Nonparametric tests for unit roots and cointegration

Transcription

1 Journal of Econometrics 18 (22) Nonparametric tests for unit roots and cointegration Jorg Breitung Institute of Statistics and Econometrics, Humboldt University Berlin, Spandauer Strasse 1, D-1178 Berlin, Germany Received 19 January 2; received in revised form 29 October 21; accepted 13 November 21 Abstract It is possible to construct unit root tests without specication of the short-run dynamics. These tests are robust against misspecication and structural breaks in the short-run components and can be used to test a wide range of nonlinear models. The variance ratio statistic is similar to the test statistic suggested by Kwiatkowski et al. (J. Econom. 15 (1992) 159) but assumes nonstationarity under the null hypothesis. A straightforward generalization of the variance ratio statistic 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 well in linear and nonlinear models with a suciently large sample size. c 22 Elsevier Science B.V. All rights reserved. JEL classication: C22; C32 Keywords: Unit roots; Cointegration; Nonlinear processes 1. Introduction Many tests for unit roots and cointegration are based on a parametric model. For example, the Dickey Fuller test (Dickey and Fuller, 1979) and the cointegration tests proposed by Johansen (1988, 1991) employ an autoregressive representation of the time series. Other tests (e.g. Phillips and Perron, 1988; Quintos, 1998) use kernel estimators for 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 specication of the short-run dynamics or the estimation of nuisance parameters. Such an approach is Tel.: ; fax: address: breitung@wiwi.hu-berlin.de (J. Breitung) /2/$ - see front matter c 22 Elsevier Science B.V. All rights reserved. PII: S (1)139-7

2 344 J. Breitung / Journal of Econometrics 18 (22) called model free in Bierens (1997a) and nonparametric in Bierens (1997b). Albeit both terms may be somewhat misleading, we follow Bierens (1997b) and use the term nonparametric. In fact, it is dicult 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=2 y [at ] converges weakly to W (a), where W (a) denotes a standard Brownian motion and 2 is the so-called long-run variance which is equal to the limit of E(T 1 yt 2 ). To obtain a test statistic that is asymptotically free of unknown parameters, may be replaced by a consistent estimate. It is, however, possible to get rid of the nuisance parameter without using an estimator of. For example, since the rank transformation R T ( ) is invariant to a scale transformation of the series we have ( ) 1 R T (y [rt ] )=R T T y [rt ] and, thus, the asymptotic theory for a unit root test based on ranks does not involve the parameters involved by the short-run dynamics of the process (see Breitung and Gourieroux, 1997; Breitung, 21). Park and Choi (1988) and Park (199) were the rst who proposed test statistics that do not require corrections for short-run dynamics. They observed that the normalized F-statistic for superuous regressors tends to a (nonstandard) limiting distribution not depending on nuisance parameters. An important problem with this approach is the choice of superuous variables. If articial stochastic variables are used, then the test decision depends on a random draw so that another set of generated variables may yield a conicting result. Moreover, superuous stochastic regressors are an additional source 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 ( T y t) 2 T 2 T y2 t = (y T y ) 2 T : (1) T y2 t Under the null hypothesis of a unit root process and suitable assumptions on the initial value, we have T 2 W (1) 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 alternative of a stationary process, the numerator and denominator are of the same order of magnitude so that T is O p (1) under the alternative as well. Bierens (1997a) resolves this problem by using the squares of the weighted sum 1 T 1=2 g(t=t) t g(1)w (1) g(r)dw (r) (2)

