%T ⇒ 0 as T →∞ and

%T ⇒ �2∫ 10 Wj(a)2 da

�2y

;

where �2 = (∑∞

j=0 �j)2�2

and �2y = �2

∑∞

j=0 �2j .

Under the local alternative yt='Tyt−1+ t with 'T=1−c=T; the limiting distributionis given by

T−1%T ⇒∫ 10 [

∫ a0 J

cj(s) ds]

2 da∫ 10 J

cj(a)2 da

;

where Wj(a) is de#ned in Proposition 3 and Jcj(r) results from the same expressions,

if W (a) is replaced by the Ornstein–Uhlenbeck process J c(a) =∫ a0 e(a−s)c dW (s).

Proof. Under a stationary alternative; we have

T−2T∑

t=1

U2t ⇒ �2

∫ 1

0Wj(a)2 da:

Using this result and T−1 ∑Tt=1 u

2t

p→ �2y; the limiting distribution in the case of a sta-

tionary alternative follows easily.Under the sequence of local alternatives, we have (cf. Phillips, 1987)

T−1=2u [aT ] ⇒ Jcj(a);

T−3=2U [aT ] ⇒∫ a

0Jcj(s) ds:

Therefore, the limiting distributions result from replacing the W (a) by J c(a) in Propo-sition 3.


5. Testing the cointegration rank

The variance ratio statistic for a nonparametric unit root test can be generalized totest hypotheses on the cointegration. It is assumed that the process can be decomposedinto a q-dimensional vector of stochastic trend components �t and a (n−q)-dimensionalvector of transitory components vt .

Assumption 2. There exists an invertible matrix Q= [�; 2]; where � and 2 are linearlyindependent n× q and n× (n− q) matrices; respectively; with 0¡q¡n such that

Q′(yt − �t) =

[�′(yt − �t)

2′(yt − �t)

]≡

[�t

vt

]= zt ;

T−1=2�[aT ] ⇒ Wq(a);

T−2T∑

t=1

vtv′t = op(1);

where �t =E(yt) and Wq(a) is a q-dimensional Brownian motion with unit covariancematrix.

It is important to notice that the matrix Q in Assumption 2 need not to be known.Therefore, the test is invariant to a “rotation” of the system y∗

t = Ayt . Furthermore,we do not assume that the linear combination vt = 2′(yt − �t) is stationary. Instead,we assume that the trend component �t is “variance dominating” in the sense that thevariance of �t diverges with a faster rate than vt . Therefore, the transitory componentcan be generated by any nonlinear process with short-memory properties.The dimension of the stochastic trend component �t is related to the cointegration

rank of the linear system by q = n − r, where r is the rank of the matrix 4 in theso-called vector error correction representation

Nyt =4yt−1 + et ; (7)

and et is a stationary error vector. In a linear system, the hypothesis on the number ofstochastic trends is equivalent to a hypothesis on the cointegration rank as in Johansen(1988). However, since we do not assume that the process is linear, the representationof the form (7) may not exist.Let u t denote the vector of least-squares residuals from a regression of yt on the

vector of deterministic terms dt . Our test statistic is based on the eigenvalues 6j

(j = 1; : : : ; n) of the problem

|6jBT − AT |= 0; (8)

where

AT =T∑

t=1

u t u′t ; BT =

T∑t=1

U tU′t ;


and U t =∑t

j=1 u j denotes the n-dimensional partial sum with respect to u t . The

eigenvalues of (8) are identical to the eigenvalues of the matrix RT = ATB−1T . For

n = 1, the eigenvalue is identical to (T %T )−1 and, thus, the test can be seen as a

generalization of the variance ratio statistic to multivariate processes.The eigenvalues of (8) can be written as

6j =8′jAT 8j

8′jBT 8j; (9)

where 8j is the eigenvector associated with the eigenvalue 6j. If the vector 8j fallsinside the space spanned by the columns of �, then 8′jAT 8j is Op(T 2) and 8′jBT 8j isOp(T 4) so that the eigenvalue is Op(T−2). On the other hand, if the eigenvector 8j

falls into the space spanned by the columns of 2, it follows that T 26j tends to in,nity,as T →∞. Therefore, the test statistic

9q = T 2q∑

j=1

6j (10)

has a nondegenerate limiting distribution, where 616 626 · · ·6 6n denote the orderedeigenvalues of the matrix RT . In contrast, if the number of stochastic trends is smallerthan q, then 9q diverges to in,nity. The following proposition presents the limitingnull distribution for the test statistic 9q.

