MAKING WALD TESTS WORK FOR. Juan J. Dolado CEMFI. Casado del Alisal, Madrid. and. Helmut Lutkepohl. Humboldt Universitat zu Berlin

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1 November 3, 1994 MAKING WALD TESTS WORK FOR COINTEGRATED VAR SYSTEMS Juan J. Dolado CEMFI Casado del Alisal, Madrid and Helmut Lutkeohl Humboldt Universitat zu Berlin Sandauer Strasse Berlin, Germany Setember 1994 Abstract Wald tests of restrictions on the coecients of vector autoregressive (VAR) rocesses are known to have nonstandard asymtotic roerties for I(1) and cointegrated systems of variables. A simle device is roosed which guarantees that Wald tests have asymtotic 2 {distributions under general conditions. If the true generation rocess is a VAR() it is roosed to t a VAR(+1) to the data and erform a Wald test on the coecients of the rst lags only. The ower roerties of the modied tests are studied both analytically and numerically by means of simle illustrative examles. (*) The research for this aer was suorted by funds from Sonderforschungsbereich 373 at Humboldt Universitt Berlin. We are indebted to R. Mestre for invaluable research assistance. We have also beneted from comments by Peter C.B. Phillis, Enrique Sentana and articiants at the Euroean Meeting of the Econometric Society, Maastricht 1994.

2 1 Introduction Wald tests are standard tools for testing restrictions on the coecients of vector autoregressive (VAR) rocesses. Their concetual simlicity and easy alicability make them attractive for alied work to carry out statistical inference on hyotheses of interest. For instance, a tyical examle is the test of Granger-causality in the VAR framework where the null hyothesis is formulated as zero restrictions on the coecients of the lags of a subset of the variables. Unfortunately, those tests may have nonstandard asymtotic roerties if the variables considered in the VAR are integrated or cointegrated. The diculties in dealing with the levels estimation of such time series are well known and they have been illustrated by means of the general asymtotic theory for inference in multile linear regressions with integrated rocesses recently develoed by Park and Phillis (1988, 1989), Sims, Stock and Watson (1990) and Toda and Phillis (1993 a,b) among others. As a byroduct of the analysis it has been found that, for instance, Wald tests for Granger-causality are known to result in nonstandard limiting distributions deending on the cointegration roerties of the system and ossibly on nuisance arameters. This means that to test such hyotheses, the limiting distributions under the null hyothesis need to be simulated in each relevant case, deending on the number of variables, cointegration rank, the number of lags and ossibly unknown nuisance arameters. This can be comutationally burdensome and may be imossible if the required information is unavailable. Faced with that roblem, a ossible solution which has been usually adoted in alied work is to condition the testing rocedure on the estimation of unit roots, cointegration rank and cointegrating vectors. Thus, for instance, a rst order dierenced VAR could be estimated if variables were known to be I(1) with no cointegration, or an error correction model (ECM) could be secied if they were known to be cointegrated. Of course, a riori, it is hardly the case that such a knowledge exists with certainty. Consequently, a retesting sequence is usually needed before estimating the VAR model in which inference is conducted. Given the low ower of those tests and their deendence on nuisance aramenters in nite samles, that testing sequence has tyically unknown overall roerties, leaving oen the ossibility of severe distortions in the inference rocedure. To overcome these diculties, we roose in this aer an extremely simle method which 1

3 leads to Wald tests with standard asymtotic 2 {distributions and which avoids ossible retest biases. With this device the tests may be erformed directly on the least squares (LS) estimators of the coecients of the VAR rocess secied in the levels of the variables. Note that although the variables are allowed to be otentially cointegrated it is not assumed that the cointegration structure of the system under investigation is known. Hence, reliminary unit root tests are not necessary and, therefore, the testing rocedure is robust to the integration and cointegration roerties of the rocess. The idea underlying the rocedure is based on the following argument. It is well known that the nonstandard asymtotic roerties of the Wald test on the coecients of cointegrated VAR rocesses are due the singularity of the asymtotic distribution of the LS estimators. Then, the simle device resented here is to get rid of the singularity by tting a VAR rocess whose order exceeds the true order. It can be shown that this device leads to a nonsingular asymtotic distribution of the relevant coecients, overcoming the roblems associated with standard tests and their comlicated nonstandard limiting roerties. In what follows, the test based uon the estimated coecients of the augmented VAR rocess will be denoted as modied Wald test. In indeendent work Choi (1993) and Toda & Yamamoto (1993) have roosed a similar device for univariate and multivariate rocesses, resectively. However, their analysis of the ower roerties of the modied tests is rather limited. This is an imortant issue since the modied aroach uses the samle ineciently and thereby may result in severe reductions of ower. Thus, in this aer we ay articular attention to analysing those cases in which the ineciency is likely to be more imortant. Also, we feel that our roof of the aymtotic distribution of the Wald statistic is more transarent than that of Toda & Yamamoto. From our result it is aarent when it is actually necessary to add an extra lag and when standard asymtotic results make that device unnecessary. The rest of the aer is lanned as follows. First, in Section 2, the simle idea underlying the testing rocedure is illustrated by means of a unit root test in an AR(1) model. Section 3 extends the revious analysis to the general VAR case with I(1) variables, since this is the most imortant one in ractice. The ower roerties of the modied test are analysed in Section 4. Some illustrating Monte Carlo simulations are oered in Section 5. Finally, some conclusions are drawn in Section 6. 2