3 J. Breitung / Journal of Econometrics 18 (22) as the numerator in (1), where g(r) denotes the derivative of g(r). 1 For some appropriate weight function g(r), it can be shown that the test is consistent against stationary 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 use functionals on the partial sum Y t = y y t. The advantage of this approach is that no weights are needed to make the test consistent. Furthermore, it turns out that our tests are more powerful than (the stylized version of) Bierens (1997a) test and may 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 component. 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 the variance ratio statistic to cointegrated systems and restrictions on the trend function are considered in Section 6. Section 7 presents the results of a Monte Carlo comparison of alternative test statistics and Section 8 concludes. 2. The null hypothesis Let {y t } T 1 be an observed time series that can be decomposed as y t = t + x t, where t =E(y t )= d t is the deterministic component modeled as a linear combination of a vector of nonrandom regressors d t. Typical components of d t are a constant, a time trend or dummy variables. Following by Davidson (22), we employ a denition of integration that is not based on a particular time series model. Assumption 1. A time series x t is integrated of order one; or x t I(1); if as T T 1=2 x [at ] W (a); where is a constant; [ ] represents the integer part; and W (a) is a Brownian motion dened on C[; 1]. Dierent sets of sucient conditions for the functional central limit theorem (FCLT) involved can be found in Herrndorf (1984), Gallant and White (1988), and Phillips and Solo (1992). It is useful to decompose the stochastic part of the time series as x t = t + v t, where t is a random walk component with uncorrelated increments and T 1=2 [at ] W (a). The transitory component v t is O p (1) and represents the short-run dynamics of the process. For a linear process y t = j= j t j with =1, E( t )= and E(t 2 )= 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 to accommodate a nonlinear mean function. However, for our purpose it is sucient to consider a simplied version of his test given by (2).

4 346 J. Breitung / Journal of Econometrics 18 (22) Beveridge Nelson decomposition, i.e., if j= j2 2 j (cf. Phillips and Solo, 1992). Similarly, a Beveridge Nelson type of decomposition is available for nonlinear processes (cf. Clarida and Taylor, 1999). Since the asymptotic properties of our tests do not depend on the transitory component, the tests are robust against a possible misspecication of the short-run dynamics represented by v t. Furthermore, v t may be fractionally integrated with (1 L) d v t = t, where L is the lag operator, d is a real number and t is white noise. From Sowell (199, Theorem 1) it follows that Assumption 1 is satised for d 1. In such situations, the augmented Dickey Fuller test is expected to have poor power, because a high 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 convenience, we will neglect the deterministic part of series, d t so that y t = x t. As already mentioned, the statistic we consider is a stylized version of the test suggested by Bierens. Since we neglect a (possibly nonlinear) time trend and consider a scalar weight function, the test statistic simplies to T = T( T g(t=t)y t) 2 : (3) T y2 t Bierens (1997a) construct the weights using Chebishev time polynomials but any other dierentiable weight function may be used as well. It is interesting to consider the eects of the weight function on the power of the test. The following proposition characterizes the asymptotic behavior of the test statistic under the alternative of a stationary process. Proposition 1. Let y t = j= j t be stationary and ergodic; E(t 2 )= 2 and the roots of the polynomial (z)= + 1 z + 2 z 2 + are all outside the complex unit circle. For T ; we have T 1 T =v 2 ; where v =1+ 2c j j ; j =E(y t y t+j )=E(yt 2 ); j=1 c j;t =T 1 T g(t=t)g[(t+j)=t ]; c j =lim T c j;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; 198) it follows that T 1=2 g(t=t)y t N (;g); 2

5 where 2 g = j= J. Breitung / Journal of Econometrics 18 (22) c j E(y t y t+j ): Furthermore; T 1 yt 2 converges in probability to E(yt 2 ); so that T 1 T =v 2 ; where v = g=e(y 2 t 2 ). This proposition shows that the power of the test crucially depends on the weight function. Thus, it is important to specify the weight function carefully. For illustration, consider the trigonometric weights g k (t=t) = cos(! k t=t ); where! k = k 2, k =1; 2;:::: A similar weight function is considered in Bierens (1997b). The main dierence between g k (t=t) and the Chebychev polynomial used in Bierens (1997a) is that the Chebychev polynomial introduces a phase shift. However, this does not have any eect on our discussion. A second order Taylor expansion gives ( ) t + j cos! k T and, thus, cos(! k t=t) + sin(! k t=t ) j! k T 1 2 cos(! kt=t ) ( ) 2 j!k T 2c j;t 1 (j! k) 2 2T 2 : If y t is white noise, we have 1 = 1 and j = for j 2. Therefore, to achieve a good power of the test, c 1;T should be as small as possible, that is, a high frequency should be used for the trigonometric weights. On the other hand, if y t is positively correlated, a low frequency is more appropriate. This example demonstrates, that there is no uniformly optimal weight function for the test and it is dicult to specify the weight function without an idea about the autocorrelation function of the series (see also 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 size such that k = T=(2q) and, therefore, g q (t=t) = cos(t=q), where q =1; 2;:::: For the maximal frequency q = 1, the weight function ips between the values 1 and 1. Note that in this case the weight function is not dierentiable. From the above reasoning, we expect 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 test is dierent. Proposition 2. Let g q (t=t ) = cos(t=q); where q and assume that y t obeys Assumption 1. Then; as T we have T 1=2 cos(t=q)y t 2f y (=q)w (1); (4) t=2 where f y (=q) denotes the spectral density of y t at frequency =q.

