Testing for the Cointegrating Rank. of a VAR Process with a Time Trend. Helmut Lutkepohl and Pentti Saikkonen. Spandauer Str. 1 P.O.

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1 October 29, 1997 Testing for the Cointegrating Rank of a VAR Process with a Time Trend by Helmut Lutkepohl and Pentti Saikkonen Institut fur Statistik und Okonometrie Department of Statistics Wirtschaftswissenschaftliche Fakultat Humboldt{University University of Helsinki Spandauer Str. 1 P.O. Box Berlin SF-14 University of Helsinki GERMANY FINLAND Tel.: Fax.: Abstract Standard tests for the cointegrating rank of a vector autoregressive (VAR) process have nonstandard limiting distributions which depend on the characteristics of intercept terms and time trends in the system. In practice these characteristics are often unknown. Therefore modied tests are proposed with limiting distributions which do not depend on the characteristics of deterministic terms under the null hypothesis. One type of tests makes use of lag augmentation, that is, a VAR process of order p + 1 is tted when the true order is p while the tests are based on the coecient matrices of the rst p lags only. It is shown that 2 limiting distributions are obtained in this way. The price for this simplicity will be reduced power, however. Therefore, we also consider LM (Lagrange multiplier) type tests. These tests are shown to have nonstandard limiting distributions which do not depend on deterministic terms and have better local power than competing LR (likelihood ratio) tests in some situations. We thank Katsuto Tanaka for helpful comments on an earlier version of this paper and Martin Schweizer for his advise on It^o integrals. Moreover, we are grateful to Ritva Luukkonen, Kirstin Hubrich and Ralf Bruggemann for performing the computations. The Deutsche Forschungsgemeinschaft, SFB 373, provided nancial support.

2 1 Introduction Determining the number of cointegrating relations of a set of integrated variables is an important part of many empirical studies. A number of tests for cointegration have been proposed in the literature. Some of them are designed for detecting whether there is some cointegrating relation in a given set of variables. These tests are typically based on single equation estimation. Examples are given by Engle & Granger (1987), Engel & Yoo (1987), Phillips & Ouliaris (199), Kremers, Ericsson & Dolado (1992). The disadvantage of this approach is that it does not answer the question of how many cointegrating relations there are. Having an answer to that question may, however, be of central importance for modelling and interpreting the DGP (data generation process) of a given set of variables. Therefore tests for determining the number of cointegrating relations have been proposed. Notable examples are Johansen (1988, 1991, 1994), Reinsel & Ahn (1992), Saikkonen & Luukkonen (1997), Stock & Watson (1988), Gregoir & Laroque (1994), Perron & Campbell (1993), Bewley & Yang (1995) and Bierens (1997). All these tests have nonstandard limiting distributions and therefore special tables with critical values were produced (see, e.g., Osterwald{Lenum (1992)). A major obstacle in using these tests is the fact that the limiting distributions of most of the test statistics depend on characteristics of the DGP which are often unknown in practice. Specically they depend on the trend characteristics of the variables. Some of the tests are not applicable if there are deterministic trends in addition to the stochastic trends in the system. For other tests dierent limiting distributions result in this case. Also the treatment and type of the intercept term is of importance for Johansen's LR (likelihood ratio) tests which are based on VAR (vector autoregressive) processes. Since these characteristics are commonly unknown in practice a consequence is that the crucial characteristics of the DGP are either assumed or pretests are carried out for determining them. The latter approach eectively results in a procedure with uncertain overall properties. Exceptions are, for instance, the LR tests proposed by Perron & Campbell (1993) which allow for a deterministic linear trend and have a limiting distribution which does not depend on the trend parameters. Unfortunately, it was found by Rahbek (1994) and Saikkonen & Lutkepohl (1997) (henceforth S&L) that these tests may have rather poor local power properties. The purpose of this study is to propose tests for the cointegrating rank which are simple 1

3 and still have reasonable power properties. In particular, they should have limiting distributions which do not depend on the presence of deterministic trends and which are independent of the actual properties of intercept terms in VAR processes. Hence, a single table with critical values may be used without making the usual distinctions regarding deterministic trends. We will use two dierent approaches for constructing tests with these desirable properties. The idea underlying the rst approach is to t a VAR(p + 1) process if the true DGP is a VAR(p) and perform the test on the rst p coecient matrices only. Toda & Yamamoto (1995), Dolado & Lutkepohl (1996) and Saikkonen & Lutkepohl (1996) have shown that in such a procedure the rst p coecient matrices have a standard nonsingular asymptotic normal distribution under general conditions. This result will be used in constructing LR type tests for the cointegrating rank which have the usual limiting 2 distributions. A similar idea has been used by Choi (1993) to construct a unit root test in the univariate case. Choi nds that the power of the test is inferior to that of the usual Dickey{Fuller type tests. Hence, it may be expected that the cointegration tests proposed in the following also have less power than, say, Johansen's LR tests. This is a consequence of tting additional parameters in the VAR(p + 1). A possibility to avoid that ineciency is to construct tests suggested by the LM (Lagrange multiplier) principle. It is shown that, under the null hypothesis, the asymptotic distributions of these tests do not depend on the properties of the deterministic trend and intercept terms. In the univariate case, LM type tests for unit roots have been proposed and investigated by Nabeya & Tanaka (199), Schmidt & Phillips (1992) and Ahn (1993). We will show how these tests can be extended to multivariate processes. Since the trend parameters are estimated under the null hypothesis, the tests may be expected to be more powerful than tests which do not use this restriction. It turns out that in many situations the local power is indeed much better than that of the Perron- Campbell LR tests, say. We also report small sample simulation results which show that the small sample power of the LM type tests is sometimes superior to that of competing LR tests. The structure of this paper is as follows. In the next section the underlying model and some of its properties are presented. In Section 3 some well-known LR tests for cointegration are reviewed and it is pointed out why their use may be problematic in practice. In Section 4 cointegration tests based on the idea of lag augmentation are discussed and in Section 5 2