Proposition 5. Assume that yt admits a decomposition as in Assumption 2 with0¡q6 n. Then; as T →∞

9q ⇒ tr

∫ 1

0W

qj (a)W

qj (a)

′ da

[∫ 1

0V

qj (a)V

qj (a)

′ da

]−1 ;

where Wqj (a) is the q-dimensional analog of Wj(a) de#ned in Proposition 3 and

Vqj (a) =

∫ a0 W

qj (s) ds.

Proof. Let Z t =∑t

j=1 zj denote the partial sum with respect to zt = Q′u t = [�′t ; v

′t]′.

Then; the eigenvalues of problem (8) also solves the problem

|6jDT − CT |= 0;

where

CT =T∑

t=1

zt z′t ; DT =

T∑t=1

Z t Z′t :

Next; we partition the corresponding eigenvectors 8j=[8′1j; 8′2j]

′ such that 8′j zt= 8′1j�t+8′2jvt ; and Z t is partitioned accordingly. We normalize the matrix of eigenvectors as

[81; : : : ; 8q] =[

Iq=T

]


so that 8′j zt = �jt + '′jvt for j6 q; where �jt denotes the jth component of the vector

�t and 'j denotes the jth column of XT . It follows that

6j =8′jCT 8j

8′jDT 8j

=

∑Tt=1 �2jt + op(T 2)∑Tt=1 Z2

jt + op(T 4)

=

∑Tt=1 �2jt∑Tt=1 Z

2jt

+ op(1);

where Zjt =∑t

s=1 �js. As T →∞; we; therefore; have

T 2q∑

j=1

6j ⇒ tr

∫ 1

0W

qj (a)W

qj (a)

′ da

[∫ 1

0V

qj (a)V

qj (a)

′ da

]−1 :

From this proposition it follows that the distribution of the q smallest eigenvaluesof problem (8) does not depend on nuisance parameters and, thus, we do not need toselect the lag order of the VAR process as in Johansen’s approach or the truncationlag as for the test of Quintos (1998).

6. Cointegrated systems with restricted trends

In cointegrated systems it is often the case that the deterministic terms are con-strained under the cointegration hypothesis. In particular, it is assumed that yt has alinear time trend, whereas the cointegrating relations 2′yt have a constant mean (e.g.Johansen, 1994). This speci,cation implies that 2′yt has to be adjusted for a constantmean, whereas the vector of permanent components �′yt is adjusted for a time trend.To impose such restrictions on the deterministic terms, estimates for the matrices 2 and� are needed. A possible way to estimate these matrices is to use the principle compo-nent estimator, which has the attractive property that � is estimated as an orthogonalcomplement of the cointegration matrix. It follows, that this matrix is estimated withthe same convergence rate as the cointegration matrix (cf. Harris, 1997).Let 2 and � denote the estimates from a principal component procedure. Then, the

adjusted vector of time series results as

z∗t =

[�′yt − a0 − a1t

2′yt − b

];

where a0 and a1 are the least-squares estimates from a regression of �′yt on a constantand a time trend and b denotes the mean of 2

′yt . Then, the statistic is computed by

using z∗t instead of yt and the critical values for a test with time trend are applied.


Note that

z∗t =[�′yt − E(�′yt) + op(T 1=2)2′yt − E(2′yt) + Op(1)

]

and, thus, the diRerences between the estimated and true nonstationary componentsare asymptotically negligible but the transitory components are measured with annonvanishing error. However, the transitory components do not aRect the asymptoticnull distribution of the test statistic so that the limiting distribution is the same as forthe case with an unrestricted linear time trend (see Table 6 for critical values).

