Point optimal tests of the null hypothesis of cointegration

Transcription

1 Journal of Econometrics 24 (25) Point optimal tests of the null hypothesis of cointegration Michael Jansson Department of Economics, University of California, Berkeley, 549 Evans Hall #388, Berkeley, CA , USA Accepted 9 February 24 Abstract This paper obtains an asymptotic Gaussian power envelope for tests of the null hypothesis of cointegration. In addition, the paper proposes a feasible point optimal cointegration test whose local asymptotic power function is found to be close to the asymptotic Gaussian power envelope. c 24 Elsevier B.V. All rights reserved. JEL classication: C22 Keywords: Cointegration; Local asymptotic power; Point optimal test; Power envelope. Introduction The concept of cointegration (Engle and Granger, 987) has attracted considerable attention in the literature and answers to a variety of questions concerning inference in cointegrated systems have been provided. Asymptotic optimality theory for the problem of estimating cointegrating vectors under normality has been developed by Phillips (99) and several asymptotically ecient estimation procedures have been proposed (for a review, see Watson, 994). Moreover, an asymptotic Gaussian power envelope for tests of the unit root assumption underlying these cointegration methods has been obtained by Elliott et al. (996). In contrast, although numerous papers have considered the problem of testing the null hypothesis of cointegration against the alternative of no cointegration (examples include Choi and Ahn, 995; Park, 99; Shin, 994; Tel.: ; fax: address: mjansson@econ.berkeley.edu (M. Jansson) /$ - see front matter c 24 Elsevier B.V. All rights reserved. doi:.6/j.jeconom.24.2.

2 88 M. Jansson / Journal of Econometrics 24 (25) 87 2 Xiao and Phillips, 22), no asymptotic optimality theory for that testing problem has been developed. This paper obtains an asymptotic Gaussian power envelope for tests of the null hypothesis of cointegration and proposes a feasible point optimal cointegration test. By construction, the point optimal test attains the asymptotic Gaussian power envelope at a prespecied alternative. Against other alternatives, the local asymptotic power of the point optimal test is close to the asymptotic Gaussian power envelope. Moreover, the point optimal test is found to perform well in a Monte Carlo experiment. In terms of the methodology employed, the present paper is related to Elliott et al. s (996) and Saikkonen and Luukkonen s (993) studies of unit root testing in autoregressive and moving average models, respectively. As do Elliott et al. (996) and Saikkonen and Luukkonen (993), this paper develops asymptotic optimality results by obtaining limiting distributions of optimal test statistics derived under a normality assumption. By accommodating stochastically trending and (possibly) endogenous regressors, the results obtained here generalize the xed-regressor results of Saikkonen and Luukkonen (993). Section 2 presents the model. In Section 3, the asymptotic Gaussian power envelope is derived. Section 4 constructs a feasible point optimal test. Finally, Section 5 reports local asymptotic power results and investigates the nite sample performance of the test proposed in this paper, while all mathematical derivations are collected in an Appendix. 2. The model and assumptions Let z t be an observed m-vector time series generated by z t = z t + z t ; 6 t 6 T; () where z t is a deterministic component and z t is a zero-mean stochastic component. For concreteness, the deterministic component is assumed to be a pth-order polynomial time trend: z t = zd t ; (2) where d t =(;:::;t p ) and z isa(p +) m matrix of parameters. The cases of particular interest are the constant mean and linear trend cases corresponding to d t = and d t =(;t), respectively. Partition zt into a scalar yt and a k-vector xt (k = m ) as zt =(yt ;xt ) and suppose zt is generated by the potentially cointegrated system y t = x t + v t ; (3) x t = u x t ; where v t is an error process with generating mechanism v t = u y t u y t ; 6 t 6 T: (4)