4 LM type tests are considered. A small Monte Carlo comparison of the properties of some of the tests is performed in Section 6 and conclusions follow in Section 7. Most proofs are given in the Appendix. 2 The Model Consider an n-dimensional time series y t = (y 1t ;... ; y nt ) ; t = 1;... ; T; generated by y t = + 1 t + x t ; t = 1; 2... ; (2:1) where (n 1) and 1 (n 1) are unknown parameters which may, of course, be zero and x t is an unobservable error process. Assume that x t follows a pth order VAR process x t = A 1 x t?1 + + A p x t?p + " t ; (2:2) where the A j are (n n) coecient matrices. This process can be written in the error correction (EC) form x t = x t?1 + p?1 X j=1? j x t?j + " t ; t = p + 1; p + 2;... ; (2:3) where =?(I n?a 1??A p ) is (nn); the? j =?(A j+1 + +A p ) (nn) are unknown parameters and is the usual dierencing operator. Here and in the following a sum is zero if the lower bound for the summation index exceeds its upper bound, e.g. if in (2.3) p = 1. For convenience, we impose the initial value condition x t =, t. Other assumptions on the initial values are possible without aecting our main asymptotic results provided the distribution of the initial values is independent of the sample size. We also assume that the error term " t is white noise, that is " t (; ) with positive denite. Later on we will make more explicit assumptions for the process " t. Suppose the components of the process x t are integrated of order one and cointegrated so that we can write = ; (2:4) where and are (n r) matrices of full column rank and r < n. Here r is the cointegrating rank. In order to exclude processes integrated of order two we require that the 3

5 characteristic equation det(i n? A 1 z?? A p z p ) = det I n? p?1 X j=1? j z j! (1? z)? z has exactly n? r roots equal to one and all other roots outside the unit circle. As is well known, x t is then an (asymptotically) stationary process with a zero mean value (see Engle & Granger (1987) and Johansen (1991)). In fact, if we dene = I n?? 1??? p?1 = I n + X p?1 ja j+1 we can conclude from Johansen's (1991) formulation of Granger's representation theorem that x t = C tx i=1 j=1! = " i + t ; t = 1; 2;... ; (2:5) where, apart from the specication of initial values, t is a stationary process and C =? (??)?1?. Here as well as below, if B is an (n m) matrix of full column rank (n > m) we let B? stand for an (n (n? m)) matrix of full column rank and such that B B? =. Note that the present form of the DGP, where the trend is added to the stochastic part, has several advantages for our purposes. Among these advantages is its emphasis on the fact that the trend is at most linear. Starting from the DGP (2.1)/(2.2) implies that y t also has a VAR(p) representation y t = + 1 t + A 1 y t?1 + + A p y t?p + " t ; t = p + 1; p + 2;... ; (2:6) where and = (I n? A 1?? A p ) + px j=1 ja j! 1 =? + (? I n ) 1 (2:7) 1 = (I n? A 1?? A p ) 1 =? 1 : (2:8) Alternatively, (2.6) may be written in error correction form y t = + 1 t + y t?1 + or y t = + ( y t?1? (t? 1)) + p?1 X j=1 p?1 X j=1? j y t?j + " t ; t = p + 1; p + 2;... ; (2:9)? j y t?j + " t ; t = p + 1; p + 2;... ; (2:1) 4

6 where =? + 1 and = 1. In addition to the above general model we shall also discuss its restricted version obtained by assuming that the trend parameter 1 belongs to the column space of the matrix?. In this case 1 =? 1 = : (2:11) Consequently, the deterministic linear trend term 1 t in (2.9) vanishes. In the above model we are interested in testing for a specic cointegrating rank r, that is, a test of the null hypothesis H(r ) : rk() = r vs. H(r ) : rk() > r (2:12) is desired. In the following sections we will discuss dierent classes of suitable tests. 3 Some Previously Considered Tests For a sample y 1 ;... ; y T, LR tests for the pair of hypotheses in (2.12) are available under dierent assumptions regarding the deterministic terms. For instance, if no prior knowledge regarding and 1 is available and thus and 1 are left unrestricted, the LR test may be computed as follows. Dene z t = (1; t) for p = 1 and z t = (1; t; y t;... ; y t?p+1) for p > 1 and M T = (T? p)?1 2 4 T X y t?1 y t?1? y t?1 z t z t z t!?1 3 z t y 5 t?1 (3:1) and let ^ ; ^ 1 ; ^A 1 ;... ; ^A p be the least squares (LS) estimators of the parameters of the model (2.6). Moreover, denote the corresponding LS residuals by ^" t and dene ^ = (T? p)?1 and ^ =?(I n? ^A1?? ^Ap ). Equivalently, ^ and ^" t may be obtained by LS estimation of the EC form (2.9). Denoting by ^ 1 ^ n the ordered generalized eigenvalues obtained as solutions of ^" t^" t det(^m T ^? ^) = ; (3:2) 5

7 the LR statistic for testing the pair of hypotheses (2.12) is given by LR P C (r ) = (T? p) nx j=r +1 log(1 + ^ j ): (3:3) We attach the superscript P C to this statistic because its asymptotic distribution was derived by Perron & Campbell (1993). Its advantage is that it does not depend on the actual values of and 1. In fact, the asymptotic distribution remains even unchanged if the actual parameter vectors are zero. Note also that leaving the parameters and 1 unconstrained means that the model (2.9) can in principle even generate quadratic trends. In the above estimation procedure no restrictions are imposed on and 1. Also the cointegrating rank specied under the null hypothesis is not taken into account in estimating the trend parameters. Critical values for the resulting LR P C test are given in Table 1 of Perron & Campbell (1993). S&L show that the local power of this test may be quite poor relative to other tests which use more knowledge on the deterministic terms. For instance, Johansen (1994) proposes to base the LR test on the EC model (2.1) and thereby imposes the restriction that the deterministic trend is at most linear. The LR statistic is obtained in a way similar to the previous case by dening z = (1; t y;... ; t y ), replacing y t?p+1 t?1 in (3.1) by [y : t? t?1 1] and using the LS estimator ^ + of + := [ :?] in (3.2) instead of ^. We will denote this test statistic by LR + (r ). Its asymptotic distribution diers from that of LR P C (r ). Critical values are given, for instance, in Table 15.4 of Johansen (1995). The local power of LR + is generally better than that of LR P C which is an advantage of the former test (see S&L). Yet another test results if 1 6= and 1 = is assumed. In that case at least some of the variables have linear trends whereas the cointegration relations are free of trend. In this situation the LR statistic is obtained by eliminating the second component (t) from z t. The resulting test statistic will be denoted by LR i (r ). It was proposed by Johansen (1991) and critical values may be found in Johansen (1995, Table 15.3). Again it has an asymptotic distribution which is dierent from the previously considered tests. The local power of this test can even be of a better order than that of LR P C and LR + (see Rahbek (1994)). Unfortunately, it is valid only if 1 6= and 1 =. A dierent asymptotic distribution results if 1 = and, hence, there is no linear trend. For that case dierent critical values are required which may be found in Johansen & Juselius (199, Table A.2). The test statistic used with these critical values will be denoted by LR i (r ). We emphasize 6