7. Small sample properties

In this section, we compare the small sample properties of the tests by means ofMonte Carlo simulations. It is not intended to give a comprehensive account of themerits and drawbacks of our test relative to other unit root tests based on a parametricor semiparametric adjustment for short-run dynamics. Rather, we try to give a roughidea of the relative performance of the tests, where the augmented Dickey–Fuller testis used as a benchmark.For the univariate tests, the data are generated by the process

xt = 'xt−1 + t − 2 t−1 (11)

and yt=�′dt+xt , where t ∼ niid(0; 1) and dt=1 (constant mean) or dt=[1; t]′ (lineartrend). The sample size is T = 200. Under the null hypothesis we have ' = 1 and2¡ 1. For 2 �=0, the process has no ,nite order AR representation and following Saidand Dickey (1984) an autoregressive approximation is employed with p = 4 and 12lagged diRerences. This test is denoted by ADF(p).The ,rst nonparametric test in the comparison is the variable addition statistic of Park

(1990). Four independently generated random walks are used as superMuous regressorsyielding the test statistic J2(4). 2 For our version of Bierens’ (1997a) test, we use atrigonometric weight function given by gk(t=T ) = cos(2$kt=T ), where k = 1; 4; 16; 32.The respective test is labeled as �T (k). The critical values with respect to a signi,cancelevel of 0.05 are obtained from 10.000 Monte Carlo runs of the model with '=1 and2 = 0. To adjust for deterministic terms, the test statistic is constructed using theresiduals from a regression of yt on a constant or a linear trend.

Finally, the variance ratio statistic is computed using the residuals from a regressionof yt on a constant or a linear trend. The respective test statistic is denoted as T−1%T .Critical values for this test can be found in Appendix A (Table 5).Table 1(a) presents the empirical sizes computed as the rejection frequencies for

H0 :'=1 and various values of 2 in a model with a constant mean. Since the criticalvalues are computed from the same random draws, the empirical sizes are exact 0.05for J2(4), �T (k) and T−1%T .

2 Park (1990, Remark c) recommends to use two or more regressors since “a single superMuous regressorseems insu8cient to discriminate the competing models for small samples”.


Table 1Rejection frequencies for a model with a constant mean

Test statistic 2 =−0:5 2 = 0 2 = 0:5 2 = 0:8

(a) Empirical size (' = 1)J2(4) 0.048 0.050 0.077 0.270�T (1) 0.050 0.050 0.042 0.020�T (4) 0.049 0.050 0.042 0.023�T (16) 0.048 0.050 0.076 0.197�T (32) 0.033 0.050 0.167 0.451T−1%T 0.047 0.050 0.072 0.223ADF(4) 0.049 0.052 0.062 0.373ADF(12) 0.058 0.060 0.055 0.074

Test statistic ' = 0:95 ' = 0:90 ' = 0:80 ' = 0:50

(b) Empirical power (2 = 0)J2(4) 0.239 0.448 0.780 0.997�T (1) 0.033 0.016 0.007 0.006�T (4) 0.179 0.212 0.170 0.052�T (16) 0.204 0.325 0.447 0.494�T (32) 0.211 0.335 0.483 0.638T−1%T 0.290 0.539 0.805 0.990ADF(4) 0.262 0.680 0.982 1.000ADF(12) 0.202 0.417 0.715 0.923Note: The entries of the table display the rejection frequencies based on 10,000 replications of model

(11), where dt = 1. The sample size is T = 200 and the nominal size of the test is 0.05. Since the criticalvalues are computed from the same random draws, the empirical sizes are exact 0.05 for J2(4), �T (k) andT−1%T .