3 M. Jansson / Journal of Econometrics 24 (25) In (3) and (4) R k and (; ] are unknown parameters, and u t =(u y t ;u x t ) satises the following assumption. A. u t = i= C i t i, where { t : t Z} is i.i.d. (;I m ), i= C i has full rank and i= i C i, where denotes the Euclidean norm. The system is initialized at t = with v =u y = and x =. The moving average speci- cation (4) implies that yt and xt are cointegrated if and only if =. Assumption A imposes the standard, but important, regularity condition that xt is a non-cointegrated integrated process. In addition, it is assumed that the cointegration between yt and xt is regular (in the sense of Park, 992) when =. It would be of interest to relax one or both of these assumptions, but doing so is beyond the scope of the present paper. Conformably with zt, partition z t and z as z t =(y t ;x t) and z =( y ; x ). Dening = y x, the model can be written in triangular form as y t = d t + x t + v t ; v t = u y t u y t ; x t = xd t + ut x : This paper considers the problem of testing H : = vs: H : ; (P) treating and as unknown nuisance parameters. Section 3 derives an asymptotic power envelope under the following strengthening of A. A. u t i.i.d. N(;), where is positive denite. As a by-product of the analysis, a point optimal test statistic is obtained. Section 4 relaxes A and constructs a feasible point optimal test which is applicable whenever A holds. Remarks. (i) Some papers in the existing literature on cointegration testing (e.g., Shin, 994) employ an alternative model for v t ; the so-called local-level unobserved components model. As does the moving average model (4), the unobserved components model parameterizes cointegration as a point. Using the fact that u y t + t has an MA () representation (e.g., Stock, 994), it can be shown that the power envelope developed in the next section coincides with power envelope for the testing problem H : 2 = vs: H : 2 in the model y t = d t + x t + u y t + t ; where x t = xd t +ut x, u t =(u y t ;ut x ) satises A, t i.i.d. N(; 2) and { t} is independent of {u t }. For concreteness, and in order to conserve space, the present paper only studies the moving average model. (ii) The triangular model formulation (3) and (4) does not treat y t and x t symmetrically. In particular, it is assumed that a linear combination of y t and x t is stationary

4 9 M. Jansson / Journal of Econometrics 24 (25) 87 2 only if it puts non-zero weight on y t. In contrast, the fact that = is permitted implies that it is possible for a linear combination of y t and x t to be stationary even if it puts zero weight on x t. As a consequence, the testing problem (P) is not invariant under orthogonal transformations of (y t ;x t) (such as a relabelling of y t and x t in the case where x t is a scalar). The assumption that one variable, y t, is known to appear with a non-zero coecient in the (potentially) cointegrating relation would appear to be satised in most empirical applications. 3. An asymptotic Gaussian power envelope Under A, the model is fully parametric and classical statistical theory can be employed to develop optimality theory for the testing problem (P). In particular, it is possible to construct an attainable upper bound on the local asymptotic power of a class of cointegration tests. The upper bound is constructed by eliminating the nuisance parameters ; ; x and from the problem and then applying the Neyman Pearson lemma. Two dierent elimination methods are used: x and are treated as known, while the principle of invariance (e.g., Lehmann, 994) is used to handle and. The power bound derived under the assumption that x and are known is attainable even when these parameters are unknown. As a consequence, it can (and will) be assumed that x and are known. The remaining nuisance parameters, and, are eliminated by proceeding along the lines of numerous other papers on tests concerning the covariance structure of the error term in a linear regression model (e.g., King, 98; King and Hillier, 985; Dufour and King, 99). Dene the matrices Y =(y ;:::;y T ), X =(x ;:::;x T ) and D=(d ;:::;d T ). Consider the groupof transformations of the form g a;b (Y; X )=(Y + D a + X b; X ); (G) where a R p+ and b R k. Each g a;b induces a transformation g a;b (; ; )=(; + a;+ b) ( G) in the parameter space. Because is invariant under ( G), the testing problem (P) is invariant under (G). It therefore seems natural to restrict attention to tests that are invariant under (G). All previously proposed tests (of which the author is aware) are invariant under this groupof transformations, so the class of tests that are invariant under (G) is quite large. Because is a maximal invariant under ( G), it follows from Theorem 6.3 of Lehmann (994) that the distribution of any invariant statistic (i.e. any statistic invariant under (G)) depends on (; ; ) only through, implying that the nuisance parameters and can be eliminated by restricting attention to statistics that are invariant under (G). A natural maximal invariant under (G) is ((R Y ) ; vec(x ) ), where R is a matrix whose columns form an orthonormal basis