4 2 Testing for a Unit Root with Standard Asymtotics Let the univariate time series fy t g be generated by the random walk: y t = y t?1 + " t ; = 0; (t = 1;... ; T ) (1) where f" t g is a sequence of i.i.d. random variables with mean zero and variance 2 " such that Ej" t j 2+ < 1 for some > 0. Further, the initial condition y 0 is assumed to be any random variable whose distribution is xed and indeendent of the samle size T. Suose that one wishes to test the null hyothesis H 0 : = 0. The standard aroach is the use of the Dickey-Fuller test based on T ^ or the t-ratio of ^. However, if the model is augmented by one lag, as in the Augmented Dickey-Fuller test 1, that is: y t = y t?1 + y t?2 + " t (2) with = 0, and (2) is estimated by OLS, then, the estimated is: ^ = (y?1m 0 2 y?1 )?1 (y?1m 0 2 y) where y?i = [y 2?i ;... ; y T?i ] 0, i = 1; 2; M 2 is the rojection matrix sanned by y?2, that is M 2 = I? y?2 (y 0 y?2?2)y 0 and y = [y?2 2;... ; y T ] 0. Using a similar notation for the " t 's and dening " = [" 2 ;... ; " T ] 0 we have: y 0 M?1 2y = (y?2 + "?1 ) 0 M 2 " = " 0 M?1 2" y 0 M?1 2y?1 = (y?2 + "?1 ) 0 M 2 (y?2 + "?1 ) = " 0 M?1 2"?1 Then, alying the convergence results for I(1) variables (see Park and Phillis (1989)) to the revious exressions, with the symbol ")" denoting weak convergence in distribution, we get: T 1=2^ = (T?1=2 " 0?1")(T?2 y 0?2y?2 )? (T?1 " 0?1y?2 )(T?1 y 0?2")T?1=2 (T?1 " 0?1"?1 )(T?2 y 0?2y?2 )? (T?2 " 0?1y?2 ) 2 T?1 = (T?1 " 0?1"?1 )?1 (T?1=2 " 0?1") + O (T?1=2 ) ) N(0; 1) (3) from the alication of standard Central Limit Theorems for i.i.d. random variables. Thus, the modied test based on T 1=2^ (or the t-ratio of ^) could be used to test for a unit root, 1 Note that (2) can be rearameterised as y t = y t?1 + ( + )y t?2 + " t. 3

5 yielding standard asymtotics. Of course, the disadvantage of this rocedure is that it uses the samle information ineciently, leading to a loss in ower. As mentioned earlier, it is well known that T ^ (or the t-ratio) in the estimation of (1) follows the Dickey-Fuller distribution (see Fuller (1976)), so the modied test is clearly disadvantageous once the true distribution is known 2;3. However, in more general cases within a VAR framework, the correct nonstandard distributions deend on the size of the system, the number of lags and the cointegration structure (see Table 1 in Toda and Phillis (1993a)) making it dicult or even imossible to use the correct tables in each secic case faced in alied work. Thus, it may be sensible to make a sacrice in terms of ower and gain the correct size in terms of an asymtotic 2 {distribution. We devote the next section to analyzing this simle idea in a more general setu. 3 Main Result for VAR Systems Consider the k-dimensional multile time series generated by a VAR () rocess: y t = A 1 y t? A y t? + " t (4) where " = (" 1t ;... ; " kt ) 0 is a zero mean indeendent white noise rocess with nonsingular covariance matrix " and, for j = 1;... ; k; Ej" jt j 2+ < 1 for some > 0. The order of the rocess is assumed to be known or alternatively it may be estimated by some consistent model selection criterion (see, e.g. Paulsen (1984) or Lutkeohl (1991, Chater 11)) 4. Let a =vec[a 1... A ] where vec denotes the vectorization oerator that stacks the columns of the argument matrix and suose that we are interested in testing q indeendent 2 Choi (1993), using a similar aroach, nds that the t-ratio for ^ suers from low ower relative to the Dickey-Fuller test. Moreover, for small samles, it suers from size distortions. However, it has reasonable roerties in constructing condence intervals for the sum of AR coecients ossibly in the resence of unit roots. 3 A similar result will hold for T 1=2^. However, as is well known, the joint distribution of T 1=2^ and T 1=2^ has a singular asymtotic covariance matrix; thus joint tests do not follow standard limiting distributions (see Banerjee and Dolado (1988) and Sims, Stock and Watson (1990)). 4 Of course this involves some retesting bias, but it is also involved in the standard rocedure. Paulsen (1984) and Toda & Yamamoto (1994) rove that if y t is I(d) (integrated of order d) the usual selection rocedures are consistent if d. Thus, if d = 1, the lag selection rocedures are always valid. In Section 5, we examine the consequences of overestimating the true VAR order. 4

