Maximum eigenvalue versus trace tests for the cointegrating rank of a VAR process

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1 Econometrics Journa (2001), voume 4, pp Maximum eigenvaue versus trace tests for the cointegrating rank of a VAR process HELMUT LÜTKEPOHL, PENTTI SAIKKONEN, AND CARSTEN TRENKLER Institut für Statistik und Ökonometrie, Humbodt-Universität zu Berin, Spandauer Str. 1, D Berin, Germany E-mai: uetke@wiwi.hu-berin.de; trenker@wiwi.hu-berin.de Department of Statistics, P.O. Box 54, FIN University of Hesinki, Finand E-mai: pentti.saikkonen@hesinki.fi Received: February 2001 Summary The properties of a range of maximum eigenvaue and trace tests for the cointegrating rank of a vector autoregressive process are compared. The tests are a ikeihoodratio-type tests and operate under different assumptions regarding the deterministic part of the data generation process. The asymptotic distributions under oca aternatives are given and the oca power is derived. It is found that the oca power of corresponding maximum eigenvaue and trace tests is very simiar. A Monte Caro comparison shows, however, that there may be differences in sma sampes. The trace tests tend to have more distorted sizes whereas their power is in some situations superior to that of the maximum eigenvaue tests. Keywords: Cointegration, Loca power anaysis, Vector autoregressive process. 1. INTRODUCTION In empirica studies of systems of economic time series, the number of cointegrating reations is often of major interest because it affects the mode setup and inference procedures at other stages of the anaysis. Therefore, the cointegrating rank of a system is usuay investigated at an eary stage. If a vector autoregressive (VAR) mode is an adequate description of the data generation process (DGP), ikeihood ratio (LR) type tests as proposed by Johansen (1988, 1995) are the most commony used inference toos in this context. Two variants of these tests are avaiabe, the so-caed maximum eigenvaue tests and the trace tests. Both types of test are frequenty appied in empirica studies. Given the ong-time coexistence of the two tests it is surprising that ony itte is known about the reative performance of the two types of LR tests. Toda (1994) reports on a Monte Caro experiment comparing the sma-sampe properties of the two types of test and finds that neither of the tests is uniformy superior but that the trace tests perform better in some situations where the power is ow. His simuation study is imited in a number of respects, however. First, he considers bivariate DGPs ony. For these processes, maximum eigenvaue and trace tests differ ony when the nu hypothesis states that the cointegrating rank is zero, which is ceary a specia situation. Second, Toda considers ony tests which aow for a deterministic inear trend in the DGP. The performance of tests which aow not for a inear trend but just for a mean term is known to be quite different. Further work. Pubished by Backwe Pubishers Ltd, 108 Cowey Road, Oxford OX4 1JF, UK and 350 Main Street, Maden, MA, 02148, USA.

2 288 Hemut Lütkepoh et a. on the powers of the maximum eigenvaue and trace tests is reported by Johansen (1995), Hansen and Johansen (1998) and Paruoo (2001). These authors consider aso the asymptotic oca power of the maximum eigenvaue tests for the case when no deterministic term is incuded. From the point of view of appied work, this case is ess important than the cases which aow for a deterministic mean term or a inear trend. The oca power of the maximum eigenvaue test for the specia case where a cointegrating rank of zero is tested against rank one is aso treated by Pesavento (2000). She compares the oca power and sma-sampe performance of the maximum eigenvaue test to other tests for a singe cointegration reation and aows for the possibiity of deterministic mean and trend terms. In this study we perform a oca power and sma-sampe comparison between a more compete set of maximum eigenvaue and trace tests. In particuar, we compare tests based on different assumptions regarding the deterministic part: that is, we consider tests for DGPs with deterministic mean and inear trend terms. Different treatments of the mean and deterministic trend term which have been proposed in the iterature are aso considered. More precisey, a number of LR-type tests reviewed in Hubrich et a. (2001) wi be incuded in the comparison. Moreover, we compare the oca power of the different tests for more genera situations than previousy. In particuar, tests for a cointegrating rank arger than zero with severa cointegration reations are examined. Finay, our sma-sampe Monte Caro study takes into account higher-dimensiona processes as we. The study is structured as foows. In the next section the mode framework and the hypotheses of interest are discussed. The tests and their asymptotic properties are presented in Section 3. A oca power comparison is performed in Section 4 and the resuts of a sma-sampe simuation study are presented in Section 5. Concusions are drawn in Section 6. Some mathematica derivations are deferred to the Appendix. The foowing terminoogy and notation is used throughout. The differencing operator is denoted by : that is, for a time series or stochastic process y t we have y t = y t y t 1. The symbo I (d) denotes an integrated process of order d: that is, the purey stochastic part of the process is stationary or asymptoticay stationary after differencing d times whie it is sti nonstationary after differencing just d 1 times. Convergence in distribution or weak convergence is signified by d. The trace, the rank and the maxima eigenvaue of the matrix A are denoted by tr(a), rk(a) and λ max (A), respectivey. If A is an (n m) matrix of fu-coumn rank (n > m), we denote an orthogona compement by A so that A is an (n (n m)) matrix of fu-coumn rank and such that A A = 0. The orthogona compement of a nonsinguar square matrix is zero and the orthogona compement of a zero matrix is an identity matrix of suitabe dimension. An (n n) identity matrix is denoted by I n. GLS is used to abbreviate generaized east squares, RR regression stands for reduced rank regression and r.h.s. is short for right-hand side. VECM abbreviates vector error correction mode. A sum is defined to be zero if the ower bound of the summation index exceeds its upper bound. 2. MODELS AND HYPOTHESES OF INTEREST 2.1. The modes An observabe n-dimensiona time series y t = (y 1t,..., y nt ) (t = 1,..., T ) is considered which is generated by the DGP y t = µ 0 + µ 1 t + x t, t = 1, 2,..., (1)