6 348 J. Breitung / Journal of Econometrics 18 (22) Proof. From Eq. (32) of Phillips and Solo (1992) we have [ ] cos(t=q)y t =Re e it=q y t t=2 t=2 [ ] =Re (e i=q ) e i=q t +O p (1) = t=2 {Re[(e i=q )] cos(t=q) Im[(e i=q )] sin(t=q)} +O p (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 dened as (!) = tan 1 { Im[(e i! )]=Re[(e i! )]}: Furthermore, a cos j! + b sin j! = cos[j! + tan 1 ( b=a)]: This gives cos(t=q)y t = cos[t=q + (=q)] t +O p (1): Using the results of Chan and Wei (1988), we have T 1=2 cos[t=q + (=q)] t 2f y(=q) W (1) 2 which yields the desired result. Note that for q 1, we have lim T T 1 T cos(t=q)2 = 1 so that if y t 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 f y (=q). To summarize, Bierens (1997a) asymptotic theory is valid for xed k in the weight function g(t=t)=cos(k2t=t ). However, if k at the rate T, then a dierent asymptotic theory applies and using the limiting 2 distribution results in severe size distortions. 4. The variance ratio statistic To test the null hypothesis that y t is I() against the alternative y i I(1), Tanaka (199) and Kwiatkowski et al. (1992) suggest an LM-type test statistic. If it is assumed that under the null hypothesis y t is white noise with zero mean, the test statistic is % T = T 2 T Y t 2 T ; (5) 1 T y2 t

7 J. Breitung / Journal of Econometrics 18 (22) where Y t = y y t denotes the partial sum process. If y t is serially correlated, the denominator is replaced by the estimated long-run variance (cf. Kwiatkowski et al., 1992). Note that % T is the (normalized) variance ratio of the partial sums and the original series. Such statistics have a long tradition in time series analysis. For example, the Durbin Watson statistic is the ratio of sample variances computed from the original and the dierenced 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 y t is I(1) against the alternative y t I(). Thus, our test ips the null and alternative hypothesis of the test suggested by Kwiatkowski et al. (1992). To adjust for a nonzero mean of the form d t = z t, the time series y t is regressed on z t and the residuals û t =y t ˆ z t are used to form the variance ratio statistic: ˆ% T = T 1 T Û 2 t T ; (6) û2 t where Û t =û 1 + +û t. For d = we let û t = y t. 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 test procedure are considered in Appendix B. It should be noted, however, that such a local-to-unity GLS procedure is based on a particular sequence of parametric alternatives, so that it does not t well to the nonparametric avor of our test. Nevertheless, the simulations reported in Appendix B suggest that such a modication can improve the 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, the following proposition presents the limiting null distribution of the test statistics. Proposition 3. Under Assumption 1 we have 1 [ a W T 1 j (s)ds ] 2 da ˆ% T 1 ; W j (a) 2 da where W (s) W (s) for d t =; W 1 (s) W (s) 1 W 2 (s) W (s) (4 6s) W (a)da for d t =1; 1 W (a)da (12s 6) 1 aw (a)da for d t =[1;t] : Proof. From Assumption 1 it follows that T 1=2 û [at ] W j (a); T 3=2 Û [st ] s W j (a)da