The results of the Monte Carlo experiment indicate that Park and Choi’s test J2(4)has severe size distortions if the moving average parameter 2 tends to one. A similarproblem was observed for the semiparametric test suggested by Phillips and Perron(1988) (e.g., Schwert, 1989; Perron and Ng, 1996). 3 The performance of Bierens’ testdepends on the frequency of the weighting function. For low frequencies the empiricalsizes are close to the nominal ones for all values of 2. For high frequencies the testshows serious size distortions even for moderate values of 2. This is due to the factthat as k →∞, the asymptotic distribution involves the parameter 2 (see Section 2).Similar results are obtained for the model with a time trend (see Table 2).The variance ratio statistic T−1%T also shows a considerable size bias for large

positive values of 2. For 2= 0:5, the size bias is moderate but for 2= 0:8, the test isseverely biased towards a rejection of the null hypothesis. A similar outcome is foundfor ADF(4), however, if the ADF test is augmented with 12 lagged diRerences, theempirical size is close to the nominal size for values up to 2 = 0:8.The empirical powers of the test procedures for diRerent values of ' are presented

in Table 1(b) for the model with a constant and Table 2(b) for the model with linear

3 Perron and Ng (1996) propose a small sample modi,cation of the test statistic and adopt a parametricapproach to estimate the nuisance parameters. Since, we do not assume a particular model for the short-rundynamics, such modi,cations are not applicable here.


Table 2Rejection frequencies for a model with a linear trend

Test statistic 2 =−0:5 2 = 0 2 = 0:5 2 = 0:8

(a) Empirical size (' = 1)J2(4) 0.044 0.050 0.076 0.345�T (1) 0.053 0.050 0.036 0.008�T (4) 0.052 0.050 0.036 0.008�T (16) 0.047 0.050 0.071 0.146�T (32) 0.033 0.050 0.169 0.399T−1%T 0.045 0.050 0.102 0.452ADF(4) 0.048 0.055 0.073 0.533ADF(12) 0.057 0.058 0.057 0.088

Test statistic ' = 0:95 ' = 0:90 ' = 0:80 ' = 0:50

(b) Empirical power (2 = 0)J2(4) 0.115 0.221 0.459 0.918�T (1) 0.013 0.002 0.001 0.001�T (4) 0.101 0.116 0.074 0.013�T (16) 0.118 0.206 0.313 0.365�T (32) 0.120 0.214 0.350 0.518T−1%T 0.182 0.420 0.788 0.995ADF(4) 0.162 0.454 0.901 1.000ADF(12) 0.127 0.260 0.491 0.765Note: The entries of the table display the rejection frequencies based on 10,000 replications of model

(11), where dt = [1; t]′. The sample size is T = 200 and the nominal size of the test is 0.05.

time trend. The MA parameter 2 is set to zero. It turns out that using a trigonometricweight function with a low frequency yields a poor performance of the Bierens’ typeof test. As expected (see Section 3) the power of the test improves with an increasingfrequency. However, since the actual size of the test increases as well, it is quitedi8cult to select an appropriate frequency. For '=0:95, the variance ratio test is evenslightly more powerful than the ADF(4) test, whereas for other values of ', the powerof the variance ratio test is larger than the power of the ADF(12) statistic but smallerthan the power of the ADF(4) statistic.Next, we consider four nonlinear processes:

“bilin” : Nyt = 0:9 t−1Nyt−1 + t ; (12)

“VCM” : Nyt = ?tNyt−1 + t with ?t = 0:9 cos(2$t=T ); (13)

“TAR” : Nyt =

{0:9Nyt−1 + t for |Nyt−1|¡ 2;

−0:9Nyt−1 + t for |Nyt−1|¿ 2;(14)

“STUR” : yt = ?tyt−1 + t where ?t = 0:1 + 0:9?t−1 + 8t : (15)

The ,rst process (12) is a bilinear process (see Granger and Anderson, 1978), wherethe correlation between t−1 and Nyt−1 implies a linear time trend in yt . The secondprocess (13) is a variable coe8cient model with a cyclical coe8cient ?t . The thirdprocess (14) is a threshold autoregressive process and the fourth process (15) is a


Table 3Empirical sizes for some nonlinear processes

Process ADF(1) T−1%T J2(4) �T (16)

Sizebilin 0.103 0.046 0.048 0.051VCM 0.182 0.077 0.067 0.013TAR 0.011 0.059 0.057 0.034STUR 0.135 0.029 0.044 0.010

Powerbilin 0.703 0.400 0.210 0.083VCM 0.013 0.231 0.134 0.165TAR 0.088 0.463 0.252 0.165STUR 1.000 0.919 0.650 0.364Note: The entries of the table display the empirical sizes computed from 10,000 replications of model