8 that it is computed exactly in the same way as LR (r ). In practice it may not be easy to decide which one of the two tests is appropriate in a particular situation. There are a number of asymptotically equivalent variants of these tests. Reinsel & Ahn (1992) and Yap & Reinsel (1995) proposed small sample corrections of the LR statistics. Moreover, Johansen (1991) also considers LR statistics for testing H(r ) : rk() = r vs. H(r + 1) : rk() = r + 1: All these tests suer from similar problems as the LR tests discussed in the foregoing. In summary, these tests leave us the choice between low power or making assumptions which may be dicult to defend in some situations of practical importance. Therefore we will consider alternative tests in the following sections. 4 A Test Based on Lag Augmentation The rst approach uses a very simple idea to obtain a test with standard limiting distribution under the null hypothesis. It turns out that a small modication of the test statistic (3.3) results in a 2 limiting null distribution which does not depend on the deterministic trend terms. It suces to replace z t in (3.1) by z t = 8 < : (1; t; y t?2) for p = 1 (1; t; y t?1;... ; y t?p+1; y t?p?1) for p > 1 ; that is, the vector z t is augmented by y t?1?p and a VAR(p + 1) is estimated instead of the VAR(p). Let us denote the resulting M T matrix by M T. Notice, that we still have ^ =?(I n? ^A 1?...? ^A p ), where the ^A j are now LS estimators of the rst p coecient matrices of a VAR(p + 1) model. Equivalently, ^ may be obtained as LS estimator of the matrix in the model y t = + 1 t + y t?1 +? 1 y t?1 + +? p?1 y t?p+1 + y t?p?1 + " t : The test statistic obtained in this way will be denoted by LR aug (r ). It is a consequence of the following proposition that it has a 2 limiting distribution under the null hypothesis. 7

9 Theorem 4.1 Suppose that = plimm T such that (n n) exists and is nonsingular and let ^ be an estimator of p T vec(^? ) d?! N(;?1 ) (4:1) where LR aug (r ) d?! denotes convergence in distribution. Then, if the true cointegrating rank is r, d?! 2 ((n? r ) 2 ). Proof: The proposition is an immediate consequence of Proposition 1 of Robin & Smith (1994); see also the discussion in their Sec. 4, Cragg & Donald (1997) and Anderson (1951). 2 If ^ is obtained as described above, that is, a VAR(p + 1) is tted and ^ is constructed from the rst p coecient matrices ^A j ; j = 1;... ; p, only, and if the DGP is a VAR(p), then (4.1) holds under quite general conditions for the white noise process " t. The reason for not stating the precise conditions for " t in the proposition lies in the fact that (4.1) is true under alternative sets of assumptions. For instance, it follows from results in Saikkonen & Lutkepohl (1996) that it remains valid for innite VAR processes without deterministic trends when suitable assumptions on the order p + 1 of the tted model are made. It is also possible that the conditions of the theorem can be established under currently unknown assumptions. For the nite order VAR case, a sucient set of assumptions is given in the following corollary. Corollary 4.1 Suppose the true DGP is a VAR(p) process as in (2.6) with cointegrating rank r and white noise process " t = (" 1t ;... ; " nt ) satisfying the following conditions: " t and " s are independent and identically distributed for s 6= t and, for all t, E(" t ) = ; E(" t " t ) = is nonsingular and Ej" jt j 2+ < 1; j = 1;... ; n; for some >. Then LR aug (r ) has an asymptotic 2 ((n?r ) 2 ) distribution. Proof: It follows from Toda & Yamamoto (1995) and Dolado & Lutkepohl (1996) that plim M T exists and (4.1) holds under the conditions of the corollary. Hence, the result follows from Theorem

10 It may be worth noting that the same results hold if 1 = and the VAR process is estimated without a deterministic trend term. In that case the regressor t is simply dropped from z t. Furthermore, the variants of the tests mentioned at the end of Setion 3 can be modied by lag augmentation and will then have asymptotic 2 distributions under the null hypothesis. One would expect that, if the conditions of Corollary 4.1 are satised and the characteristics of the deterministic terms in the process (2.6) are known, the standard LR type tests have better power properties because an ineciency is built into the modied statistics by adding an extra lag in the estimation procedure. Although adding one extra lag may seem like a minor modication, in this case it has rather severe consequences for the properties of the estimators and, hence, for the tests of the cointegrating rank. Consider, for instance, a univariate I(1) process. Then the estimator ^ is superconsistent (^? 1 = O p (T?1 )) if the true AR order is used whereas it is just p T -consistent (^? 1 = O p (T?1=2 )) if an extra lag is added (see Choi (1993)). Hence, a unit root test based on the latter estimator is asymptotically inecient relative to one based on the former estimator. More precisely, the local power of the lag augmentation test of the hypothesis H : = 1 against local alternatives of the form H : = 1? =T will equal the size of the test whereas LR tests without overtting will have larger power against such alternatives (see S&L). In small samples the loss in power may not be quite as severe in multivariate cointegration tests as in univariate unit root tests. Still some loss in power as a price for adding an additional lag has to be expected even in small samples. In the next section we will therefore develop alternative tests which do not require lag augmentation and still have asymptotic null distributions which are invariant to the properties of the deterministic terms. 5 LM Type Tests 5.1 Preliminaries The following derivation of LM type tests for the cointegrating rank is implicitly based on Gaussian assumptions where necessary. For instance, when we talk about maximum likelihood (ML) estimation, the Gaussian likelihood function is assumed. The asymptotic 9