8 35 J. Breitung / Journal of Econometrics 18 (22) (e.g. Park and Phillips; 1988). Thus; we obtain T 1 % T = T 4 T Û 2 t T 2 T û2 t 1 [ a W j (s)ds ] 2 da 1 W j (a) 2 da : It is important to note that the null distributions do not depend on nuisance parameters. This is due to the fact that the parameter 2 cancels from the variance ratio. Simulated critical values of the asymptotic null distributions are provided in Appendix A. The following proposition shows that the test is consistent against stationary alternatives and considers the usual class of local alternatives (e.g. Phillips, 1987). Proposition 4. Let y t be stationary with Wold representation y t = j= j t j ; where =1; j= 2 j ; and t is white noise with E( t )= and E(t 2 )= 2. Under this alternative; we have T 1 ˆ% T as T and ˆ% T 2 1 W j (a) 2 da ; 2 y where 2 =( j= j) 2 2 and y 2 = 2 j= 2 j. Under the local alternative y t = T y t 1 + t with T =1 c=t; the limiting distribution is given by 1 T 1 ˆ% T [ a J c j(s)ds] 2 da 1 J c ; j(a) 2 da where W j (a) is dened in Proposition 3 and J c j(r) results from the same expressions, if W (a) is replaced by the Ornstein Uhlenbeck process J c (a)= a e(a s)c dw (s). Proof. Under a stationary alternative; we have 1 T 2 Û 2 t 2 W j (a) 2 da: Using this result and T 1 T û2 p t y; 2 the limiting distribution in the case of a stationary alternative follows easily. Under the sequence of local alternatives, we have (cf. Phillips, 1987) T 1=2 û [at ] J c j(a); T 3=2 Û [at ] a J c j(s)ds: Therefore, the limiting distributions result from replacing the W (a) byj c (a) in Proposition 3.

9 5. Testingthe cointegration rank J. Breitung / Journal of Econometrics 18 (22) The variance ratio statistic for a nonparametric unit root test can be generalized to test hypotheses on the cointegration. It is assumed that the process can be decomposed into a q-dimensional vector of stochastic trend components t anda(n q)-dimensional vector of transitory components v t. Assumption 2. There exists an invertible matrix Q =[; ]; where and are linearly independent n q and n (n q) matrices; respectively; with q n such that [ ] [ ] Q (y t t ) t (y t t )= = z t ; (y t t ) T 1=2 [at ] W q (a); T 2 v t v t =o p (1); v t where t =E(y t ) and W q (a) isaq-dimensional Brownian motion with unit covariance matrix. 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 yt = Ay t. Furthermore, we do not assume that the linear combination v t = (y t t ) is stationary. Instead, we assume that the trend component t is variance dominating in the sense that the variance of t diverges with a faster rate than v t. Therefore, the transitory component can 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 in the so-called vector error correction representation y t = y t 1 + e t ; and e t is a stationary error vector. In a linear system, the hypothesis on the number of stochastic 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 representation of the form (7) may not exist. Let û t denote the vector of least-squares residuals from a regression of y t on the vector of deterministic terms d t. Our test statistic is based on the eigenvalues j (j =1;:::;n) of the problem where j B T A T =; A T = û t û t; B T = Û t Û t; (7) (8)

10 352 J. Breitung / Journal of Econometrics 18 (22) and Û t = t j=1 û j denotes the n-dimensional partial sum with respect to û t. The eigenvalues of (8) are identical to the eigenvalues of the matrix R T = A T B 1 T. For n = 1, the eigenvalue is identical to (T ˆ% 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 j = ja T j j B T j ; (9) where j is the eigenvector associated with the eigenvalue j. If the vector j falls inside the space spanned by the columns of, then ja T j is O p (T 2 ) and jb T j is O p (T 4 ) so that the eigenvalue is O p (T 2 ). On the other hand, if the eigenvector j falls into the space spanned by the columns of, it follows that T 2 j tends to innity, as T. Therefore, the test statistic q q = T 2 (1) j=1 j has a nondegenerate limiting distribution, where n denote the ordered eigenvalues of the matrix R T. In contrast, if the number of stochastic trends is smaller than q, then q diverges to innity. The following proposition presents the limiting null distribution for the test statistic q. Proposition 5. Assume that y t admits a decomposition as in Assumption 2 with q6n. Then; as T [ 1 ] 1 1 q tr W q j (a) W q j (a) da Ṽ q j (a)ṽ q j (a) da ; where W q j (a) is the q-dimensional analog of Ṽ q j (a)= a W q j (s)ds. W j (a) dened in Proposition 3 and Proof. Let Ẑ t = t j=1 ẑj denote the partial sum with respect to ẑ t = Q û t =[ˆ t; ˆv t]. Then; the eigenvalues of problem (8) also solves the problem where j D T C T =; C T = ẑ t ẑ t; D T = Ẑ t Ẑ t: Next; we partition the corresponding eigenvectors j =[ 1j ; 2j ] such that jẑt = ˆ 1j t + 2j ˆv t; and Ẑ t is partitioned accordingly. We normalize the matrix of eigenvectors as [ ] Iq [ 1 ;:::; q ]= T