6 linear restrictions: H 0 : Ra = s vs: H 1 : Ra 6= s (5) where R is a known (q k 2 ) matrix of rank (rk) q and s is a known (q 1) vector. For examle, if y t is artitioned in m and (k? m){dimensional subvectors y 1 t and y 2 t and the A i matrices are artitioned conformably, then y 2 t does not Granger-cause y 1 t i the hyothesis H 0 : A 12;i = 0 for i = 1... is true. The standard Wald test is as follows. Get an asymtotically normal estimator ^a satisfying: T 1=2 (^a? a ) ) N(0; ) and use the statistic: w = T (R^a? s) 0 (R^ R 0 )?1 (R^a? s) (6) where ^ is some consistent estimator of. The Wald statistic w has an asymtotic 2 { distribution with q degrees of freedom if is nonsingular. If the VAR() rocess fy t g is I(0), invertibility holds for the usual estimators (LS or ML) and Wald tests may be alied in the usual manner. However, this is not true if fy t g is I(d), d > 0. An exosition of the revious result for I(1) rocesses can be found in Lutkeohl (1991, Chater 11) and we summarize the main arguments in what follows. Consider the EC (error correction) reresentation of (4): y t = D 1 y t? D?1 y t?+1? y t? + " t (7) where D i =?(I k? A 1?...? A i ); i = 1;... ;? 1; and = (I k? A i?...? A ) with rk() = r. Therefore, can be written as the roduct = BC where B is (k r) and C is (r k) with rk(b) = rk(c) = r. Dening: 2 3 Y = [y 1... y T ] ; X t = 6 4 y t. y t? ; X = [X 0... X T?1 ] D = [D 1... D?1 ] ; Y? = [Y 1? ;... ; Y T? ] ; E = [" 1 ;... ; " T ] (7) can be rewritten in comact form as: Y = DX? BCY? + E (8) 5

7 Then, denoting as ~ D and ~ B ~ C the ML estimators of D and BC (see Johansen (1991)) we have: h i T 1=2 vec( ~D;? B ~ C ~? [D;?BC]) ) N(0; co ) with and co I k(?1) 0 I k(?1) 0 A " (9) 0 C 0 0 C = lim T?1 XX 0 XY 0 C 0? CY? X 0 CY? Y?C A (10) Note that the dimension of co is (k 2 k 2 ), whereas the dimension of is [(k(? 1) + r) (k(? 1) + r)]. Thus, the rank of co cannot be greater than k(k(?1)+r) which is smaller than k 2 unless r = k (the stationary case). Hence, if fy t g is I(1), co singular. It has been shown by various authors (Park and Phillis (1989) and Sims, Stock and Watson (1990)) that the ML estimators ~ A i of A i obtained via the EC reresentation (7) with known cointegrating matrix C and the unrestricted LS estimators ^A i obtained from (4) have the same asymtotic distribution. Therefore we assume that C is known and that ~ B and ~ D i are the ML estimators of the remaining arameters in (7). Then it follows that: where EC is a nonsingular [k(k(? 1) + r)] covariance matrix. Note that ~ B and B disaear for r = 0. h T 1=2 vec ( D; ~? B) ~ i? (D;?B) ) N(0; EC ) From (7) we get the arameters of the VAR reresentation (4) as is A 1 = I k + D 1 A i = D i? D i?1 (i = 2;... ;? 1) A =?D?1? Hence, [A 1 ;... ; A ] = [B; D 1 ;... ; D?1 ] W + [I k ; ] (11) 6