11 distribution of the test statistic obtained in this way holds under more general conditions, however. They are stated explicitly in Theorem 5.1 below. The idea is to take into account all restrictions available under the null hypothesis in estimating the trend parameters. In particular, it is taken into account that the trend can be at most linear and not quadratic. Moreover, the cointegrating rank r is assumed under H(r ). It is hoped that a test which respects the restrictions has more power than tests which do not incorporate all possible restrictions. In Section 5.5 it will be shown that there are indeed local power gains of the LM type tests presented in the following over LR tests. Recalling that = ; where and are (n r), and using that ( )?1 +? (??)?1 = I? n we can write (2.3) as x t = u t?1 + v t?1 + p?1 X j=1? j x t?j + " t ; t = p + 1; p + 2;... (5:1) where u t = x t ; v t = x? t, = ( )?1 and =? (??)?1. If r = r = rk() and thus the null hypothesis in (2.12) holds, we clearly have =. However, under the alternative some columns of? can be identied with cointegrating vectors so that 6=. A further simplication is obtained by premultiplying (5.1) by? and considering? x t = v t?1 + p?1 X j=1? j x t?j + " t ; t = p + 1; p + 2;... (5:2) and testing the null hypothesis that =? =. Therefore, our test procedures for the hypothesis H(r ) are based on testing the restriction = in a feasible version of (5.2). It may be worth noting that LR tests can also be formulated in such a way that a similar transformation as in (5.2) is used for the EC model on which the tests are based (see S&L). The rst step is to trend-adjust the observations y t. For that purpose we estimate the trend parameters and 1 and then consider ~x t = y t? ~? ~ 1 t. As mentioned earlier, the approach for estimating and 1 assumes that r = r as specied in H(r ). It has some similarities to the procedure employed by Saikkonen & Luukkonen (1997) in the special case where (2.1) involves the a priori restriction 1 =. It was shown in that paper that a generalized least squares (GLS) estimator of the level parameter is consistent of order O p (T? 1 2 ) in the direction of cointegrating vectors whereas in other directions the estimator is not consistent but bounded in probability. A similar situation appears in the present context. Suppose the hypothesis H(r ) holds and that the parameters in (2.1) are 1

12 estimated by Johansen's (1994) reduced rank method. Instead of estimating the complete trend parameter 1 we thereby get an estimator of the parameter = 1 only, that is, 1 in the direction of the cointegrating vectors can be estimated. However, the considered model does not contain 1 in the direction of? so that this component of 1 is not identied in this context. Therefore, estimation of the complete trend parameter 1 is problematic. In this situation one may consider estimating and 1 simply by regressing y t on a constant and the time variable t as proposed by Stock & Watson (1988) for computing their test statistics. Such a procedure is inecient here, however, because it ignores not only the short-term dynamics but also the long-term dynamics and the cointegrating rank assumed under the null hypothesis. Moreover, in Stock & Watson (1988) it is seen to have an impact on the asymptotic distribution of the test statistics. Therefore a dierent procedure is used in the following. The rst step of our test procedure consists of using the reduced rank method to estimate the parameters in (2.1). The resulting estimators will be indicated by the symbol \" and, under the null hypothesis H(r ), they are consistent. Based on these estimators we will then obtain suitable estimators of and 1 and use the resulting detrended y t in a feasible version of (5.2) to construct an `LM type test' of the null hypothesis =. We use the term LM type test here primarily because the trend parameters are estimated under the null hypothesis which distinguishes our procedure from the LR tests discussed in Section Estimation of Trend Parameters Our test procedures require estimators of the trend parameters and 1 with appropriate asymptotic properties under the null hypothesis. As already pointed out above, a major problem is to obtain reasonable estimators in the direction of?. Thus, we shall rst discuss the estimation of the parameters =? and =? 1. One possible estimation method is based on the fact that the expectation of y? t is + for t = 1 and for t > 1. Note that here it is assumed that y = and, if needed, we shall later also assume that y t = for t <. Thus, we consider estimating and by LS from the regression model ~?y t = + t + e t ; t = 1;... ; T; (5:3) where 1 = 1 and t = for t > 1. The error term equals e t = x? t + ~ y? t? y? t. 11

13 Of course, since t = for t > 1, is essentially just estimated from the rst observation in (5.3) and therefore it cannot be estimated consistently. On the other hand, although the parameter is also unidentied in the model (2.1), it can be identied as the expectation of y? t and, hence, it can be estimated consistently. The precise properties of the estimators ~ and ~ resulting from (5.3) are given in Lemma 1 in the Appendix. These considerations lead to an estimator of the parameters 1 given by ~ 1 = ( ~ ~ ) ~?1 ~ + ~? ( ~ ~?? )?1 ~ : (5:4) Note that the right hand side is invariant to normalizations of ~ and ~?. Constructing an estimator for is slightly more complicated because, unlike 1, an estimator of in the direction of is not directly obtained from the estimation of the null model (see (2.7)). Denote = and notice that =? + 1 ; where, and can be estimated using (2.1). Thus, using the estimators ~ 1 and ~ = I n? ~? 1?? ~? p?1 we introduce the estimator ~ = (~ ~?1 ~)?1 ~ ~?1 ( ~ ~ 1? ~): (5:5) This estimator is motivated by the analogous estimator of the intercept term discussed in Saikkonen (1992). Similarly to (5.4) we now dene ~ = ( ~ ~ ) ~?1 ~ +? ~ ( ~ ~?? )?1 ~ : (5:6) Thereby we have a rst method to obtain estimators of the trend parameters. As is well known, the LS estimator of (5.3) is asymptotically as ecient as optimal GLS estimation. From an asymptotic point of view the estimators ~ and ~ are thus quite reasonable. However, since they ignore all short-run dynamics of the process y? t it may be preferable to consider alternative estimators which also try to allow for the short-run dynamics. One possibility is to t an autoregressive model to the error term in (5.3) and apply a feasible GLS estimation method. Although in this situation a nite order autoregressive model will only be an approximation, this approach might still have useful nite sample properties. Below we shall consider an alternative approach in which an autoregressive approximation of this kind is not involved. The ideas are somewhat similar to those employed in the GLS estimation of Saikkonen & Luukkonen (1997). 12