5 M. Jansson / Journal of Econometrics 24 (25) for the orthogonal complement of the column space of R =(X; D). The distributional properties of this maximal invariant are developed next. Partition in conformity with u t as ( yy ) xy = xy xx and for any, dene = =2 =2, where a lower triangular T T matrix with ones on the diagonal and below the diagonal: = = : Under A, vec(x ) N(( x I T )vec(d); xx ) and =2 R Y X N((R =2 X )xx xy ; yy x (R R )); where yy x = yy xy xx xy and X signies the conditional distribution relative to the -algebra generated by X. Apart from an additive term that does not depend on, minus 2 times the log density of the maximal invariant can be written as L T () = log R where Y = Y R (R R ) R = R + yy xy ( R(R R) R )Y ; (6) =2 X xx xy and the derivation of (6) makes use of the relations R(R R) R and R R = R R R R. Expression (6) diers from its xed-regressor counterpart (e.g., King, 98) in two respects. In xed-regressor settings, the term corresponding to log R R is non-random and can be omitted. Moreover, the denition of Y reects the fact that correlations between u y t and ut x must be taken into account when X is random. As in the xed-regressor case, the term Y ( R(R R) R )Y in (6) can be interpreted as the weighted sum of squared residuals from a GLS regression (of Y on R using the covariance matrix ). It follows from the Neyman Pearson Lemma that the test which rejects when L T () L T ( ) is large is the most powerful invariant test of = vs. =. An asymptotic analogue of that optimality result can be obtained by deriving the limiting distribution of L T () L T ( ) under a local-to-unity reparameterization of and in which = T( ) and = T ( ) are held constant as T increases without bound. A formal statement is provided in Theorem, the proof of which represents

6 92 M. Jansson / Journal of Econometrics 24 (25) 87 2 the limiting distribution of L T () L T ( ) in terms of the random functional (; )=2 V dv 2 (V ) 2 ( ) ( ) ( ) + Q dv Q Q Q dv log ( ) ( ) ( ) Q dv QQ Q dv + log QQ ; Q Q where V (s)= s e (s t) dv (t), V (s)=v (s)+ s V (t)dt, Q (s)= s e (s t) dq(t), Q(s)=(;:::;s p ;W(s) ), and V and W are independent Wiener processes of dimensions and k; respectively. Theorem. Let z t be generated by () (4) and suppose A holds. An upper bound on the local asymptotic power of any invariant test of = against = T = T is given by ()=[ (; ) c ()], where is the asymptotic level of the test, c () satises [ (; ) c ()] = and invariance is with respect to transformations of the form (G). Under the reparameterization employed in Theorem, the null and alternative hypotheses are = and, respectively. The upper bound provided by the Gaussian power envelope is sharp in the sense that () can be attained for any given by the test which rejects for large values of the corresponding point optimal test statistic, viz. L T () L T ( T ). Indeed, previous research on special cases of the testing problem considered here (e.g., Saikkonen and Luukkonen, 993) suggests that a test based on L T () L T ( ) should have a local asymptotic power function close to ( ) if is chosen appropriately. The function ( ) therefore constitutes a useful benchmark against which the local power function of any invariant test of the null hypothesis of cointegration can be compared. The power envelope depends on p, the order of the deterministic component, and k, the dimension of x t. On the other hand, although the form of the point optimal test statistic depends on, the power envelope does not depend on the covariance matrix of the underlying errors u t. In particular, the power envelope does not depend on the extent to which the regressors are endogenous in the sense that ut x is correlated with the latent error u y t. Jansson (24a) derives the asymptotic Gaussian power envelope in a model isomorphic to the present model under the assumption that is known. That power envelope depends on yy xy xx xy, the squared coecient of multiple correlation between u y t and ut x, and substantial power gains over conventional tests are available when the correlation is non-zero. In contrast, it follows from Theorem that it is impossible to exploit any correlations between u x t and u y t when testing the null hypothesis that y t and x t are cointegrated with an unknown cointegrating vector.