8 where ?C I k?i k I W = k?i k ?I 4 k I k?i k [(r + k(? 1)) k] Since rk(w ) = r + k(? 1) and T 1=2 (^a? a ) ) N(0; ) it follows that the [k 2 k 2 ] covariance matrix = (W 0 I k ) EC (W I k ) has obviously rank k[k(? 1) + r]. Thus, it is singular for r < k. However, if A i is omitted from (11), k columns of W are deleted which results in a [(r + k(? 1)) k(? 1)] matrix W with rank k(? 1). Denoting by ^a?1 ^a, we get: the estimator of the remaining [k 2 (? 1)] elements of a obtained from T 1=2 (^a?1? a?1 ) ) N [0; (W ; I k ) EC (W I k )] N 0;?1 (12) where the [k 2 (? 1) k 2 (? 1)] covariance matrix?1 test can be imlemented in the usual way. Therefore, the following theorem holds: has now full rank and the Wald Theorem 1 Let the k{dimensional I(1) rocess fy t g be generated by the VAR() rocess in (7) and let ^A i (i = 1;... ; ) be the LS estimators and ^a?1 the [k 2 (? 1)]{dimensional vector consisting h i of the k 2 (? 1) elements of ^a = vec ^A1;... ; ^A that are obtained by deleting one of the ^A i matrices. Then: T 1=2 (^a?1? a?1 ) ) N(0;?1 ) where the [k 2 (? 1) k 2 (? 1)] covariance matrix?1 is nonsingular. Moreover, given a consistent estimator ^?1, the Wald test of the null hyothesis H 0 : Ra?1 = s, w = T (R^a?1? s) 0 (R^?1 R 0 )?1 (R^a?1? s) (13) 7

9 has an asymtotic 2 (q){distribution.2 The theorem imlies that whenever the elements in at least one of the comlete coecient matrices A i are not restricted under H 0, the Wald statistic has its usual 2 {distribution. Thus, if elements from all A i, i = 1;... ; ; are involved in the restrictions as, for instance, in noncausality hyotheses, we may just add an extra lag in estimating the arameters of the rocess and thereby ensure standard asymtotics for the Wald test. Of course, if the true DGP is a VAR() rocess, then a VAR( + 1) with A +1 = 0 is also an aroriate model. Using the revious notation, in this case the modied Wald test will be based on the h estimator ^a +1, namely the rst [k 2 ] elements of vec ^A1... ^A i +1. Notice that for this rocedure to work it is obviously neither necessary to know the cointegration roerties of the system nor the order of integration of the variables. Thus, if there is uncertainty whether the variables are I(1) or I(0), one may simly add the extra lag and then erform the test to make sure to be on the safe side. Of course there will be a loss of ower, given that in the nonstationary case some VAR coecients or linear combinations of them can be estimated more eectively with larger rate of convergence than in the I(0) case. Nevertheless, one may argue about the accetability of the resulting loss in ower. In general, we will exect the loss in ower to be of little relevance if the true order is large and the dimension k is small or moderate, since in this case the relative reduction in the estimation recision due to one extra VAR coecient matrix will be small. However, if the true order is small and k is large, an extra lag of all variables may lead to a sizeable decline in the ower of the modied Wald test. To get a feeling for the trade-o between size and ower of the roosed rocedure, a small Monte-Carlo analysis is carried out in Section 5. It may be worth noting that the theorem remains valid if an intercet term or other deterministic terms, like seasonal dummies or time trends, are included in the VAR model. This follows from the results in Park and Phillis (1989) and Sims, Stock & Watson (1990) who demonstrate that the asymtotic roerties of the VAR coecients are essentially unaected by such terms. Moreover, a similar result can be obtained for VAR systems with I(d) variables where d > 1. In that case, d coecient matrices A i must be unrestricted under H 0. Alternatively, d lags must be added if all arameter matrices of the original rocess are restricted. This is also a consequence of results given in Sims, Stock & Watson (1990). 8

10 4 Power Proerties To analyse the ower roerties of the modied Wald test, we rst notice that it is consistent. Suose, for instance, that the alternative hyothesis is: H 1 : Ra = s + ; 6= 0 (14) Then, under H 1, we have: w = T (R^a? +1 s)0 (R^ +1 R0 )?1 (R^a? s) +1 = T (R^a +1? s? ) 0 (R^ +1R 0 )?1 (R^a +1? s? ) +T 0 (R^ +1R 0 )?1 + 2T 1=2 0 (R^ +1R 0 )?1 T 1=2 (R^a +1? s? ) = O (1) + O (T ) + O (T 1=2 ) = O (T ) Thus, for any ositive number M, Pr[ w > M]! 1 as T " 1, i.e. the test is consistent. To study the local ower roerties, consider the local alternative: H 1 : Ra = s + T?1=2 for xed (15) Then, w ) 2 (q; 2 ), i.e. a non-central 2 {distribution with non-centrality arameter given by: 2 = 0 (R +1R 0 )?1 (16) Following Kendall and Stuart (1961, Chater 24) (see Mizon and Hendry (1980)), the rst two moments of the non-central 2 {distribution can be aroximated by a central 2 (with dierent degrees of freedom). More recisely: 2 (q; 2 ) ' h 2 (m; 0) (17) where h = (q )=(q + 2 ) and m = (q + 2 ) 2 =(q ). Consequently, for any M, the aroximate and large samle ower P of w is given by: (q + P = Pr [ w > M] ' P r 2 2 ) 2 > M q + 2 (18) q q Note that if H 0 is true, = 0, so that P r [ w > M]! P r 2 (q) > M 9