3 Maximum eigenvaue vs. trace tests 289 where µ 0 and µ 1 are the (n 1) parameter vectors of the deterministic term. The first one, µ 0, wi be referred to as the mean term and µ 1 is caed the trend parameter. The process x t represents the stochastic part which has mean zero and is unobservabe uness µ 0 and µ 1 are known. Thus, the deterministic and stochastic parts are simpy added in (1). It is assumed that the components of x t are at most I (1) and do not have roots on the unit circe other than 1. This impies that the same is true for y t. Of course, there may be cointegration among the variabes of x t and, hence, of y t. Moreover, x t is assumed to be generated by a VAR process. For convenience, we write the process in VECM form, p 1 x t = x t 1 + Ɣ j x t j + ε t, t = 1, 2,..., (2) j=1 where and the Ɣ j are (n n) matrices of unknown parameters. Assuming that rk( ) = r, the matrix can be written as a product = αβ, where α and β are (n r) matrices. Our requirement that a variabes are at most I (1) impies that α (I n p 1 j=1 Ɣ j)β has to be nonsinguar (see Johansen (1995, p. 49)). The rank r is the cointegrating rank of x t and, hence, of y t. It is the quantity of interest in what foows. The error process ε t i.i.d.(0, ) is independenty, identicay distributed white noise with zero mean and nonsinguar covariance matrix. Moreover, in setting up LR-type tests it is assumed that the ε t are Gaussian, that is ε t i.i.d.n(0, ) and, hence, x t and y t are aso Gaussian processes. The sum on the r.h.s. of (2) represents the short-term dynamics of the process. It disappears if p = 1. The initia vaues x t (t = p + 1,..., 0) are assumed to be zero for convenience. The asymptotic anaysis remains vaid if they are assumed to be any random variabes which do not depend on the sampe size. Defining Y t p t 1 = (y t 1,..., y t p ), it is easy to see that the process y t aso has a VECM representation which can be written as y t = ν 0 + ν 1 t + y t 1 + Ɣ Y t p+1 t 1 + u t = ν + + y + t p+1 t 1 + Ɣ Yt 1 + u t, t = p + 1, p + 2,..., (3) where ν 0 = µ 0 + (I n + p 1 j=1 Ɣ j)µ 1, ν 1 = µ 1, ν = ν 0 + ν 1, + = [ν 1 : ], y + t 1 = [t 1 : y t 1 ], Ɣ = [Ɣ 1 : : Ɣ p 1 ] and Y t p+1 t 1 = Y t p+1 t 1 Y t p t 2. Depending on the assumptions for µ 0 and µ 1, different restrictions wi be imposed on ν, ν 0 and ν 1. On the other hand, the parameter µ 0 cannot in genera be recovered uniquey from (3) due to the singuarity of. A brief discussion of the different cases wi be given in Section 3, where the cointegration tests are presented. The hypotheses of interest are considered beow Hypotheses of interest It is we known (see, for exampe, Johansen (1995)) that the number of ineary independent cointegrating reations of y t is equa to the rank of which wi be denoted by r in what foows. This number is the quantity of interest in the tests considered in this study. We discuss tests designed for checking the pairs of hypotheses H(r 0 ) : rk( ) = r 0 vs. H(r 0 ) : rk( ) > r 0 (4)