(12)–(15). “bilin” denotes the bilinear process (12). “VCM” is the variable coe8cient model (13), “TAR”is a threshold unit root process (14), and “STUR” is a stochastic unit root process (15). The sample size isT = 200 and the nominal size of the test is 0.05. For “bilin”, “VCM” and “TAR” the power is computedby testing the series generated as y∗

t = 0:9y∗t−1 + Nyt and y∗

0 = 0. For the “STUR” model, the alternativeis an autoregressive process with ?t = 0:1 + 0:85?t−1 + 8t and ?0 = 0:95.

stochastic unit root process as considered in Granger and Swanson (1997), where forall processes t is white noise with E( 2t ) = 1 and E(82t ) = 0:052. For this speci,cationwe have E(?t) = 1.The empirical sizes were computed from 10,000 realizations with T = 200. All

tests allow for a linear trend. From the results in Table 3, it can be concluded thatthe nonparametric tests are more robust against nonlinear short-run dynamics than theADF(1) test that assumes a linear autoregressive process. Among the nonparametrictests, the variable addition test of Park and Choi (1988) is slightly more robust thanits competitors. On the other hand, the variance ratio test is much more powerful thanthe variable addition test and the stylized version of Bierens’ test. For the VCM andTAR process, the variance ratio test also outperforms the ADF(1) test.To investigate the properties of the nonparametric cointegration test, we generate

data according to the “canonical” process (Toda, 1994) with MA(1) errors[Nx1tNx2t

]=['1 00 '2

] [x1; t−1

x2; t−1

]+[ 1t 2t

]−[0:5 00 0:5

] [ 1; t−1

2; t−1

]; (16)

where yt = � + xt , E( 21t) = E( 22t) = 1 and E( 1t 2t) = @. To test the hypothesis r = 1,we let '1 = 0 and '2 = −0:2. Under the alternative we set '1 ∈{−0:05;−0:1;−0:2}.Furthermore, we let @ = 0 and 0.8 to investigate the impact of the error correlation.The sample size is T =200 and 10,000 samples are generated to compute the rejectionfrequencies of the tests.For Johansen’s LR trace test, the process is approximated by a VAR(p) process,

where p is 4 and 12, respectively. The respective tests are denoted by LR(4) andLR(12) in Table 4. Unrestricted constants are included in each equation. The non-parametric test statistic is denoted by 9q and the critical values are taken fromTable 6 in the appendix. First, consider the results for testing H0 : q= r = 1. From the


Table 4Testing hypotheses on the cointegration rank

Test statistic '1 = 0 '1 =−0:05 '1 =−0:10 '1 =−0:20

H0: r = 1, '2 =−0:2@ = 091 0.059 0.346 0.604 0.853LR(4) 0.072 0.428 0.894 0.999LR(12) 0.048 0.180 0.389 0.636

@ = 0:891 0.043 0.295 0.566 0.853LR(4) 0.057 0.310 0.793 0.999LR(12) 0.063 0.190 0.382 0.636

H0: r = 0, '2 = 0@ = 090 0.107 0.300 0.582 0.900LR(4) 0.083 0.241 0.558 0.962LR(12) 0.094 0.166 0.290 0.506

@ = 0:890 0.107 0.240 0.508 0.854LR(4) 0.083 0.511 0.949 1.000LR(12) 0.094 0.352 0.581 0.768Note: The entries of the table report the rejection frequencies based on 10,000 replications of model (16),

where Dt is constant.