11 conrming the aroriate nominal and large samle size of the test. Moreover, since 2 = T (Ra +1?s) 0 (R +1R 0 )?1 (Ra +1?s); 2 " 1 with T so that h " 2 and m " 1, i.e. P " 1. Similarly, if takes higher values, for xed T, 2 and m increase and so does the ower. To summarise, equation (18) oers an analytical formula to examine the eects of the factors (a +1; ; T; k) = on the large samle ower of w to reject H 0 against the sequence (15). We devote the next section to analysing some of those eects in nite samles. 5 A Small Monte-Carlo Analysis To illustrate the revious discussion on the use of Granger-causality tests in VAR systems with I(1) variables, we have generated 1000 relications of the bivariate VAR(2) cointegrated rocess y t = (y 1t ; y 2t ) 0 given by: y t = 4? 5 y t? :5 0:3 5 y t?1 + " t (19) 0 0 T?1=2 0:5 where " t N(0; I 2 ). The rocess has cointegration rank r = 1 (= 0) i 6= 0 ( = 0). If = 0, y 1t is Granger noncausal for y 2t and if 6= 0, y 1t causes y 2t. Therefore, = 0 is used to study the size of the test and = 1; 2 are used to analyse ower. For each time series 50 resamle values are generated with zero initial conditions, taking net samle sizes of T = 50, 100 and 200. The tted rocesses include a constant term, that is the model y t = +A 1 y t?1 +A 2 y t?2 +" t is tted for the standard rocedure and an analogous VAR(3) rocess for the modied rocedure. Table 1(a) resents the relative rejection frequencies for tests with asymtotic 5% significance level of a 2 (2){distribution when = 1, i.e. there is cointegration. In this case it is not dicult to see that the standard Wald test has an aymtotic 2 (2){ distribution under H 0. Thus, this case is favourable for the standard test. To assess whether the rejection rates are signicantly dierent from the theoretical rate the following 95% condence interval is useful: [3:6%; 6:4%]. The test rejects slightly too often for small and moderate samles (T = 50 and 100) 5. With resect to the ower, it is clear that it is higher when the true VAR(2) rocess is estimated. In other words, the modied test wastes information by estimating 5 This is in agreement with the slow convergence of the standard t-ratio in the univariate case analysed by Choi (1993). 10

12 extra coecients. However, the assumtion that the true order is known might be too otimistic, so in Table 1(b) we retend that the data are generated by a VAR(3) rocess and reeat the tests which now have asymtotic 2 (3){null{distributions. The corresonding modied Wald test is obtained from a VAR(4) rocess. In this case the owers of the two tests are found to be almost identical. Thus, even under this minor deviation from the ideal conditions for the standard test, the loss in eciency for the modied rocedure almost disaears. Table 1(c) reorts the size and ower for = 0, i.e. the case where there is no cointegration. In ractice, the cointegration rank is unknown and has to be determined in a retesting rocedure. In this case the standard test does not have an asymtotic 2 (2){distribution under the null hyothesis. Hence, this examle illustrates the consequences of using the standard Wald test incorrectly with a 5% critical value from a 2 (2){distribution. As in the rst examle, VAR(2) and VAR(3) rocesses are tted to the variables in levels. We nd that the standard test rejects too often under H 0 even for large samles (see Ohanian (1988), and Toda and Phillis (1993a)) while the modied test converges to its correct nominal size for T = 200. Hence, the standard test is clearly misleading while the modied test maintains roughly the same roerties in large samles as for the cointegrated rocess (19) with 6= 0. Consequently, in terms of size, the modied rocedure is clearly referable if the cointegration rank is unknown 6. Next, in order to check the loss in ower of the modied Wald test for given values of the dimension k of the rocess and the true order of the VAR, we carry out two tyes of exeriments. First, to analyse the eect of enlarging for given k, the DGP(19) is generalised to: y t = 4? 0:5 0:3 5 y t? y t?+1 + " t (20) 0 0 T?1=2 0:5 where " N(0; I 2 ), = 1, = 1 and = (2; 3;... ; 6). The emirical owers were calculated out of 1000 relications for a net samle size of 100 and are reorted in Table 2. The null hyothesis is again H 0 : = 0. The gures without arentheses and those with arentheses denote the relative ineciency (measured by the ratio of owers) of the modied with resect 6 Note that the ower of the standard test in this case is uwards biased since it has a larger size than the nominal 5% level. Comutation of the size adjusted ower for T =200 and = 1; 2 yields rejection frequencies 23.4% and 61.3% for the standard test and 18.7% and 54.6% for the modied test, resectively. 11