14 First note that the levels parameters of the VAR process (2.2) may be obtained as A 1 = I n + +? 1 ; A j =? j?? j?1 ; j = 2;... ; p? 1; A p =?? p?1 : The corresponding sample analogs A ~ 1 ;... ; A ~ p are dened similarly by using the estimators ~; ~ and? ~ j (j = 1;... ; p? 1). Hence, the A ~ j satisfy the restrictions implied by the cointegrating rank. To determine the estimators ~ and ~ 1 proposed in the following, consider the trend adjusted series ~x t = y t? ~? ~ 1 t for t 1 and ~x t = for t and dene ~" t = (I n? A ~ 1 L?? Ap ~ L p )~x t = A(L)~xt ~ ; where L is the lag operator and A(L) ~ is dened in the obvious way. Now consider the identity ~" t = ~ A(L)(~xt? x t ) + ( ~ A(L)? A(L))xt + " t = ~ G t (? ~ ) + ~ H t ( 1? ~ 1 ) + e aux t (5:7) where e aux t = ( ~ A(L)? A(L))x t + " t ; ~ Gt = ~ A(L)a t and ~ H t = ~ A(L)b t with a t = 8 < : 1 for t 1 for t ; b t = 8 < : t for t 1 for t One might estimate the dierences? ~ and 1? ~ 1 from the auxiliary regression model in (5.7) by appropriate feasible GLS and use the resulting estimators to correct the estimators ~ and ~ 1. However, since we wish to estimate the parameters and 1 in the direction of? we shall apply this approach after a further transformation given by ~k t = ~ K 1t ' + ~ K 2t ' 1 + e (1) t ; t = 1;... ; T; (5:8) where ~ k t = ~? ~" t, ~ K1t = ~? ~ G t ~? ( ~? ~? )?1, ~ K2t = ~? ~ H t ~? ( ~? ~? )?1, ' = ~? (? ~ ) and ' 1 = ~? ( 1? ~ 1 ). Note that it follows from the denitions that ~ K1t = and ~ K2t = ~? ~ ~? ( ~? ~? )?1 for t p + 1. The denition of the error term e (1) t Appendix where it is seen that we approximately have e (1) t : can be found in the ~? " t. Thus, it seems reasonable to estimate the parameter vector ' = [' ; ' 1] in (5.8) by the GLS estimator ~' = 2 4 ~' ~' = ~K t (~? ~ ~? )?1 ~ Kt!?1 T X 13 ~K t (~? ~ ~? )?1 ~ kt

15 where ~ K t = [ ~ K 1t ; ~ K 2t ]. Then new estimators of and can nally be dened by and ~ (1) = ~ + ~' ~ (1) = ~ + ~' 1 : The estimators ~ (1) and ~ (1) can be thought of as two-step estimators obtained by correcting the initial estimators ~ and ~. Asymptotically, this correction has no eect (see Lemma 2 of the Appendix). This result is quite reasonable because, as already pointed out, the asymptotic properties of the estimators ~ and ~ cannot be improved by taking the covariance structure of the error term e t in (5.3) into account. In nite samples the situation may be dierent, however, and the fact that the error term e (1) t in (5.8) is approximately white noise may make the two-step estimator preferable to ~ and ~. Of course, it is also possible to use the estimators ~ (1) and ~ (1) as new initial estimators and obtain similar asymptotically equivalent three-step estimators or even higher order iterates. In what follows we shall use the notation ~ and ~ to signify any of these asymptotically equivalent estimators. The same notational convention applies to the estimators ~ and ~ 1 dened in (5.6) and (5.4). We close this subsection by noting that the above estimators can be straightforwardly modied to the case where it is assumed that 1 = and, hence, no deterministic trend term appears in the model (2.6), i.e. 1 =. Then we have = and in the above formulas we simply dene ~ =. It is easy to check that the derivations used in the proofs in the Appendix remain valid under this assumption. 5.3 Test Statistics As discussed in Section 5.1, a reasonable approach for testing the hypothesis H(r ) is to test the constraint = in a feasible version of the regression model (5.2). Hence, we consider ~?~x t = ~v t?1 + p?1 X j=1? j ~x t?j + t ; t = p + 1;... ; T; (5:9) where ~x t = y t? ~? ~ 1 t with ~ and ~ 1 being any of the estimators considered in the previous subsection. Conventional test statistics for the hypothesis = may be based on an LS estimation of (5.9). For instance, the resulting LM statistic is o LM(r ) = tr n^ MvvX ~ ^ (~? ~ ~? )?1 ; (5:1) 14

16 where ^ is the LS estimator of from (5.9), ~? ~ ~? ~ is the residual covariance estimator of the error term in (5.9) and with ~M vvx = 2 4 T X ~v t?1 ~v t?1? ~v t?1 ~ X t?1 ~ X t?1 = ~x t?1. ~x t?p+1 ~ X t?1 ~ X t? :!?1 An asymptotically equivalent test statistic results by estimating from ~? ~x t = ~u t?1 + ~v t?1 + p?1 X j=1 ~ X t?1 ~v t?1 3 5 (5:11)? j ~x t?j + ~ t ; t = p + 1; p + 2;... ; (5:12) where ~u t = ~ ~x t and = is estimated unrestrictedly together with the other parameters. Denoting the corresponding estimator of by ~ gives a test statistic Here with ~M vvx = 2 4 T X ~v t?1 ~v t?1? LM (r ) = tr n ~ ~ M vvx ~ (~? ~ ~? )?1 o : (5:13) ~v t?1 ~ X t?1 ~ X t?1 = ~u t?1 ~x t?1. ~x t?p+1 ~ X t?1 ~ X t? :!?1 ~ X t?1~v t?1 3 5 (5:14) The following theorem gives the limiting distribution of the test statistics under the null hypothesis. We use the symbol B(s) to signify an (n? r )-dimensional standard Brownian motion and B (s) = B(s)? sb(1) ( s 1) denotes an (n? r )-dimensional standard Brownian bridge. Theorem 5.1 Suppose that the " t in (2.2) are a martingale dierence sequence with E(" t j" t?1 ;... ; " 1 ) =, 15