4 290 Hemut Lütkepoh et a. Determ. Tabe 1. Modes and LR-type tests. terms Mode statistic Reference µ 0 = µ 1 = 0 y t = y t 1 + Ɣ Y t p+1 t 1 + u t L R 0 Johansen (1988, 1995) µ 0 arbitrary y t = ν 0 + y t 1 + Ɣ Y t p+1 t 1 + u t L R i0 Johansen (1991) [ ] µ 1 = 0 y t = 1 + Ɣ Y t p+1 t 1 + u t L R Johansen and Juseius (1990) y t 1 y t = (y t 1 µ 0 ) + Ɣ Y t p+1 t 1 + u t L R SL Saikkonen and Luukkonen (1997) Test Saikkonen and Lütkepoh (2000a) [ ] t 1 µ 0, µ 1 y t = ν Ɣ Y t p+1 t 1 + u t y t 1 L R + Johansen (1992, 1994, 1995) arbitrary y t = ν 0 + ν 1 t + y t 1 + Ɣ Y t p+1 t 1 + u t L R PC Perron and Campbe (1993) y t µ 1 = (y t 1 µ 0 µ 1 (t 1)) L R GLS Saikkonen and Lütkepoh (2000a) + p 1 j=1 Ɣ j ( y t j µ 1 ) + u t Lütkepoh and Saikkonen (2000) and H(r 0 ) : rk( ) = r 0 vs. H(r 0 + 1) : rk( ) = r (5) Tests of the former pair of hypotheses are often referred to as trace tests and those for the atter pair are known as maximum eigenvaue tests. We are aso interested in the power of the tests. If H(r 0 ) is true, the matrix in the VECM form (3) can be written as a product = αβ, where α and β are (n r 0 ) matrices of rank r 0. Thus, we may write the nu hypothesis as H(r 0 ) : = αβ with α, β (n r 0 ) and rk(α) = rk(β) = r 0. Furthermore, if rk( ) = r > r 0, there exist (n r) matrices [α : α 1 ] and [β : β 1 ] of rank r, such that [ ] β = [α : α 1 ] β 1 = αβ + α 1 β 1. (6) In the next section, tests for different assumptions regarding the deterministic terms wi be discussed briefy. 3. THE TESTS In this section we wi consider LR-type tests for the hypotheses in (4), (5) under aternative assumptions for the deterministic mean and trend terms. An overview of the assumptions for µ 0 and µ 1 and the associated modes and tests is given in Tabe 1, which is adopted from Hubrich et a. (2001). Under Gaussian assumptions, the LR-type statistics reated to the modes in Tabe 1 may be obtained by concentrating out the short-run dynamics and performing a RR regression. The LR statistics of interest can then be obtained by soving a suitabe eigenvaue probem. For exampe, for the case where no deterministic terms appear in the mode (µ 0 = µ 1 = 0) the LR statistics can be obtained as foows. Denote the residuas from regressing y t and y t 1 on Y t p+1 t 1 by R 0t and R 1t, respectivey, and define S i j = T 1 T t=1 R it R jt (i, j = 0, 1).