13 to the standard Wald test and the absolute emirical ower of the latter, resectively. In agreement with the conjecture oered in Section 3 we nd that, for k = 2, the relative ineciency of the modied test, based uon the estimation of a VAR( + 1) rather than a VAR(), decreases with the true order. For instance, we nd that, for > 3, the loss in ower becomes less than 10%. Hence, if a VAR system has a small number of variables with a long lag length, as is often the case in ractice, then the ineciency caused by adding a few more lags would be relatively small. Second, to examine the eect of enlarging k for given, the DGP in (19) is generalised to: ?... 0:5 a 12 a a 1k T?1=2 0:5 a a 2k y t = y t? : y t?1 + " t (21) :5 where now y t = (y 1t ; y 2t ;... ; y kt ) 0, k = (2; 3;... ; 6), " t N(0; I k ), = 1, = 1, a 1l = 0:3=(l? 1) (l = 2;... ; k) and a 2l = 0:3=(l? 2) (l = 3;... ; k). Having generated 1000 relications for T = 100, the numbers in Table 3 have the same meaning as in Table 2, with the null hyothesis being again H 0 : = 0. We conclude from this exeriment that if the VAR system has many variables and the true lag length is short (=2 in this case), then the ineciency caused by adding even one extra lag would be relatively big. For instance, for k = 6, the modied Wald test has only a little more than one-fourth of the ower of the standard test. However, given that the absolute ower of the latter is around 20%, the absolute loss of ower is not that large after all. Finally, in order to make analytical comarisons of the relative ower roerties of both tests by means of the aroximate ower function derived in (18), we have used a simler illustrative bivariate DGP based uon a VAR(1) system with I(1) variables. In this way, the analysis becomes tractable and it can be used to shed light on the eect of some of the incidental arameters of the DGP. In articular, we focus attention on the following set of arameters = [; ; V (" 1t ); V (" 2t ); Cov(" 1t ; " 2t )]. 12

14 We consider the following DGP: y t = 4? 5 y t?1 + " t ; " t N 4@ 0? 0 1 A 1 13 A5 (22) with = T?1=2. As in the DGP's considered above, = 0 corresonds to the case where y 1t is Granger noncausal for y 2t. Given the simlicity of the DGP, it is easy to comute the non-centrality arameter 2 in the VAR(1) system (standard rocedure) which is given by (see Aendix): 2 1 = 2 (1 +? 2)=(1? 2 ) (23) where = 1??. Similarly, in the VAR(2) model (modied rocedure), the corresonding exression is (see Aendix): 2 = 2 2 (1? 2 =)= (24) For jj < 1, it is easy to show that 2 1 > 2 2, as exected 7. Moreover, since h and m are increasing in 2 this means that the ower of the standard test is larger than the ower of the modied test. Note, also, that for = 0 and = 0, i.e. r = 1, 2 1 is not dened, reecting the non-standard distribution of the standard Wald test in the absence of cointegration. Nevertheless, the modied test has a non-centrality arameter which does not deend uon, reecting that it has the correct size under the null hyothesis and that its limiting distribution is a non-central 2 even when cointegration does not exist. To check how well the analytical aroximate large samle ower comares to the emirical rejection frequencies, 2000 relications were conducted for T = 100 of the following four exeriments, (arameter congurations in arentheses): Exeriment 1 ( = 1; = 1; = 1); Exeriment 2 ( = 0:2; = 1; = 1); Exeriment 3 ( = 1; = 2; = 1); and Exeriment 4 ( = 1; = 1; = 0:1). For each exeriment, the correlation between " 1t and " 2t (corr = = 1=2 ) takes three values, i.e. corr = (0:0; 0:5 and?0:5). This is done to control for the deendence of the ower functions on the covariance as exemlied by exressions (23) and (24). Thus, Exeriment 1 is the base exeriment; Exeriment 2 examines the eect of a reduction in with resect to the base exeriment. Similarly, Exeriments 3 and 4 examine the eect of an increase in and a decrease in, resectively. 7 Since 2 1 > 2 1 > 2 (1 +? 2)= and (1 +? 2) 1? 2 =. Thus, 2 1 >