17 E(" t " t j" t?1 ;... ; " 1 ) = is positive denite and the fourth moments are bounded. Then, under the null hypothesis, LM(r ); LM (r ) d! tr (Z 1 Z 1 B (s)db (s) B (s)b (s) ds?1 Z 1 ) B (s)db (s) where db (s) = db(s)? dsb(1) and an integral with respect to the semimartingale B (s) is understood to be an It^o integral. 2 Note that the integral R 1 B (s)db (s) may be interpreted as a short-hand notation for R 1 B(s)dB(s)? B(1) R 1 sdb(s)? R 1 B(s)dsB(1) B(1)B(1). The proof of the theorem is given in the Appendix. Notice that the limiting distribution of the test statistics is free of unknown nuisance parameters. This means in particular that it does not depend on the actual trending properties of the process and contrasts with some of the LR tests. Note that we have not assumed that and 1 are actually nonzero. The percentiles of the asymptotic distributions can be readily found by simulation. We will provide the details in Subsection 5.4. In the special case where n? r = 1 the limiting distribution simplies, as shown in Schmidt & Phillips (1992), because in that case the stochastic integral in Theorem 5.1 equals? 1. The following corollary deals with this special case. 2 Corollary 5.1 If n? r = 1 then, under the null hypothesis, LM(r ); LM (r ) d! 4 Z 1 B (s) 2 ds?1 : 2 The limiting distribution in Corollary 5.1 was previously obtained by Ahn (1993, Theorem 1) who developed LM tests for testing the univariate autoregressive unit root hypothesis in the presence of a time trend. He also simulated asymptotic critical values of the test statistic in this special case. Now consider the case where the restriction (2.11) is assumed so that 1 = and a deterministic trend term does not appear in (2.6). Then ~ = and the estimator ~ 1 is dened accordingly. It can be seen from the proof of Theorem 5.1 that in this case the same proof applies and we get the same limiting distribution of the resulting test statistics in this 16

18 case. However, we do not persue this testing strategy under the assumption = because in that case the LR i test is available which has potentially better local power properties since it is designed especially for this situation. One might also augment the auxiliary regression equations (5.9) and (5.12) by including an intercept term. In the related univariate case this approach has been used by Schmidt & Phillips (1992) and Ahn (1993). In the same way as in these special cases, the limiting distribution of the test statistics will then change and involve a demeaned Brownian bridge. 5.4 Percentiles of the Asymptotic Distributions In Table 1 simulated percentage points of the asymptotic distribution given in Theorem 5.1 are presented for dierent values of d = n? r. These percentage points are simulated in the following manner. We have generated T = 1 d-dimensional independent standard normal variates t N(; I d ) and have computed the quantities and A T = 1 T 2 B T = 1 T " t?1 X k=1 " t?1 ( k? ) X k=1 ( k? ) #" t?1 X # k=1 ( k? )# (5:15) ( t? ) ; (5:16) where = T?1 P T t. The quantity in (5:15) converges weakly to R 1 B (s)b (s) ds whereas R 1 (5:16) converges to B (s)db (s) : Hence, the quantity tr(b T A?1B T T) has the desired asymptotic distribution. The percentiles in Table 1 are obtained from 2 replications of this experiment. Independent drawings are used for each dimension d. 5.5 Local Power Analysis Since we found in Section 4 that the lag augmentation tests have power against local alternatives of order O(T?1=2 ) and they are therefore inferior to standard LR tests in this respect, it is of interest to also derive the local power of the LM type tests to check their performance relative to the LR tests. We will show that they indeed have quite favorable local power compared to the LR tests LR + and LR P C. We consider local alternatives of the form H T (r ) : = + T?1 1 1; (5:17) 17

19 Table 1. Percentage Points of the Distribution of tr Z 1 Z 1 B (s)db (s) B (s)b (s) ds?1 Z 1 B (s)db (s) where B (s) is a d-dimensional Brownian Bridge. Dimension d % Percentage point 95% % where and are xed (n r ) matrices of rank r and 1 and 1 are xed (n (r? r )) matrices of rank r? r and such that the matrices [ : 1 ] and [ : 1 ] have full column rank r. In the following we use the assumption from Johansen (1995) and Rahbek (1994) that the VAR order is p = 1 and the eigenvalues of the matrix I r + are less than 1 in modulus. The power of our LM type tests against alternatives of the form (5.17) may be obtained from a general result given in S&L. To apply that result we consider the reduced rank (RR) regression model ~x t = ~x t?1 + ~e t ; t = 1;... ; T; (5:18) where ~x t = y t? ~? ~ 1 t, ~e t = " t? (~? ) + (~ 1? 1 )(t? 1) + T?1 1 1 (~? ) + T?1 1 1(~ 1? 1 )(t? 1)? (~ 1? 1 ) + T?1 1 1~x t?1 def = r t + T?1 1 1~x t?1 and ~ and ~ 1 are any of the estimators from Section 5.2. Choosing Y t = ~x t, X t = ~x t?1, E t = ~e t, A = and B =, it is shown in the Appendix that Assumption 1 of S&L holds for the model (5.18). Hence, we may obtain the asymptotic distribution of our LM type statistic from Theorem 1 of S&L. It can be shown that for this particular case the limiting distribution reduces to the form given in the next theorem. Theorem 5.2 Let W(u) denote a Brownian motion with covariance matrix and K(t) the Ornstein- Uhlenbeck process dened by the integral equation Z u K(u) =?W(u) +? 1 1?(??)?1 K(s)ds ( u 1) (5:19) 18