11 J. Breitung / Journal of Econometrics 18 (22) so that jẑt = ˆ jt + jv t for j 6 q; where ˆ jt denotes the jth component of the vector ˆ t and j denotes the jth column of T. It follows that j = j C T j j D T j T = 2 jt +o p (T 2 ) T Z2 jt +o p(t 4 ) = T 2 jt T Z2 jt +o p (1); where Z jt = t s=1 js. AsT ; we; therefore; have q [ 1 ] 1 1 T 2 j tr W q j (a) W q j (a) da Ṽ q j (a)ṽ q j (a) da : j=1 From this proposition it follows that the distribution of the q smallest eigenvalues of problem (8) does not depend on nuisance parameters and, thus, we do not need to select the lag order of the VAR process as in Johansen s approach or the truncation lag 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 constrained under the cointegration hypothesis. In particular, it is assumed that y t has a linear time trend, whereas the cointegrating relations y t have a constant mean (e.g. Johansen, 1994). This specication implies that y t has to be adjusted for a constant mean, whereas the vector of permanent components y t is adjusted for a time trend. To impose such restrictions on the deterministic terms, estimates for the matrices and are needed. A possible way to estimate these matrices is to use the principle component estimator, which has the attractive property that is estimated as an orthogonal complement of the cointegration matrix. It follows, that this matrix is estimated with the same convergence rate as the cointegration matrix (cf. Harris, 1997). Let ˆ and ˆ denote the estimates from a principal component procedure. Then, the adjusted vector of time series results as [ ] ˆ ẑ y t â â 1 t t = ˆ ; y t ˆb where â and â 1 are the least-squares estimates from a regression of ˆ y t on a constant and a time trend and ˆb denotes the mean of ˆ y t. Then, the statistic is computed by using ẑ t instead of y t and the critical values for a test with time trend are applied.

12 354 J. Breitung / Journal of Econometrics 18 (22) Note that ẑ t = [ ] y t E( y t )+o p (T 1=2 ) y t E( y t )+O p (1) and, thus, the dierences between the estimated and true nonstationary components are asymptotically negligible but the transitory components are measured with an nonvanishing error. However, the transitory components do not aect the asymptotic null distribution of the test statistic so that the limiting distribution is the same as for the 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 of Monte Carlo simulations. It is not intended to give a comprehensive account of the merits and drawbacks of our test relative to other unit root tests based on a parametric or semiparametric adjustment for short-run dynamics. Rather, we try to give a rough idea of the relative performance of the tests, where the augmented Dickey Fuller test is used as a benchmark. For the univariate tests, the data are generated by the process x t = x t 1 + t t 1 (11) and y t = d t +x t, where t niid(; 1) and d t =1 (constant mean) or d t =[1;t] (linear trend). The sample size is T = 2. Under the null hypothesis we have = 1 and 1. For, the process has no nite order AR representation and following Said and Dickey (1984) an autoregressive approximation is employed with p = 4 and 12 lagged dierences. This test is denoted by ADF(p). The rst nonparametric test in the comparison is the variable addition statistic of Park (199). Four independently generated random walks are used as superuous regressors yielding the test statistic J 2 (4). 2 For our version of Bierens (1997a) test, we use a trigonometric weight function given by g k (t=t) = cos(2kt=t), where k =1; 4; 16; 32. The respective test is labeled as T (k). The critical values with respect to a signicance level of.5 are obtained from 1. Monte Carlo runs of the model with = 1 and =. To adjust for deterministic terms, the test statistic is constructed using the residuals from a regression of y t on a constant or a linear trend. Finally, the variance ratio statistic is computed using the residuals from a regression of y t on a constant or a linear trend. The respective test statistic is denoted as T 1 ˆ% 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 H : = 1 and various values of in a model with a constant mean. Since the critical values are computed from the same random draws, the empirical sizes are exact.5 for J 2 (4), T (k) and T 1 ˆ% T. 2 Park (199, Remark c) recommends to use two or more regressors since a single superuous regressor seems insucient to discriminate the competing models for small samples.