15 Table 4 reorts the results of the revious set of exeriments in terms of analytical (P ) and emirical (P ) rejection frequencies, together with the values of the roortion factor (h?1 = (q + 2 )=(q +2 2 )), the number of degrees of freedom (m) and the relative ower (R) comuted in terms of the ratio of emirical rejections. To comute the analytical ower, the degrees of freedom of the aroximate central 2 {distributions were roxied by the closest integer art of m. Several results are worth mentioning. First, the analytical and emirical rejection frequencies yield broadly similar results with their dierences never exceeding 10 ercentage oints in the least favourable cases. Thus, the asymtotic local ower analysis roves to be useful in interreting the relative ower outcomes in nite samles. Second, within each exeriment, the ower of the standard test is highest for corr =?0:5 and lowest for corr = 0:5, reecting the fact that 2 1 decreases with increasing correlation between the error terms. At the same time, the ower of the modied Wald test does not deend on the sign of the correlation coecient, as shown in (24). Therefore, the more negative is the correlation coecient the larger will be the relative ineciency of the modied test, i.e. the smaller is R. The intuition behind this result lies in the form of the cointegrating vector in DGP (22), i.e. (1,-1). This imlies that the variance of deviations from the cointegrating relationshi, (y 1t? y 2t ), deends uon V (" 1t? " 2t ) (see Aendix). Thus if < 0, V (y 1t? y 2t ) will increase. Since in the standard Wald test the null hyothesis = 0 can be solely exressed as a restriction on the coecient of (y 1t?1? y 2t?1 ), the higher the variance of that variable, the more eciently the coecient will be estimated and, hence, the larger will be the ower of the test. Once we condition on further lags of y 1t and y 2t, as in the modied rocedure, that direct eect disaears. This is reected by the deendence of 2 2 on 2 rather than. Had the cointegrating vector been (1,1), the "residual" (y 1t + y 2t ) would have a variance which deends on V (" 1t + " 2t ). Therefore, in this case, the oosite result holds, that is, > 0 will increase 2 1 and the ower of the standard test. Third, the owers of the two tests decrease with, reecting the fact that a lower variance of the error term in the equation of interest results in a higher ower. Fourth, the owers of the two tests obviously increase towards unity as increases. Lastly, the lower is, namely, the less cointegrated are the variables and the higher is the variance of (y 1t? y 2t ), the larger is the ower of the standard test relative to the ower of the modied test, since 2 2 does not 14

16 deend on. Overall, we conclude that the loss in ower entailed by the use of the modied rocedure, for the articular DGP under study, will be larger the more negative is the correlation coecient between the error terms and the less cointegrated are the variables. Note, however, that low values of could lead to otential size distortions (over-rejections) of the standard test and thereby exaggerate the loss of ower of the modied test (see footnote 4) 8. 6 Concluding Remarks In this aer a device is roosed that guarantees standard 2 asymtotics for Wald tests erformed on the coecients of cointegrated VAR rocesses with I(1) variables if at least one coecient matrix is unrestricted under the null hyothesis. By the same token, if all the matrices are restricted, it is shown that adding one extra lag to the rocess and concentrating on the original set of coecients results in Wald tests with standard asymtotic distributions. This leads to a number of interesting imlications which stem from the ossibility of exressing null hyotheses as restrictions on coecients of stationary variables (see Sims, Stock and Watson (1990)). First, for I(1) variables (with or without cointegration), if a VAR(2) is tted, all t-ratios are asymtotically normal. Second, a VAR() can be tested against a VAR(+1), 1, with a standard Wald test. Third, if the true DGP is a VAR() and a VAR( + 1) is tted, standard Wald tests can be alied to the rst VAR coecient matrices. These results do not deend on the resence of deterministic terms in the DGP as long as the restrictions are conned to the VAR coecients. Furthermore, nonlinear restrictions can be tested in the same way. As regards the reduction in ower entailed by the inecient use of the samle in the modied rocedure, our Monte Carlo simulations show that it will be more severe in high dimensional VARs with a small true lag length. Moreover, in bivariate systems, ossibly cointegrated, we nd that a negative correlation between the error terms in the equations seems to cause larger ineciency when the cointegrating relationshi is of the form (1,-1), while a ositive correlation causes larger ineciency if it is of the form (1,1). 8 As in DGP(19), we retended that the data were generated by a VAR(2) and reeated the tests with 2 (2) critical values in VAR(2) and VAR(3) models. As in the revious case, we found that the relative ineciency in terms of ower was minor, with R 0:90 in all cases. 15

17 However, we nd that when there are serious doubts about the series being cointegrated, the size distortions of the modied rocedure are much smaller in nite samles. Thus, the ower disadvantage is likely to be outweighed by the ease of alicability of the modied rocedure. Finally, it is imortant to note that the revious results could be generalised to VAR systems with I(d) variables, d > 1. In that case, the modied rocedure involves adding d extra lags. 16

18 APPENDIX Given the DGP(22), the univariate reresentations of y 1t and y 2t are given by: y 1t = (" 1t? (1? )" 1t?1 + " 2t?1 )=1? L y 2t = (" 2t? (1? )" 2t?1 + " 1t?1 )=1? L (A:1) (A:2) while the deviation from the cointegrating relationshi, u t, follows the rocess: u t = (y 1t? y 2t ) = (" 1t? " 2t )=1? L (A:3) where = 1??, such that jj < 1. Here L is the lag oerator. Then, the standard test is based uon the regression model: y t = ^A1 y t?1 + ^e t ; E(e t e 0 t) = e or y t = ^By t?1 + ^e t : In articular, the second equation of the system, to which the noncausality test is alied, can be written as: y 2t = b 21 y 1t?1 + b 22 y 2t?1 + e 2t (A:4) Using (A.3), (A.4) can be rearameterised as: y 2t = b 21 u t?1 + (b 21 + b 22 )y 2t?1 + e 2t (A:5) That is, the rearameterisation makes it ossible to exress the arameter of interest, b 21, as a coecient on an I(0) variable. Obviously, estimation of (A.4) by OLS yields consistent estimators of and in DGP(22), such that lim ^b 21 = and lim ^ 2 e2 =. Moreover, since u t?1 is asymtotically orthogonal to y 2t?1, (being I(0) and I(1) variables, resectively), the asymtotic variance of ^b 21, V (^b 21 ) deends only on E(u 2). Indeed, V (^b 2 21 ) = =E(u 2) = t (1? 2 )=(1 +? 2). Thus, the non-centrality arameter of the standard test is given by: 2 1 = 2 =V (^b 21 ) = 2 [1 +? 2] =(1? 2 ) (A:6) In the modied Wald test, the regression model is: y t = ^A 1 y t?1 + ^A 2 y t?2 + ^" t 17