20 Furthermore, dene N(s) = (?? ) K(s), N (s) = N(s)? sn(1) and dn (s) = dn(s)? dsn(1). With this notation, if = + T?1 1 1 in (2.3), LM(r ) d! tr (Z 1 Z 1 N (s)dn (s) N (s)n (s) ds?1 Z 1 ) N (s)dn (s) : (5:2) 2 This theorem is also proven in the Appendix. Obviously, the local power depends on n? r only and not on the dimension n and r separately. Moreover, it does not depend on the actual values of the mean and trend parameters and 1 as in the case of the LR tests studied in S&L. Furthermore, it is straightforward to check that Z s N(s) = B(s) + ab K(u)du; (5:21) where B(s) is again an (n? r )-dimensional standard Brownian motion and the quantities a and b are given by a = (??)?1=2? 1 and b = (??) 1=2 (??)?1? 1 (5:22) (cf. Johansen (1995, pp )). Hence, the limiting distribution depends on,,, 1 and 1 only through a and b. This implies, for instance, for the case r? r = 1, where 1 and 1 are just vectors, that the limiting distribution only depends on the two parameters f = b a and g 2 = a ab b? (a b) 2 (5:23) (see Johansen (1995, Corollary 14.5)). This fact will be used later. Since the asymptotic distribution of the LM type test given in Theorem 5.2 diers from those of the LR tests for the cointegrating rank given in Corollary 1 of S&L we have simulated the resulting local power in the same way as in S&L. In other words, we consider the case where 1 and 1 are (n 1) vectors and simulate the discrete time counterpart of the (n? r )-dimensional Ornstein-Uhlenbeck process N(s) as with t iid N(; I n?r ), N =, N t = 1 T 1 1N t?1 + t ; t = 1;... ; T = 1; 1 = 8 >< >: 1 for n? r = 1 (1; ) for n? r = 2 (1; ; ) for n? r = 3 19

21 and 1 = 8 >< >: From these generated N t we compute and A T = 1 T 2 B T = 1 T " t?1 X k=1 " t?1 f for n? r = 1 (f; g) for n? r = 2 (f; g; ) for n? r = 3 (N k? N) X k=1 (N k? N) #" t?1 X # k=1 : (N k? N)# (5:24) (N t? N) ; (5:25) where N = T?1 P T N t. The quantity in (5:24) converges weakly to R 1 N (s)n (s) ds R 1 and (5:25) converges to N (s)dn (s) : Hence, the quantity tr(b T A?1B T T) has the desired asymptotic distribution. The resulting rejection frequencies for dierent values of f and g are plotted in Figures 1, 2 and 3 together with the corresponding rejection frequencies of the competing LR tests which allow for a deterministic trend in the DGP. The rejection frequencies of the LR tests are taken from S&L. In fact, we have used the same random numbers in generating the N t values. It is seen that the LM test has generally more local power than LR P C. The power gains can be substantial. For instance, if n? r = 1 and f =?15, LR P C rejects in 36% of the replications whereas the rejection frequency for LM is 57%. In many situations LM also outperforms LR +. Especially in Figure 1 it is seen that the local power of LM can be considerably greater than that of LR +. In other parts of the parameter space LR + outperforms LM, however. In summary, the LM type tests can be superior to LR tests when a linear trend is present in the DGP. It should be understood, however, that local power properties are informative about the performance of the tests in large samples when alternatives close to the null hypothesis are of interest. In small samples the situation may be dierent. Therefore we present some small sample simulations in the following section where also comparisons between the LM and lag augmentation tests are made. 2

22 6 Small Sample Properties of the Tests A limited Monte Carlo experiment was performed to study the small sample properties of our tests and to compare them to other tests for the cointegrating rank. A two-dimensional VAR(1) DGP from Toda (1994, 1995) of the form y t = is used to compare the tests. 3 5 yt?1 + " t ; " t iidn ; A (6:1) Toda shows that this type of process may be regarded as a \canonical form" for investigating the properties of LR tests for the cointegrating rank. Other VAR(1) processes of interest in practice can be obtained by linear transformations of y t which leave the tests invariant. If 1 = 2 = 1 the cointegrating rank is r = and the process consists of two nonstationary components. In that case the second component has a deterministic linear trend if 6= and there is no linear trend if =. For = the two components are independent whereas they are instantaneously correlated for 6=. If 2 = 1 and j 1 j < 1 the cointegrating rank is r = 1. Again there will be a linear trend in this case if 6=. If both 1 and 2 are less than 1 in absolute value the process is stationary (r = 2). In that case a nonzero cannot generate a linear trend. It will therefore be set to zero for stationary processes. We have generated samples of sizes T = 1 and 2 plus 5 presample values starting with an initial value of zero. The last presample values are used for estimation purposes. Selected rejection frequencies of the tests LR i, LR i, LR +, LR P C, LR aug, LM and LM are given in Tables 2-5. We have included the former two tests although their underlying assumptions are violated for some of the DGPs. For instance, if r = or 1 and 6=, LR i is not applicable. Because in practice the trend characteristics of the DGP may be unknown it is of interest to understand the implications of an incorrect application of the tests when their underlying assumptions are violated. Therefore we have included LR i and LR i although they are strictly speaking no serious competitors for some of the DGPs. For the LM type tests the trend parameters are estimated as ~ (1) = ( ~ ~ ~ )?1~ +? ~ ( ~ ~?? )?1 ~ (1) and ~ (1) 1 = ~ ( ~ ~ )?1 ~ + ~? ( ~? ~? )?1 ~ (1) (see Section 5.2). The rejection frequencies in the tables are based on asymptotic critical values for a test level of 5%. We have not computed rejection frequencies corrected for the actual small sample sizes because these will also not be available in practice. Hence, a test is useful for 21