13 J. Breitung / Journal of Econometrics 18 (22) Table 1 Rejection frequencies for a model with a constant mean Test statistic = :5 = =:5 =:8 (a) Empirical size ( =1) J 2 (4) T (1) T (4) T (16) T (32) T 1 ˆ% T ADF(4) ADF(12) Test statistic = :95 = :9 = :8 = :5 (b) Empirical power ( = ) J 2 (4) T (1) T (4) T (16) T (32) T 1 ˆ% T ADF(4) ADF(12) Note: The entries of the table display the rejection frequencies based on 1, replications of model (11), where d t = 1. The sample size is T = 2 and the nominal size of the test is.5. Since the critical values are computed from the same random draws, the empirical sizes are exact.5 for J 2 (4), T (k) and T 1 ˆ% T. The results of the Monte Carlo experiment indicate that Park and Choi s test J 2 (4) has severe size distortions if the moving average parameter tends to one. A similar problem 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 test depends on the frequency of the weighting function. For low frequencies the empirical sizes are close to the nominal ones for all values of. For high frequencies the test shows serious size distortions even for moderate values of. This is due to the fact that as k, the asymptotic distribution involves the parameter (see Section 2). Similar results are obtained for the model with a time trend (see Table 2). The variance ratio statistic T 1 ˆ% T also shows a considerable size bias for large positive values of. For =:5, the size bias is moderate but for =:8, the test is severely biased towards a rejection of the null hypothesis. A similar outcome is found for ADF(4), however, if the ADF test is augmented with 12 lagged dierences, the empirical size is close to the nominal size for values up to =:8. The empirical powers of the test procedures for dierent 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 modication of the test statistic and adopt a parametric approach to estimate the nuisance parameters. Since, we do not assume a particular model for the short-run dynamics, such modications are not applicable here.

14 356 J. Breitung / Journal of Econometrics 18 (22) Table 2 Rejection frequencies for a model with a linear trend Test statistic = :5 = =:5 =:8 (a) Empirical size ( =1) J 2 (4) T (1) T (4) T (16) T (32) T 1 ˆ% T ADF(4) ADF(12) Test statistic = :95 = :9 = :8 = :5 (b) Empirical power ( = ) J 2 (4) T (1) T (4) T (16) T (32) T 1 ˆ% T ADF(4) ADF(12) Note: The entries of the table display the rejection frequencies based on 1, replications of model (11), where d t =[1;t]. The sample size is T = 2 and the nominal size of the test is.5. time trend. The MA parameter is set to zero. It turns out that using a trigonometric weight function with a low frequency yields a poor performance of the Bierens type of test. As expected (see Section 3) the power of the test improves with an increasing frequency. However, since the actual size of the test increases as well, it is quite dicult to select an appropriate frequency. For =:95, the variance ratio test is even slightly more powerful than the ADF(4) test, whereas for other values of, the power of the variance ratio test is larger than the power of the ADF(12) statistic but smaller than the power of the ADF(4) statistic. Next, we consider four nonlinear processes: bilin : y t =:9 t 1 y t 1 + t ; (12) VCM : y t = t y t 1 + t with t =:9 cos(2t=t); (13) { :9yt 1 + t for y t 1 2; TAR : y t = (14) :9y t 1 + t for y t 1 2; STUR : y t = t y t 1 + t where t =:1+:9 t 1 + t : (15) The rst process (12) is a bilinear process (see Granger and Anderson, 1978), where the correlation between t 1 and y t 1 implies a linear time trend in y t. The second process (13) is a variable coecient model with a cyclical coecient t. The third process (14) is a threshold autoregressive process and the fourth process (15) is a