7 4. Feasible point optimal tests M. Jansson / Journal of Econometrics 24 (25) This section constructs a test statistic which can be computed without knowledge of any nuisance parameters and has a limiting distribution of the form (; ) under A. Let ˆ; ˆ and ˆ denote estimators of = E(u u ), = t=2 E(u tu ) and = + +, respectively. Partition the matrices ˆ and ˆ in conformity with u t and let ˆ x =(ˆ xy ; ˆ xx ), ˆ! yy x = ˆ ˆ ˆ and ˆ yy x = ˆ ˆ ˆ, where ˆ=(;! ˆ xy ˆ xx ). Let Y D =M D Y and X D = M D X, where M D = I D(D D) D, and dene R + =(D; X + Û ˆ ˆ x ), =2 where Û =(Y D X D ˆ; X D ) and ˆ =(X D X D) X D Y D. Finally, let Y + = Y =2 X ˆ xx ˆ! xy + Û ˆ ˆ ˆ x for any. The proposed test statistic is P T ( )=L + T () L+ T ( T ) 2 ˆ! yy x ˆ yy x ; (7) where is a prespecied constant and L + T ( )=log R + R + +ˆ! yy xy + ( R + (R + R + ) R + )Y + : Under the assumptions of Theorem, P T ( ) is asymptotically equivalent to L T () L T ( ). As Theorem 2 shows, P T ( ) has a limiting distribution of the form (; ) even when u t exhibits serial correlation of the form permitted under A. This robustness property, not shared by L T () L T ( ), is achieved by employing two serial correlation corrections. The rst correction, employed in the construction of Y + and R +, is similar to the correction proposed by Park (992) in the context of estimation of a cointegrating regression. Indeed, the purpose of this correction is to remove serial correlation bias from the limiting distribution of the estimator of appearing in L T + ( ). The second correction term, 2 ˆ! yy x ˆ yy x,in(7) resembles the correction term in Phillips (987) Z test for an autoregressive unit root and accommodates serial correlation in u y t! xy xx ut x. Theorem 2. Let z t be generated by () (4), suppose A holds and suppose = T = T for some. If ( ˆ; ˆ ; ˆ) p (; ; ), then P T ( ) d (; ). A consistent estimator of is ˆ = T U ˆ Û, while and can be estimated consistently by means of conventional kernel estimators (for reviews, see den Haan and Levin (997) and Robinson and Velasco (997)). Primitive sucient conditions for consistency can be found in previous work by the author (Jansson, 22, 24a; Jansson and Haldrup, 22). Suce it to say that the consistency requirement of Theorem 2 is met by a variety of estimators ˆ; ˆ and ˆ. To implement the feasible point optimal test, a value of must be specied. Following Elliott et al. (996), the approach advocated here is to choose in such a way that the asymptotic local power against the alternative = T is approximately equal to 5% when the 5% test based on P T ( ) is used. That is, the recommendation is to use the test which is (asymptotically).5-optimal, level.5 in the sense of Davies (969). Table a (b) tabulates the recommended values of for k =;:::;6

8 94 M. Jansson / Journal of Econometrics 24 (25) 87 2 Table Percentiles of P T ( ) k = k =2 k =3 k =4 k =5 k =6 (a) Constant mean: d t = % % % % (b) Linear trend: d t =(;t) % % % % Note: The percentiles were computed by generating 2, draws from the discrete time approximation (based on 2 steps) to the limiting random variables. regressors in the constant mean (linear trend) case and reports selected percentiles of the asymptotic null distributions of the corresponding P T ( ) statistics. 5. Power properties 5.. Local asymptotic power Fig. plots the 5% asymptotic Gaussian power envelope :5 ( ) along with the local asymptotic power functions of two feasible cointegration tests in the constant mean case with scalar x t. The two feasible tests, denoted P T (9) and S T, are the tests proposed in Section 4 and the test due to Shin (994), respectively. The latter test appears to be the most widely used cointegration test in applications. Moreover, S T is known to enjoy local optimality properties under the assumptions of Theorem (Harris and Inder, 994). In addition, previous research (Jansson and Haldrup, 22; Jansson, 24b) indicates that none of the tests due to Choi and Ahn (995), Park (99) and Xiao and Phillips (22) dominate Shin s (994) test in terms of local asymptotic power. For these reasons, it seems natural to use the performance of S T as a benchmark when evaluating new tests such as P T (9). As might be expected, the local asymptotic power of S T is close to the envelope for small values of (smaller than 5, say). For larger values of, on the other hand, the local asymptotic power of the locally optimal test is well below the envelope. In contrast, the local asymptotic power of P T (9) is close to the envelope for all values of. In particular, the local asymptotic power properties of P T (9) are similar to those The curves were generated by taking 2, draws from the distribution of the discrete approximation (based on 2 steps) to the limiting random variables.

9 M. Jansson / Journal of Econometrics 24 (25) Power Envelope. P (9) T S T λ Power Curves Fig.. 5% Level tests, constant mean, scalar x Power Envelope P T (3.5) S T λ Power Curves Fig. 2. 5% Level tests, linear trend, scalar x. 5% Level tests; (a) Constant mean, and (b) linear trend scalar x. of S T for small values of (when the latter is optimal) and P T (9) dominates S T in terms of local asymptotic power for larger values of. As is apparent from Fig. 2, the situation is similar in the linear trend case although the magnitude of the dierences is smaller.