23 applied work only if it respects roughly the nominal signicance level. Otherwise the Type I error cannot be controlled in practice and a test which does not control this error is of limited value even if it has good power properties. The results for all test statistics are based on the same generated time series. Hence the entries for a given DGP in a single column of the tables are not independent but can be compared directly. Still it may be worth recalling that for a true rejection probability p the standard error of an estimator based on 1 replications of the experiment is s p = p p(1? p)=1. For example, s:5 = :7, s :5 = :16 and s :95 = :7. Note also that the tests are not performed sequencially. Thus, the test results for r = 1 are not conditioned on the outcome of the test of r =. In Table 2 results for processes with true cointegrating rank r = ( 1 = 2 = 1) are presented. It is seen that LR P C and LR + reject the true null hypothesis slightly too often but still have roughly the correct size. This contrasts with both versions of the LM tests which are somewhat conservative in this case and thus their size is bounded by the desired 5%. The LR i test which is based on the assumption that the process does not have a deterministic trend also overrejects a bit when = and, hence, its assumptions are satised. On the other hand, if = 1 and, thus, there is a linear trend in the DGP, LR i becomes conservative. The tests LR aug and LR i have severe problems in some cases. LR aug has a totally unacceptable small sample rejection rate far in excess of the nominal 5% when r = is tested. Thus, these simulations show that the test clearly has a small sample problem even for samples of the size T = 2. This, of course, is a further reason for not recommending it for applied work despite its simple and, hence, attractive foundations. In fairness it should perhaps be noted that we are presently considering a particularly unfavorable situation for the lag augmentation test because the process is far from beeing stationary and has a small order. In this situation, adding an extra lag is a rather severe modication which doubles the number of autoregressive coecients to be estimated. The LR i test also overrejects H() in an unacceptable manner if = and, hence, there is no deterministic linear trend. Of course, the test is not constructed for this situation and it is therefore used incorrectly here. The problem is, however, that a user may not know whether the conditions for LR i are fullled. As a result, the test should be avoided if no denitive knowledge is available that there is a deterministic trend. 22

24 In the previous sections we have not derived asymptotic results for testing r = 1 when the true rank is r =. Therefore we cannot speculate on what to expect in the present situation when testing H(1). For the present processes most tests turn out to be conservative in this case. Therefore, nding too large a cointegrating rank is generally not very likely. An exception occurs for LR i if this test is used incorrectly for a DGP whithout a linear trend ( = ). Of course, in practice one would not even perform a test of H(1) if H() is not rejected. In Table 3 results are given for a DGP with cointegrating rank r = 1, sample size T = 1 and no error correlation ( = ). Now the rejection frequencies for r = indicate the power of the tests. Conservative tests may be expected to have reduced power. Therefore it is not surprising that the LM tests are less powerful than LR P C, LR + and LR i in this situation. Since the latter test was seen to have a severe size distortion for processes without trend, it is not a real competitor here unless it is known with certainty that a linear trend is present. If a deterministic linear trend is present, LR i has the lowest power of all tests for 1 values close to one (see 1 = :95; :9). Hence, when uncertainty regarding the trend exists, it is not a good idea to use either LR i or LR i. Due to the extreme size distortion for a DGP with cointegrating rank r =, the LR aug test cannot be regarded as a serious candidate here either although it has the best power of all the tests. Clearly this is just a reection of its substantial overrejection for the DGP with r =. The columns in Table 3 headed r = 1 contain the empirical sizes of the tests when the correct null hypothesis is tested. Now it is seen that LR P C and LR + are a bit more conservative than the LM tests. The performance of the two LM tests is quite similar here. Analogous results are obtained when a residual correlation of = :8 is assumed. This can be seen in Table 4. In this case the two components of the process are no longer independent. There are some notable dierences to the case without correlation between the stationary and nonstationary components. First, it is seen that for r = 1, LM is a bit more conservative than LM. Second, under H(1), LR + rejects more often than LR P C. In fact, the size of the two tests is not very stable for this DGP. For 1 close to one, both tests are conservative whereas they reject in more than the nominal 5% of the cases for 1 = :8 and.7. This contrasts with the LM tests which are conservative in all cases. The other tests are again no serious competitors because they fail in some situations. We have also repeated the 23

25 simulations for the DGPs with r = 1 for sample size T = 2 and found similar results. In that case the sizes of the LR + and LR P C tests become more stable for varying 1, however. In Table 5 the results for DGPs with cointegrating rank r = 2 ( 2 = :5 and 1 varying) are given. Now the processes are stationary and hence an intercept term does not generate a deterministic trend. Therefore = is used throughout. The LM tests are now clearly more powerful than LR + and LR P C for testing r = 1 when 1 = :95, that is, if the alternative is close to the null. This may be a reection of the superior local power of the tests. Since there is no deterministic trend now, it is not surprising that LR i does quite well because this test is designed for this situation. The fact that LR i has the highest power has to be discounted by the fact that its actual size was found to be much larger than the nominal 5% for processes without linear trend. Hence, this test should not be used for the presently considered DGPs. Although we already found that the lag augmentation test cannot be recommended, it may be interesting to note its overall performance. For stationary processes or processes close to stationarity with cointegrating rank r = n? 1 its performance seems to be better than for the processes in Table 2 where r = (see Tables 3-5). For the stationary processes in Table 5 the power is in fact quite impressive for T = 1 compared to the power of the other tests. However, for T = 2 the power becomes inferior again. Such a behaviour was to be expected given that the LR aug test has power against local alternatives of a smaller rate than the other tests. The overall conclusion from these simulations is that LR aug can have quite severe size problems in samples of realistic size. Therefore the test cannot be recommended for use in applied work. On the other hand, the LM tests tend to be conservative and thus their size can at least be bounded by the signicance level. The conservativeness generally has a negative impact on the small sample power of the tests. Overall LM is slightly superior to LM in some situations in terms of size and power. If it is not clear whether there is a linear trend in the DGP the only serious competitors for the LM tests are LR + and LR P C. For sample size T = 1 the variation in the sizes of these tests is quite substantial. Sometimes they are conservative and in other situations they reject too frequently. In our simulations, the rejection frequency is never much larger than the nominal signicance level under the null hypothesis, however. In parts of the parameter space the LM tests have better power 24