15 J. Breitung / Journal of Econometrics 18 (22) Table 3 Empirical sizes for some nonlinear processes Process ADF(1) T 1 ˆ% T J 2 (4) T (16) Size bilin VCM TAR STUR Power bilin VCM TAR STUR Note: The entries of the table display the empirical sizes computed from 1, replications of model (12) (15). bilin denotes the bilinear process (12). VCM is the variable coecient model (13), TAR is a threshold unit root process (14), and STUR is a stochastic unit root process (15). The sample size is T = 2 and the nominal size of the test is.5. For bilin, VCM and TAR the power is computed by testing the series generated as yt =:9yt 1 +yt and y =. For the STUR model, the alternative is an autoregressive process with t =:1+:85 t 1 + t and =:95. stochastic unit root process as considered in Granger and Swanson (1997), where for all processes t is white noise with E(t 2 )=1 and E( 2 t )=:5 2. For this specication we have E( t )=1. The empirical sizes were computed from 1, realizations with T = 2. All tests allow for a linear trend. From the results in Table 3, it can be concluded that the nonparametric tests are more robust against nonlinear short-run dynamics than the ADF(1) test that assumes a linear autoregressive process. Among the nonparametric tests, the variable addition test of Park and Choi (1988) is slightly more robust than its competitors. On the other hand, the variance ratio test is much more powerful than the variable addition test and the stylized version of Bierens test. For the VCM and TAR 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 [ ] [ ][ ] [ ] [ ][ ] x1t 1 x1;t 1 1t :5 1;t 1 = + ; (16) x 2t 2 x 2;t 1 2t :5 2;t 1 where y t = + x t,e(1t 2 )=E(2 2t )=1 and E( 1t 2t )=. To test the hypothesis r =1, we let 1 = and 2 = :2. Under the alternative we set 1 { :5; :1; :2}. Furthermore, we let = and.8 to investigate the impact of the error correlation. The sample size is T = 2 and 1, samples are generated to compute the rejection frequencies 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) and LR(12) in Table 4. Unrestricted constants are included in each equation. The nonparametric test statistic is denoted by q and the critical values are taken from Table 6 in the appendix. First, consider the results for testing H : q = r = 1. From the

16 358 J. Breitung / Journal of Econometrics 18 (22) Table 4 Testing hypotheses on the cointegration rank Test statistic 1 = 1 = :5 1 = :1 1 = :2 H : r =1, 2 = :2 = LR(4) LR(12) =: LR(4) LR(12) H : r =, 2 = = LR(4) LR(12) =: LR(4) LR(12) Note: The entries of the table report the rejection frequencies based on 1, replications of model (16), where D t is constant. empirical sizes (see Table 4), it turns out that for =, a VAR(4) model is not sucient to approximate the innite VAR process, whereas a VAR(12) approximation yields an accurate size. The nonparametric statistic 1 possesses a negligible size bias, only. The power of 1 is substantially smaller than the power of LR(4) but clearly higher than the power of LR(12). Similar results apply for the tests letting =:8. However, the LR(12) statistic now possesses a moderate size bias, whereas 1 is nearly unbiased. Moreover, the power of 1 is closer to the (favorable) LR(4) statistic than in the case of =. We now turn to the test of H : r =. Under the null hypothesis the dierences of the variables are generated by a multivariate MA process. In this case, all three test statistic are substantially biased, where the size bias does not depend on the parameter. Although the sizes bias diers for the three test, the dierences are moderate and some general conclusions with respect to the relative power of the tests can be drawn. For = and 1 close to unity, the nonparametric test is slightly more powerful than the LR(4) test, whereas for 1 = :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 =:8 a dierent picture emerges. The relative power of drops substantially and for 1 close to one, the power is even lower than the power of the LR(12) test. The results for a model with a linear time trend are qualitatively similar and are not presented for reasons of space.