10 96 M. Jansson / Journal of Econometrics 24 (25) Finite sample evidence To gauge the extent to which the predictions from the asymptotic power results of Section 5. can be expected to be borne out in sample sizes encountered in practice, a small Monte Carlo experiment is conducted. Samples of size T = 2 are generated according to the bivariate version of () (4) with ; and x normalized to zero. The errors u t are generated by the bivariate model ( u y t u x t ) ( = (L) 2 )( y t x t ) ; (8) where ( y t ;t x ) i.i.d. N(;I 2 ) and (L)=( a) i= ai L i, corresponding to an AR () model for u y t and ut x. When the errors are generated according to (8), the long-run covariance matrix of u t is given by ( ) = : As a consequence, the parameter in (8) is the correlation coecient computed from and therefore controls the endogeneity of the regressors. In the simulations, the endogeneity parameter and the persistence parameter a both take on values in the set {; :5; :8}, while the parameter of interest,, takes on values in the set {; :975; :95; :925; :9}. The matrix is estimated by ˆ = T U ˆ Û, while and are estimated using VAR() prewhitened kernel estimators. 2 Tables 2a (2b) reports observed rejection rates of 5% level tests using the constant mean (linear trend) versions of the test statistics P T ( ) and S T. To facilitate comparisons, the P T ( ) and S T test are implemented using the same estimation strategy. That is, both tests use the same estimators ˆ and ˆ and both tests are based on a correction in the spirit of Park (992). The version of S T constructed in this fashion is described in Choi and Ahn (995), where it is denoted SBDH II. For moderate values of the persistence parameter, the Monte Carlo evidence is consistent with the asymptotic results of Section 5.. Specically, when a {; :5} rejection rates are reasonably close to 5% under the null in and the point optimal test dominates the locally optimal test in terms of power in most cases. Indeed, the simulation results suggest that non-trivial power gains can be achieved by employing the test proposed in this paper. For a =:8, on the other hand, both tests fail to control the null rejection probability and power is disappointingly low. The size distortions of the point optimal test appear to be slightly smaller than those of the locally optimal test, but the locally optimal test is superior to the point optimal test in terms of (size-adjusted) power. 2 These prewhitened kernel estimators, developed in Jansson (24a), are modied versions of Andrews and Monahan s (992) estimators.

11 M. Jansson / Journal of Econometrics 24 (25) Table 2 Monte carlo rejection rates P T (9), = S T, = a (a) 5% Level tests, constant mean, T = P T (3:5), = S T, = a (b) 5% Level tests, linear trend, T = Note: Based on 5 Monte Carlo replications.

12 98 M. Jansson / Journal of Econometrics 24 (25) 87 2 Acknowledgements This paper draws on material in Chapter 2 of the author s Ph.D. dissertation at University of Aarhus, Denmark. Helpful suggestions from Peter Boswijk, Bent J. Christensen, Graham Elliott, Niels Haldrup, Svend Hylleberg, Nick Kiefer, Ulrich Muller, Jim Powell, Peter Robinson, Michael Svarer, and the three referees are gratefully acknowledged. The paper has beneted from the comments of seminar participants at University of Aarhus, University of California, Berkeley, University of British Columbia, Cambridge University, Tilburg University, the 999 Econometric Society European Meeting, the 999 CNMLE Conference, and the 22 NBER Summer Institute. A MATLAB program that implements the tests proposed in this paper is available at Appendix Throughout the Appendix, denotes the integer part of the argument and all functions are understood to be CADLAG functions dened on the unit interval (equipped with the Skorohod topology). The following Lemma, adapted from Jansson (24a), will be used in the proof of Theorems and 2 to derive limiting representations of sample moments of GLS transformed data from limiting representations of the original data. Indeed, the following relation, in which {F Tt ( )} is expressed in terms of {F Tt }, motivates the transformations considered in Lemma 3: (F T ( );F T2 ( );:::;F TT ( )) = =2 T (F T;F T2 ;:::;F TT ) : Lemma 3. Let {F Tt :6 t 6 T; T } and {g Tt :6 t 6 T; T } be triangular arrays of (vector) random variables with F T = for all T. Let be given and dene F Tt ( )=F Tt +( T )FT;t ( ) and g Tt ( )=g Tt +( T )gt;t ( ) with initial conditions F T ( )=F T and g T ( )=g T. If F T; T t= T T T 2 t=2 t= ( t i= g Tt F Tt g Tt ) g Ti g Tt d F( ) G( ) F(s)dG(s) + FG ( ) ; (9) G(s)dG(s) + GG ( )