5 Maximum eigenvaue vs. trace tests 291 Moreover, denote the ordered (generaized) eigenvaues from soving det(λs 11 S 10 S 1 00 S 01) = 0 by λ 1 λ n. Then the trace statistic for testing the pair of hypotheses in (4) can be shown to be n L Rtrace 0 (r 0) = T og(1 λ j ) (7) j=r 0 +1 and the maximum eigenvaue statistic for testing (5) is L R 0 max (r 0) = T og(1 λ r0 +1) (8) (see Johansen (1995, Ch. 6)). If µ 1 = 0 and µ 0 is unrestricted, a DGP with possiby nonzero mean term is considered whereas a deterministic inear trend term is excuded by assumption. In this case the VECM form of the mode for y t reduces to y t = yt 1 t p+1 + Ɣ Yt 1 + u t, where is an (n (n + 1)) matrix with rank r and yt = [1, y t ]. In other words, the mean term becomes an intercept in the cointegration reations. Three variants of LR-type tests have been considered in the iterature for this situation. The first test does not take into account the fact that the mean term can be absorbed into the cointegrating reations and incudes an unrestricted intercept term in the VECM (see Johansen (1991)). The second test enforces the restriction on the constant term by incuding it in the cointegration reations (see Johansen (1995, Ch. 11)). RR regression can be appied for both mode variants in computing the corresponding LR statistics. Finay, in the third testing setup the mean term µ 0 is estimated in a first step by a GLS procedure and then it is subtracted from y t. The tests are performed on the mean-adjusted data, x t = y t µ 0, noting that x t = y t. Suitabe estimators µ 0 are proposed by Saikkonen and Luukkonen (1997) and Saikkonen and Lütkepoh (2000a). We use the atter variant which proceeds by estimating the VECM parameters by RR regression as in the previous mode with the intercept restricted to the cointegration reations and imposing the cointegrating rank which is specified under H 0. These estimates are then used in accounting for the autocovariance structure of x t in a feasibe GLS estimation of µ 0 in the mode y t = µ 0 + x t. Finay the cointegrating rank test is performed using the VECM (2) where x t is repaced by the mean-adjusted observations x t = y t µ 0. We use the estimator µ 0 proposed by Saikkonen and Lütkepoh (2000a) in this context in the simuations reported in Section 5 because it resuted in sighty better size properties of the test than the Saikkonen and Luukkonen estimator in preiminary simuations. If both µ 0 and µ 1 are unrestricted a deterministic inear trend is aowed for. Again three different LR-type tests have been proposed for this situation. The first mode is set up in such a way so as to impose the inearity of the trend term as in the atter part of (3). This test version was proposed and anaysed by Johansen (1992, 1994, 1995). The second mode incudes the trend term outside the cointegration term as in the first VECM version in (3). In principe, such a mode can generate quadratic trends if no restrictions are imposed on the deterministic parameters. The corresponding test was considered by Perron and Campbe (1993). In both modes the LR statistics for the cointegrating rank are readiy avaiabe by RR regression. Finay, the ast test in Tabe 1 is based on prior trend adjustment via a feasibe GLS procedure. For this purpose the mode is estimated by RR regression under the nu hypothesis with restricted trend term as in the Johansen test. The resuting parameter estimators of and Ɣ are used to construct the feasibe GLS estimators for µ 0 and µ 1 based on the mode y t = µ 0 + µ 1 t + x t. Denoting the estimators by µ 0 and µ 1, the cointegration test is then based on RR regression appied to the mode (2) with x t repaced by the trend-adjusted observations

6 292 Hemut Lütkepoh et a. Tabe 2. Limiting Distributions of Tests Based on ( 10 FdN ) ( 10 FF ds) 1 ( 10 FdN ). Test statistic F(s) F t L R 0 N(s) N t 1 L R i0 N(s) 1 0 N(u)du N t 1 T 1 T t=1 N t 1 L R [N(s) : 1] [N t 1 : 1] L R SL N(s) N t 1 L R [(N(s) N(u)du) : s 1 ] [ 2 (N t 1 T 1 T t=1 N t 1 ) : s 1 ] 2 L R PC trend adjusted N(s) trend adjusted N t 1 x t = y t µ 0 µ 1 t. This test was first proposed by Saikkonen and Lütkepoh (2000a). Critica vaues for a these tests may be found in the references given in Tabe 1. The notation for the different tests aso corresponds to that used in the survey by Hubrich et a. (2001). It may be worth noting that there is aso another group of tests which is appied frequenty in practice. They assume that the variabes may have deterministic inear trends whereas a trend in the cointegrating reations is excuded. These tests are discussed in detai in Saikkonen and Lütkepoh (2000b). Despite their popuarity in practice, it is questionabe that the necessary assumptions can be justified easiy in many situations. Moreover, their asymptotic distributions under oca aternatives resut in more compicated expressions. Therefore, we do not incude them in the present comparison. The imiting distributions of the test statistics isted in Tabe 1, under the oca aternatives H T (r 0 ) : = αβ + 1 T α 1β 1, (9) have been derived for DGPs with order p = 1 ony. The resuts are aso vaid for higher-order processes. The proofs can be extended to that case aong the ines of Hansen and Johansen (1998, Ch. 12) who consider the case of a process without deterministic terms. For our VAR(1) case we assume that the parameter matrices α, β, α 1 and β 1 are such that the eigenvaues of I r0 + β α and I r + [β : β 1 ] [α : α 1 ] are ess than one in moduus. This assumption ensures that a variabes are at most I (1) under the nu and aternative hypotheses (see Johansen (1995, p. 202)). Under H T (r 0 ), most trace and maximum eigenvaue statistics have imiting distributions L R trace (r 0 ) d tr(d) and L R max (r 0 ) d λ max (D), respectivey, where ( 1 ) ( 1 D = FdN 0 0 ) 1 ( 1 FF ds FdN ), 0 F(s) is an (n r 0 )-dimensiona stochastic process and N(s) is the Ornstein Uhenbeck process defined by N(s) = B(s) + ab s 0 N(u)du, where B(s) is a standard Brownian motion, a = (α α ) 1/2 α α 1 and b = (α α ) 1/2 (β α ) 1 β β 1. The specific form of F(s) varies with the assumptions regarding the deterministic term. The different F(s) processes are given in Tabe 2 together with discrete counterparts which wi be discussed in the foowing section when simuations are considered. For exampe, L Rtrace 0 (r d 0) tr(d 0 ) and L Rmax 0 (r d 0) λ max (D 0 )