empirical sizes (see Table 4), it turns out that for @ = 0, a VAR(4) model is notsu8cient to approximate the in,nite VAR process, whereas a VAR(12) approximationyields an accurate size. The nonparametric statistic 91 possesses a negligible size bias,only. The power of 91 is substantially smaller than the power of LR(4) but clearlyhigher than the power of LR(12). Similar results apply for the tests letting @ = 0:8.However, the LR(12) statistic now possesses a moderate size bias, whereas 91 is nearlyunbiased. Moreover, the power of 91 is closer to the (favorable) LR(4) statistic thanin the case of @= 0.We now turn to the test of H0 : r = 0. Under the null hypothesis the diRerences of

the variables are generated by a multivariate MA process. In this case, all three teststatistic are substantially biased, where the size bias does not depend on the parameter @.Although the sizes bias diRers for the three test, the diRerences are moderate and somegeneral conclusions with respect to the relative power of the tests can be drawn. For@= 0 and '1 close to unity, the nonparametric test 90 is slightly more powerful thanthe LR(4) test, whereas for '1 =−0:2 the power of LR(4) is slightly higher. Finally,the power of LR(12) is much smaller than the power of the other two tests. For @=0:8a diRerent picture emerges. The relative power of 90 drops substantially and for '1

close to one, the power is even lower than the power of the LR(12) test. The resultsfor a model with a linear time trend are qualitatively similar and are not presented forreasons of space.


8. Concluding remarks

Following Park and Choi (1988), Bierens (1997a, b) and Vogelsang (1998a, b),unit root tests can be constructed which, asymptotically, do not depend on parametersinvolved by the short-run dynamics of the process. The variance ratio statistic has theadvantage that the outcome of the test does not depend on a random draw of super-Muous variables or the frequency of the weight function. Moreover, our Monte Carlosimulations suggest that the variance ratio test has favorable small sample properties.For practical applications of the tests, several points deserve attention. First, the

invariance to the short-run dynamics of the process is an asymptotic property that neednot be encountered in small samples. In particular, if the variance of the transitorycomponent is important relative to the variance of the random walk component, thetest may suRer from severe size distortions. Second, it has been shown that under thealternative of a stationary process, the appropriately normalized test statistics convergeto a random variable as T tends to in,nity. In contrast, the normalized Dickey–Fullertest converge to a constant under the null hypothesis and, therefore, the test generallyhas more favorable properties than the nonparametric counterparts. Finally, in manyempirical applications it seems not di8cult to select an appropriate augmentation lagand the test statistic turns out to be quite robust against diRerent lag orders.However, there are a number of situations, where the nonparametric approach may be

attractive. Since the short-run component does not aRect the asymptotic null distributionof the test statistic, the test is robust against deviations from the usual assumption oflinear short-run dynamics. This property is important in large samples, where smalldeviations from the underlying (parametric) assumptions may have a substantial eRecton the behavior of the parametric test statistic. Furthermore, when the sample size islarge, there is reason to expect that the random walk component dominate the samplingbehavior of the nonparametric test statistic and the asymptotic theory provides a reliableapproximation to the actual null distribution. If, in addition, a high augmentation lagis needed or the results depend sensitively on the number of lags included in theDickey–Fuller regression, it may be useful to apply nonparametric tests.

Acknowledgements

The research for this paper was carried out within Sonderforschungsbereich 373 atthe Humboldt University Berlin and the Training and Mobility of Researchers Pro-gramme of the European Commission (contract No. ERBFM-RXCT980213). I thankUwe Hassler, Rolf Tschernig, the associate editor and an anonymous referee for helpfulcomments.

Appendix A. Critical values

The critical values are computed from the empirical distribution of 10,000 realizationsof the limiting expressions of the test statistics, with Gaussian random walk sequencesinstead of Brownian motions (Tables 5 and 6).


Table 5Critical values for T−1%T

T 0.10 0.05 0.01

No deterministics100 0.03126 0.02150 0.01090250 0.02932 0.01999 0.00974500 0.02920 0.01986 0.00998

Mean adjusted100 0.01435 0.01004 0.00551250 0.01433 0.01003 0.00561500 0.01473 0.01046 0.00536

Trend adjusted100 0.00436 0.00342 0.00214250 0.00442 0.00344 0.00223500 0.00450 0.00355 0.00225Note: The hypothesis of a unit root process is rejected if the test statistic falls below the respective critical

values reported in this table.