17 8. Concludingremarks J. Breitung / Journal of Econometrics 18 (22) Following Park and Choi (1988), Bierens (1997a, b) and Vogelsang (1998a, b), unit root tests can be constructed which, asymptotically, do not depend on parameters involved by the short-run dynamics of the process. The variance ratio statistic has the advantage that the outcome of the test does not depend on a random draw of super- uous variables or the frequency of the weight function. Moreover, our Monte Carlo simulations 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 need not be encountered in small samples. In particular, if the variance of the transitory component is important relative to the variance of the random walk component, the test may suer from severe size distortions. Second, it has been shown that under the alternative of a stationary process, the appropriately normalized test statistics converge to a random variable as T tends to innity. In contrast, the normalized Dickey Fuller test converge to a constant under the null hypothesis and, therefore, the test generally has more favorable properties than the nonparametric counterparts. Finally, in many empirical applications it seems not dicult to select an appropriate augmentation lag and the test statistic turns out to be quite robust against dierent lag orders. However, there are a number of situations, where the nonparametric approach may be attractive. Since the short-run component does not aect the asymptotic null distribution of the test statistic, the test is robust against deviations from the usual assumption of linear short-run dynamics. This property is important in large samples, where small deviations from the underlying (parametric) assumptions may have a substantial eect on the behavior of the parametric test statistic. Furthermore, when the sample size is large, there is reason to expect that the random walk component dominate the sampling behavior of the nonparametric test statistic and the asymptotic theory provides a reliable approximation to the actual null distribution. If, in addition, a high augmentation lag is needed or the results depend sensitively on the number of lags included in the Dickey Fuller regression, it may be useful to apply nonparametric tests. Acknowledgements The research for this paper was carried out within Sonderforschungsbereich 373 at the Humboldt University Berlin and the Training and Mobility of Researchers Programme of the European Commission (contract No. ERBFM-RXCT98213). I thank Uwe Hassler, Rolf Tschernig, the associate editor and an anonymous referee for helpful comments. Appendix A. Critical values The critical values are computed from the empirical distribution of 1, realizations of the limiting expressions of the test statistics, with Gaussian random walk sequences instead of Brownian motions (Tables 5 and 6).

18 36 J. Breitung / Journal of Econometrics 18 (22) Table 5 Critical values for T 1 ˆ% T T No deterministics Mean adjusted Trend adjusted Note: The hypothesis of a unit root process is rejected if the test statistic falls below the respective critical values reported in this table. Table 6 Critical values for q q = n r Mean adjusted Trend adjusted Note: The hypothesis r = r is rejected if the test statistic exceeds the respective critical value. The simulation are based on a sample size of T = 5. Appendix B. Local-to-unity GLS detrending As suggested by Elliott et al. (1996), the power of the test can be improved by estimating the nuisance parameters under the sequence of local alternatives c T =1+

19 J. Breitung / Journal of Econometrics 18 (22) Table 7 Critical values for a model with time trend c :1 :5 :1 13: Note: 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 of approximately.5 under the sequence of local alternatives. The sample size is T = 2. (c=t) in the autoregressive model y t = d t + c T y t 1 + u t. Let c denote the estimator for that is obtained from a regression of the quasidierences s y t = y t c T y t 1 on s d t =d t c T d t 1. The adjusted series is denoted by ũ c t =y t d c t and the respective partial sum is constructed as Ũ c t =ũ c 1 + +ũ c t. Finally, the test statistic is computed as T 1 % c T = T 2 T (Ũ c t ) 2 T : (ũc t) 2 Elliott et al. (1996) use c = 7 (c = 13:5) for the Dickey Fuller t-statistic in a model with a constant (linear trend). These values yield a power of approximately.5 for the Dickey Fuller t-statistic under the sequence of local alternatives. Although the variance ratio statistic is not optimal for this type of alternatives, we can nevertheless follow Elliott et al. (1996) and use quasidierences to estimate the nuisance parameters. 4 For the nonparametric test based on % c T, simulations suggest that the values c = 17 (constant mean) and c = 21 (linear trend) yield tests with an approximate power of.5. The following results are based on the original values of c suggested by Elliott et al. (1996) and the values that correspond to a power of approximately.5. The asymptotic null distribution can be derived by using the results in Elliott et al. (1996). For d t = 1 we have c =O p (1) and, therefore, T 1=2 ũ c [at ] W (a). It follows that for a constant mean the asymptotic distribution is the same as the asymptotic distribution for a test without intercept (see Table 5). For a model with linear time trend it follows that under the null hypothesis T 1=2 ũ c [at ] V (a; c), where V (a; c) is dened in Appendix B of Elliott et al. (1996, p. 835). It follows that under the null hypothesis T 1 % c T { 1 [ a V (s; c)ds] 2 da}={ 1 V (a; c) 2 da}. The simulated critical values obtained from the nite sample analog of this limiting distribution are reported in Table 7. Since the critical values are only slightly dierent for other sample sizes, we only report the critical values for T = 2. Table 8 reports the empirical power of the original test procedure based on OLSdetrending and the test using the local-to-unity GLS approach. It turns out that the latter procedure can be much more powerful than the original test procedure, in particular in a model with intercept. 4 I am grateful to a referee for suggesting this test procedure.

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