7 Maximum eigenvaue vs. trace tests 293 where ( 1 ) ( 1 D 0 = NdN 0 0 ) 1 ( 1 ) NN ds NdN 0 under the oca aternatives (9). Obviousy, the oca power depends on the number of common trends, n r 0, under the nu hypothesis and, via a and b, on the distance from the nu hypothesis. The nu distribution resuts by setting a = 0. The asymptotic distributions of the other statistics, except L R GLS, are obtained in an anaogous way. Most of the imiting distributions in Tabe 2 were derived by Johansen (1995) and Saikkonen and Lütkepoh (1999). The other imiting distributions for trace statistics are aso given in the references presented in Tabe 1. The corresponding resuts for some of the maximum eigenvaue statistics have not been derived expicity to the best of our knowedge (see, however, Hansen and Johansen (1998, Exercise 12.1) for the case of no deterministic term). Therefore proofs for those cases which are not avaiabe in the iterature are given in the Appendix. For the L R GLS tests, the asymptotic distributions are based on ( 1 ) ( 1 ) 1 ( 1 ) N dn N N ds N dn, 0 0 where N (s) = N(s) sn(1) and 1 0 N dn abbreviates 1 0 NdN 1 0 NdsN(1) N(1) 1 0 sdn sdsn(1)n(1). This imiting distribution is given in Lütkepoh and Saikkonen (2000) LOCAL POWER COMPARISON From the imiting distributions in Tabe 2 it foows that the oca power, that is, the reative rejection frequency if H(r 0 ) is tested and H T (r 0 ) is true, depends on α, β,, α 1 and β 1 ony through a and b. This impies, for instance, for the case r r 0 = 1, where α 1 and β 1 are (n 1) vectors, that the imiting distributions can be written as functions of the two parameters 2 = a ab b and d 2 = (b a) 2 /(a ab b). A different parametrization is used in Johansen (1995) and Saikkonen and Lütkepoh (1999). The present one is used in Hubrich et a. (2001) because it simpifies the interpretation of the resuts. 1 Note that 2 = 0 if and ony if the nu hypothesis hods. Hence, = 2 may be thought of as the distance of the oca aternative from the nu hypothesis. Moreover, 0 < d 2 1. Reca our assumption that a eigenvaues of I r + [β : β 1 ] [α : α 1 ] are ess than one in moduus which ensures that the variabes are at most I (1). In turn, if this eigenvaue condition is not satisfied, the process has components with more than one root on the compex unit circe. If the eigenvaues are a within the compex unit circe, then the matrix b a = β 1 β (β α ) 1 α α 1 is nonsinguar (see Johansen (1995, p. 204)). Hence, in our specific case where b a is a scaar, b a 0 is necessary to ensure that y t is an I (1) process, as required by our assumptions. In this sense, d 2 cose to zero may be viewed as representing processes cose to ones with higherorder integration. The quantity d = d 2 may therefore be interpreted as the distance from the 1 Using the notation in Johansen (1995), the reation to the parameters used in the present study is as foows: 2 = g 2 + f 2, d 2 = f 2 /( f 2 + g 2 ).