19 or y t = ^By t?1 + ^Cy t?2 + ^" t In articular, the second equation of the system will be: y 2t = b 21 y 1t?1 + b 22 y 2t?1 + c 21 y 1t?2 + c 22 y 2t?2 + e 2t (A:7) which can be rearameterised as: y 2t = b 21 u t?1 + c 21 u t?2 + (b 21 + b 22 )y 2t?1 +(b 21 + b 22 + c 21 + c 22 )y 2t?2 + e 2t (A:8) Using similar arguments as in the VAR(1) case, lim ^b 21 =, lim ^ 2 e2 regressors fu t?1 ; u t?2 ; y 2t?1 g are asymtotically orthogonal to y 2t?2. = and the I(0) Thus, in this case the asymtotic variance of ^b 21, V (^b 21 ) is given by the (1,1) element of: B A 33 where f ij g is the covariance matrix of fu t ; u t?1 ; y 2t g. From (A.1) - (A.3), we get: 11 = E(u 2 t ) = 22 = (1 +? 2)=(1? 2 )?1 12 = E [u t ; u t?1 ] = = E [u t ; y 2t ] = [? (1? ) +? + (1? )? ] =(1? 2 ) 23 = E [u t?1 ; y 2t ] = [? (1? ) +? + (1? )? ] =(1? 2 ) 33 = E y2t 2 = (1 + (1? )2 ) + 2? 2(1? )? 2(1? ) + 2 =(1? 2 ) From these results we can obtain the much simlied exression V (^b 21 ) = 2 =(? 2 ). Thus, the non-centrality arameter of the modied Wald test, is given by: 2 2 = 2 =V (^b 21 ) = 2 (1? 2 =)= (A:9) 18

20 Table 1 Relative Rejection Frequencies (%) (a) ( = 1, AM = VAR(2), 5% CV = 5.99) Standart Test Modied Test T (b) ( = 1, AM = VAR(3), 5% CV = 7.81) Standart Test Modied Test T (c) ( = 0, AM = VAR(2), 5% CV = 5.99) Standart Test Modied Test T (23.4) (61.3) (18.7) (54.6) Note: AM denotes assumed model; the 5% C.V. in arts (a) and (c) corresond a 2 (2){ distribution while that in art (b) corresonds to a 2 (3){distribution; Figures in arenthesis 19

21 in block (c) corresond to size-adjusted owers; Number of relications = 1000; Comutations erformed using MATLAB. Table 2 Rejection Frequencies (%) (DGB:(20) T = 100, = 1, k = 2) Lag /Power (40.9) (35.6) (33.2) (30.5) (28.2) Note: Numbers without arentheses and those within arentheses denote the relative ower of the modied to the standard Wald test and the emirical ower of the latter, resectively. Table 3 Rejection Frequencies (%) (DGB:(21) T = 100, = 1, = 2) Dimension k/power (40.9) (37.1) (34.3) (29.8) (23.2) Note: See Note in Table 2. 20

22 Table 4 Analytical and Emirical Power (%) (DGP:(22) T=100) Standard Test [V AR(1)] Modied Test [V AR(2)] Exeriment 1 [ = 1; = 1; = 1] Corr h 1 m P P h 1 m P P R Exeriment 2 [ = 0:2; = 1; = 1] Exeriment 3 [ = 1; = 2; = 1] Exeriment 4 [ = 1; = 1; = 0:1] Note: P and P are the analytical and emirical rejection frequencies, resectively; R is the ratio between the emirical owers of the modied and standard tests. 21

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24 Toda, H. and P.C.B. Phillis (1993a) "The Surious Eect of Unit Roots on Vector Autoregressions: An Analytical Study", Journal of Econometrics 59, Toda, H. and P.C.B. Phillis (1993b) "Vector Autoregressions and Causality", Econometrica 61, Toda, H. and T. Yamamoto (1993) "Statistical Inference in Vector Autoregressions with Possibly Integrated Processes" University of Tsubuka (mimeo). 23