13 M. Jansson / Journal of Econometrics 24 (25) where F and G are continuous semimartingales and FG and GG are continuous, then F T; ( ) g T; g T; ( F ( ) ) G ( ) T F Tt ( )g Tt ( ) d t= F (s)dg (s) + FG ( ) ; () ( T (g Tt g Tt ( ))g ) Tt G (s)dg(s) + GG ( ) t= where F (s)= s exp( (s t)) df(t) and G (s)= s exp( (s t)) dg(t). Joint convergence in (9) and () also applies. Proof of Theorems and 2. Under the assumptions of Theorem, = and =. Theorem therefore follows from the Neyman Pearson Lemma, Theorem 2 and the fact that the distribution of (; ) is continuous. Next, to prove Theorem 2, let = T = T and for {; }, let (y + ( );:::; y T + ( )) = Y +. Let x + t = x t ˆ x ˆ û t, where (û ;:::;û T ) = Û. Now, y t + ( ) can be written as y t + ( )=( ) d t + x t + + v t + ( ); where ( ) satises ( ) d t =( + x xx! xy ( )) d t! ˆ xy ˆ xx xd t and v t + ( )=v t ++ +ˆv ++ t ( ), where v t ++ =( ) t uy x s= s + u y x t, u y x t = u y t! xy xx ut x and ˆv ++ t ( )= ( ˆ xx ˆ! xy xx! xy ) ut x ( ˆ ( )xx! xy )û t. Moreover, x t + =x t ++ +ˆx ++ t, where x t ++ =x t x u t and ˆx ++ t =( x ˆ x ˆ ) u t + ˆ x ˆ (u t û t ). Similarly, r t + =(d t;x t + ) = r t ++ +ˆr ++ t, where r t ++ =(d t;x t ++ ) and ˆr ++ t =(; ˆx ++ t ). By proceeding as in the proof of Jansson and Haldrup(22,Lemma 6), standard weak convergence results for linear processes (e.g., Phillips and Solo, 992; Phillips, 988; Hansen, 992) can be used to show that the following hold jointly: T =2 T r + = T =2 T r ++ T + o p() d Q( ) () T =2 t= T t= ( t T t= v t + ( )=T =2 r t + v t + ( )= T s= t= t= v ++ t + o p () d! =2 yy xv ( ); (2) r ++ t v ++ t + o p () d! =2 yy x ) T v s + ( ) v +t ( ) = T t= ( t s= v ++ s ) Q(s)dV (s); (3) v ++ t + o p () d! yy x V (s)dv (s)+ yy x ds; (4)

14 2 M. Jansson / Journal of Econometrics 24 (25) 87 2 for {; }, where ( diag(t =2 ;:::;T (p+=2) ) ) T = : T xx =2 x T xx =2 Since L T + ( ) is invariant under transformations of the form Y + Y + + D a + X + b, P T ( ) can be written as PT ( )+P T 2 ( )+P T 3 ( ), where PT ( ) = log R + R + log R + R+, PT 2 ( )= ˆ! yy x(v + V + V + V + 2 ˆ yy x ), P 3 T ( )= ˆ! yy xv + R+ (R + R+ ) R + ˆ! yy xv + R + (R + R + ) R + V + ; and V + =(v + ( );:::;v T + ( )) for {; }. Using (), Lemma 3, the continuous mapping theorem (CMT) and standard manipulations, PT ( ) = log T R + R + T log T R + R+ T d log QQ log Q Q : Similarly, using (2) and (4), ( ˆ! yy x ; ˆ yy x ) p (! yy x ; yy x ), Lemma 3 and CMT, V + P 2 T ( ) = 2 ˆ! yy x((v + =2 V + ) V + ˆ yy x ) ˆ! yy x(v + =2 V + ) (V + =2 V + ) = 2! yy x((v ++ =2 V ++ ) V ++ yy x )! yy x(v ++ =2 V ++ ) (V ++ =2 V ++ )+o p () d 2 V dv 2 (V ) 2 ; where V ++ =(v ++ ;:::;v T ++ ). Finally, using () and (3), ˆ! yy x p! yy x, Lemma 3 and CMT, ( ) ( ) ( ) PT 3 ( ) d Q dv Q Q Q dv ( ) ( ) ( ) Q dv QQ Q dv : The convergence results in the preceding displays hold jointly. Combining these results, Theorem 2 follows.