8 294 Hemut Lütkepoh et a. parameter space associated with higher-order integration. We wi use the quantities and d in comparing the oca power of the tests. For a oca power comparison we consider first the case where α 1 and β 1 are (n 1) vectors and simuate the discrete time counterpart of the (n r 0 )-dimensiona Ornstein Uhenbeck process N(s) as N t = 1 T ab N t 1 + ɛ t (t = 1,..., T = 1000) with ɛ t i.i.d.n(0, I n r0 ), N 0 = 0, b = { (1, 0) for n r0 = 2 (1, 0, 0) for n r 0 = 3 and a = { ( 2 d 2, 2 (1 d 2 ) ) for n r 0 = 2 ( 2 d 2, 2 (1 d 2 ), 0 ) for n r 0 = 3. From the N t we compute A T = T 2 T t=1 F t F t and B T = T 1 T t=1 F t N t, where the F t are the discrete counterparts of the F(s) given in Tabe 2. The imiting distributions of the trace and maximum eigenvaue test statistics are then simuated as tr(b T A 1 T B T ) and λ max (B T A 1 T B T ), respectivey. Furthermore, using and A T = 1 T 2 B T = 1 T T t=1 [ t 1 ( N k N) T t=1 k=1 [ t 1 ][ t 1 ( N k N) k=1 ] ( N k N) ( N t N), k=1 with ( N = T 1 T t=1 N t ) gives the imiting distributions of the L R GLS statistics in an anaogous fashion. The resuting rejection frequencies for the cases n r 0 = 2 and n r 0 = 3 obtained from repications for different vaues of d and are potted in Figures 1 and 2. Note that the two types of tests are identica for n r 0 = 1 and, hence, a comparison is not meaningfu for this case. Note aso that L R 0 and L R SL have the same asymptotic distributions. Therefore ony the oca power curves for the atter tests are depicted in the figures. Figures 1 and 2 revea that the resuts for the trace and the maximum eigenvaue versions of the tests are quite simiar. Ceary, this resut is not surprising because, if there is just one additiona cointegration reation, both tests check against the appropriate aternative hypothesis. Notice, however, that due to the arge number of repications used in evauating the rejection probabiities, even reativey sma power differences in these figures are significant. Using repications, the standard error of an estimator of a true rejection probabiity P is s P = P(1 P)/ so that, in the worst case where P = 0.5, s 0.5 = 05, ignoring the Monte Caro variation due to estimation of the critica vaues. Moreover, for a given set of parameters, the simuations are based on the same random numbers and are therefore not independent. As a consequence, quite sma differences in the rejection probabiities are statisticay significant. For practica purposes, the differences between the maximum eigenvaue and associated trace tests are so sma, however, that they are of itte importance. Comparing the different proposas for treating the deterministic terms, it can be seen that the tests assuming no inear trend outperform the tests aowing for a trend. However, the reative performance of the test variants is not infuenced by the incusion of a trend (see panes A and B in Figures 1 and 2). The same can be said about the parameter d measuring the distance from ]

9 Maximum eigenvaue vs. trace tests 295 Pane A: Tests with a time trend Pane B: Tests without a time trend d = 5 d = 5 LR * max LR * trace d = 0.50 d = 0.50 LR * max LR * trace d = 0.75 d = 0.75 LR * max LR * trace Figure 1. Loca power of LR-type tests for n r 0 = 2. Pane A: Tests with a time trend Pane B: Tests without a time trend d = d = 5 LR * max LR * trace d = d = 0.50 LR * max LR * trace d = 0.75 d = 0.75 LR * max LR * trace Figure 2. Loca power of LR-type tests for n r 0 = 3.

10 296 Hemut Lütkepoh et a. c = Pane A: Tests with a time trend Pane B: Tests without a time trend c = 2.5 LR * max LR * trace c = 5.0 c = 5.0 LR * max LR * trace c = c = LR * max LR * trace Figure 3. Loca power of LR-type tests for n r 0 = 2 and d = 5. the region with a higher order of integration. These resuts are in ine with those of Saikkonen and Lütkepoh (1999) and Hubrich et a. (2001). For the specia case of a bivariate process, n r 0 = 2 corresponds to testing H(0) : r 0 = 0 which is aso the case treated by Pesavento (2000). She finds that under specific conditions, gains in oca power are possibe in this situation if singe-equation cointegration tests are used. In other words, the knowedge that there is at most one cointegrating reation and both variabes are I (1) can be used to obtain oca power gains. Because the maximum eigenvaue tests consider the aternative hypothesis that there is just one extra cointegration reation whereas the trace variants test against a more genera aternative, we have aso determined the oca power for the case r r 0 = 2, (where under the oca aternative, the process has two extra cointegration reations) to investigate whether our resuts are robust with respect to this feature of the DGP. In this case we use { b I2 for n r = 0 = 2 [I 2 : 0] for n r 0 = 3 and ( ) 2 d 2 2 (1 d 2 ) a c 0 = ( ) 2 d 2 2 (1 d 2 ) 0 c 0 0 for n r 0 = 2 for n r 0 = 3. The resuting oca power of the tests for d = 5 and different vaues of c and is depicted in Figures 3 and 4 where the number of repications is again and, hence, the precision of the simuations is as before.