Table 6Critical values for 9q

q0 = n − r0 0.10 0.05 0.01

Mean adjusted1 67.89 95.60 185.02 261.0 329.9 505.83 627.8 741.1 10244 1200 1360 17025 2025 2255 27616 3177 3460 40457 4650 5049 59058 6565 7061 8032

Trend adjusted1 222.4 281.1 443.62 596.2 713.3 976.13 1158 1330 16894 1972 2184 26995 3107 3429 41206 4572 4954 57807 6484 6984 80128 8830 9388 10714Note: The hypothesis r = r0 is rejected if the test statistic exceeds the respective critical value. The

simulation are based on a sample size of T = 500.

Appendix B. Local-to-unity GLS detrending

As suggested by Elliott et al. (1996), the power of the test can be improved byestimating the nuisance parameters under the sequence of local alternatives 'c

T = 1 +


Table 7Critical values for a model with time trend

c 0:10 0:05 0:01

−13:5 0.00948 0.00715 0.00429−21 0.00726 0.00567 0.00359Note: Critical values for the test with a local-to-unity GLS detrending procedure. c =−13:5 is the value

for the Dickey–Fuller statistic taken from Elliott et al. (1996) and c=−21 is chosen to achieve a power ofapproximately 0.5 under the sequence of local alternatives. The sample size is T = 200.

(c=T ) in the autoregressive model yt = d′t�+'c

Tyt−1 + ut . Let �cdenote the estimator

for � that is obtained from a regression of the quasidiRerences Nsyt = yt −'cTyt−1 on

Ns dt =dt −'cTdt−1. The adjusted series is denoted by uc

t =yt −d′t �

cand the respective

partial sum is constructed as Uct = uc

1 + · · ·+ uct . Finally, the test statistic is computed

as

T−1%cT =

T−2 ∑Tt=1(U

ct )

2∑Tt=1(u

ct)2

:

Elliott et al. (1996) use c=−7 (c=−13:5) for the Dickey–Fuller t-statistic in a modelwith a constant (linear trend). These values yield a power of approximately 0.5 for theDickey–Fuller t-statistic under the sequence of local alternatives. Although the varianceratio statistic is not optimal for this type of alternatives, we can nevertheless followElliott et al. (1996) and use quasidiRerences to estimate the nuisance parameters. 4

For the nonparametric test based on %cT , simulations suggest that the values c = −17

(constant mean) and c =−21 (linear trend) yield tests with an approximate power of0.5. The following results are based on the original values of c suggested by Elliott etal. (1996) and the values that correspond to a power of approximately 0.5.The asymptotic null distribution can be derived by using the results in Elliott et al.

(1996). For dt =1 we have �c=Op(1) and, therefore, T−1=2uc

[aT ] ⇒ �W (a). It followsthat for a constant mean the asymptotic distribution is the same as the asymptoticdistribution for a test without intercept (see Table 5). For a model with linear timetrend it follows that under the null hypothesis T−1=2uc

[aT ] ⇒ �V0(a; c), where V0(a; c)is de,ned in Appendix B of Elliott et al. (1996, p. 835). It follows that under thenull hypothesis T−1%c

T ⇒ {∫ 10 [

∫ a0 V0(s; c) ds]2 da}={

∫ 10 V0(a; c)2 da}. The simulated

critical values obtained from the ,nite sample analog of this limiting distribution arereported in Table 7. Since the critical values are only slightly diRerent for other samplesizes, we only report the critical values for T = 200.Table 8 reports the empirical power of the original test procedure based on OLS-

detrending and the test using the local-to-unity GLS approach. It turns out that the latterprocedure can be much more powerful than the original test procedure, in particular ina model with intercept.

4 I am grateful to a referee for suggesting this test procedure.


Table 8Power comparison

' With constant With linear trend

T−1%T T−1%(−7)T T−1%(−17)

T T−1%T T−1%(−13:5)T T−1%(−21)

T

0.95 0.160 0.266 0.393 0.091 0.104 0.1010.90 0.299 0.426 0.570 0.182 0.215 0.2200.80 0.562 0.593 0.783 0.446 0.420 0.489Note: The entries of the table report the rejection frequencies based on 10,000 replications of the modelyt=

'yt−1 + t , with T =100. The signi,cance level is 0.05 and the critical values are taken from Tables 5 and7, respectively.

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