15 M. Jansson / Journal of Econometrics 24 (25) References Andrews, D.W.K., Monahan, J.C., 992. An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator. Econometrica 6, Choi, I., Ahn, B.C., 995. Testing for cointegration in a system of equations. Econometric Theory, Davies, R.B., 969. Beta-optimal tests and an application to the summary evaluation of experiments. Journal of the Royal Statistical Society B 3, den Haan, W.J., Levin, A.T., 997. A practitioner s guide to robust covariance matrix estimation. In: Maddala, G.S., Rao, C.R. (Eds.), Handbook of Statistics, Vol. 5. North-Holland, New York, pp Dufour, J.-M., King, M.L., 99. Optimal invariant tests for the autocorrelation coecient in linear regressions with stationary or nonstationary AR() errors. Journal of Econometrics 47, Elliott, G., Rothenberg, T.J., Stock, J.H., 996. Ecient tests for an autoregressive unit root. Econometrica 64, Engle, R.F., Granger, C.W.J., 987. Cointegration and error correction: representation, estimation, and testing. Econometrica 55, Hansen, B.E., 992. Convergence to stochastic integrals for dependent heterogeneous processes. Econometric Theory 8, Harris, D., Inder, B., 994. A test of the null hypothesis of cointegration. In: Hargreaves, C. (Ed.), Nonstationary Time Series Analysis and Cointegration. Oxford University Press, Oxford, pp Jansson, M., 22. Consistent covariance matrix estimation for linear processes. Econometric Theory 8, Jansson, M., 24a. Stationarity testing with covariates. Econometric Theory 2, Jansson, M., 24b. Tests of the null hypothesis of cointegration based on ecient tests for a unit MA root. In: Andrews, D.W.K., Stock, J.H. (Eds.), Identication and Inference in Econometric Models: Essays in Honor of Thomas J. Rothenberg. Cambridge University Press, Cambridge, forthcoming. Jansson, M., Haldrup, N., 22. Regression theory for nearly cointegrated time series. Econometric Theory 8, King, M.L., 98. Robust tests for spherical symmetry and their application to least squares regression. Annals of Statistics 8, King, M.L., Hillier, G.H., 985. Locally best invariant tests of the error covariance matrix of the linear regression model. Journal of the Royal Statistical Society B 47, Lehmann, E.L., 994. Testing Statistical Hypotheses, 2nd Edition. Chapman & Hall, New York. Park, J.Y., 99. Testing for unit roots and cointegration by variable addition. In: Fomby, T.B., Rhodes, G.F. (Eds.), Advances in Econometrics, Vol. 8. JAI Press, Greenwich, CT, pp Park, J.Y., 992. Canonical cointegrating regressions. Econometrica 6, Phillips, P.C.B., 987. Time series regression with a unit root. Econometrica 55, Phillips, P.C.B., 988. Weak convergence of sample covariance matrices to stochastic integrals via martingale approximations. Econometric Theory 4, Phillips, P.C.B., 99. Optimal inference in cointegrated systems. Econometrica 59, Phillips, P.C.B., Solo, V., 992. Asymptotics for linear processes. Annals of Statistics 2, 97. Robinson, P.M., Velasco, C., 997. Autocorrelation-robust inference. In: Maddala, G.S., Rao, C.R. (Eds.), Handbook of Statistics, Vol. 5. North-Holland, New York, pp Saikkonen, P., Luukkonen, R., 993. Point optimal tests for testing the order of dierencing in ARIMA models. Econometric Theory 9, Shin, Y., 994. A residual-based test of the null of cointegration against the alternative of no cointegration. Econometric Theory, 9 5. Stock, J.H., 994. Unit roots, structural breaks and trends. In: Engle, R.F., McFadden, D.L. (Eds.), Handbook of Econometrics, Vol. 4. North-Holland, New York, pp Watson, M.W., 994. Vector autoregressions and cointegration. In: Engle, R.F., McFadden, D.L. (Eds.), Handbook of Econometrics, Vol. 4. North-Holland, New York, pp Xiao, Z., Phillips, P.C.B., 22. A CUSUM test for cointegration using regression residuals. Journal of Econometrics 8, 43 6.