11 Maximum eigenvaue vs. trace tests 297 Pane A: Tests with a time trend Pane B: Tests without a time trend c = 2.5 c = 2.5 LR * max LR * trace c = 5.0 c = 5.0 LR * max LR * trace c = 7.5 c = 7.5 LR * max LR * trace Figure 4. Loca power of LR-type tests for n r 0 = 3 and d = 5. It can be seen that the trace and the maximum eigenvaue tests again perform quite simiary. On the other hand, the differences between the aternative proposas for treating deterministic terms are more pronounced now. L R + and L R SL have the highest oca power among the tests in their respective groups, whereas L R i0 is generay outperformed by its competitors. As in the case of one extra cointegrating reation, the parameter d has no important effect on the reative properties of the test versions. Therefore we do not present the resuts for vaues of d other than 5. In summary, based on oca power, no cear recommendations regarding the preferred use of either the trace or the maximum eigenvaue tests can be given. Notice, however, that oca power properties are informative about the performance of the tests in arge sampes when aternatives cose to the nu hypothesis are considered. Therefore a sma-sampe comparison of the tests is performed in the next section in order to get further insights into the reative performance of the two different test versions. 5. SMALL-SAMPLE COMPARISON OF TESTS We use the foowing process x t to compare the maximum eigenvaue and trace tests in sma sampes: x t = [ 0 ] ([ 0 x 0 I t 1 + ε t, ε t i.i.d. N n r 0 ] [ Ir ]),, (10) I n r

12 298 Hemut Lütkepoh et a. Tabe 3. Reative rejection frequencies of tests for H(0) : r = 0 based on a bivariate DGP with cointegrating rank r = 0, VAR order p = 1, θ = 0, sampe size T = 100, nomina significance eve 5. Test statistic Test statistic L Rmax i0 69 L R max + 63 L Rtrace i0 63 L R trace + 55 L Rmax 62 L Rmax PC 58 L Rtrace 62 L Rtrace PC 58 L Rmax SL 49 L Rmax GLS 53 L Rtrace SL 53 L Rtrace GLS 46 where = diag(ψ 1,..., ψ r ) is an (r r) diagona matrix and is (r (n r)). This process was aso used in simuations by Toda (1994, 1995) and a number of other authors. For the purposes of cointegration testing it represents a canonica form from which other processes can be obtained by inear transformations. The LR tests for cointegration are invariant under such transformations. (see Toda (1994)). Bivariate, three- and four-dimensiona processes wi be considered. As mentioned in the introduction, Toda (1994) reports resuts for bivariate processes ony. In the bivariate case, if r = 0, and vanish and the process consists of two nonstationary components. If the cointegrating rank is 1, = ψ 1 with ψ 1 < 1 and = θ is a scaar which represents the instantaneous correation between the two components. For threeand four-dimensiona processes the matrices and have anaogous interpretations. In the simuations the parameter vaues of the deterministic components are a zero, i.e. µ i = 0 (i = 0, 1) throughout. In other words, the deterministic part is actuay zero. This choice is not restrictive because the test statistics are invariant to the specific parameter vaues of their respective deterministic terms. In other words, L R i0, L R and L R SL are invariant to the choice of µ 0 and L R +, L R PC and L R GLS are invariant with respect to the vaues assigned to µ 0 and µ 1. Notice that this invariance hods aso in sma sampes so that the same sma-sampe resuts are obtained for any choice of deterministic parameters. The sampe size used in the simuations is T = 100. In addition, 50 presampe vaues were generated, starting with an initia vaue of zero. By discarding the start-up vaues we average out the effects of initia vaues. This is done athough the initia vaues may have an impact on the test resuts if, for exampe, these vaues are very unusua. In practice, outiers at the sampe beginning can usuay ony be detected after a mode has been fitted. If they are detected, it may be preferabe to excude them from the anaysis. Therefore, we sha not anayse the impact of unusua initia vaues and prefer to average out the impact of initia vaues by discarding the start-up vaues for each time series. The number of repications is m = The rejection frequencies given in Tabes 3 5 and Figures 5 9 are based on asymptotic critica vaues for a test eve of 5%. The rejection frequencies are not size corrected because a size correction is generay not avaiabe in practice. As in the oca power simuations, the resuts for the test statistics are based on the same generated time series for a given set of parameter vaues and a given sampe size. Hence, the corresponding entries in the tabes and the figures are not independent. Again the standard error of an estimator of a true rejection probabiity P is s P = P(1 P)/ so that, in the worst case where

13 Maximum eigenvaue vs. trace tests 299 Tabe 4. Reative rejection frequencies of tests for three-dimensiona DGPs with cointegrating rank r = 0 or 1, VAR order p = 1, sampe size T = 100, nomina significance eve 5. Test r = 0 a statistic (H(0) : r = 0) r = 1 (H(1) : r = 1) r = 1 (H(1) : r = 1) ψ 1 = 1 ψ 1 = 0.9 ψ 1 = ψ 1 = 0.7 ψ 1 = 0.9 ψ 1 = ψ 1 = 0.7 = (0, 0) = (, ) L Rmax i L Rtrace i L R max L R trace L Rmax SL L Rtrace SL L R + max L R + trace L Rmax PC L Rtrace PC L Rmax GLS L Rtrace GLS a The resuts for r = 0 are invariant with respect to the innovation correation. Tabe 5. Reative rejection frequencies of tests for H(1) : r = 1 based on four-dimensiona DGPs with cointegrating rank r = 1, = (,, ), VAR order p = 1, sampe size T = 100, nomina significance eve 5. Test r = 1 Test r = 1 statistic ψ 1 = 0.9 ψ 1 = ψ 1 = 0.7 statistic ψ 1 = 0.9 ψ 1 = ψ 1 = 0.7 L Rmax i L R max L Rtrace i L R trace L Rmax L Rmax PC L Rtrace L Rtrace PC L Rmax SL L Rmax GLS L Rtrace SL L Rtrace GLS P = 50%, a precision of ±1% is obtained, ignoring the Monte Caro variation due to estimation of the critica vaues. Note aso that the resuts for testing H(1) : rk( ) = 1 are not conditioned on the outcome of the test of H(0) : rk( ) = 0 etc. In other words, the test resuts do not refer to sequentia test procedures. Notice that the properties of a sequentia testing procedure as it is required here are studied in genera terms by Johansen (1992) and Paruoo (2001). The statistica properties of the

14 300 Hemut Lütkepoh et a. Pane A: θ = 0 Pane C: θ = 0 LR* max LR* trace Pane B: θ = Pane D: θ = LR* max LR* trace Figure 5. Reative rejection frequencies of tests for H(0) : r = 0 based on bivariate DGPs with r = 0 or r = 1, T = Pane A: θ = 0 LR Pane C: θ = 0 LR io 1.1 LR PC 1.1 LR* LR GLS LR SL Pane B: θ = LR LR io Pane D: θ = 1.1 LR PC 1.1 LR* LR GLS LR SL Figure 6. Reative power of tests for H(0) : r = 0 based on bivariate DGPs with r = 0 or r = 1, T = 100.

15 Maximum eigenvaue vs. trace tests 301 Pane A: Η = (0 0) Pane C: Η = (0 0) LR* max LR* trace Pane B: Η = ( ) Pane D: Η = ( ) LR* max LR* trace Figure 7. Reative rejection frequencies of tests for H(0) : r = 0 based on three-dimensiona DGPs with r = 1 or r = 2, ψ 2 = 0.9, T = 100. Pane A: Θ = (0 0) Pane C: Θ = (0 0) LR* max LR* trace Pane B: Θ = ( ) Pane D: Θ = ( ) LR* max LR* trace Figure 8. Reative rejection frequencies of tests for H(1) : r = 1 based on three-dimensiona DGPs with r = 1 or r = 2, ψ 2 = 0.9, T = 100.

16 302 Hemut Lütkepoh et a. Pane A: H(0): r = 0 Pane C: H(1): r = Pane B: H(0): r = 0 Pane D: H(1): r = 1 LR* max LR* trace LR* max LR* trace Figure 9. Reative rejection frequencies of tests for three-dimensiona DGPs with r = 1 or r = 2, ψ 2 = 0.7, = (0, 0), T = 100. sequentia procedure depend essentiay on the properties of the individua tests. Consequenty, studying the tests individuay may be viewed as a prerequisite for investigating the sequentia procedure. We wi focus on the properties of the individua tests. The sma-sampe properties of L R 0 are not presented here because the test s assumptions regarding the deterministic terms are not very reaistic for appied work and are therefore not of much interest from that perspective. The sizes of the tests for the bivariate DGP with r = 0 are shown in Tabe 3. Because the trace and the maximum eigenvaue tests are identica for testing H(1) : r = 1 we just present the resuts for H(0) : r = 0. A tests have roughy the correct size. Obviousy, the observed sizes for both the trace and the maximum eigenvaue tests are very simiar irrespective of the specific test proposa. Notice, however, that we are working under idea conditions here by considering VAR(1) processes ony. Pesavento (2000) finds considerabe size distortions of the maximum eigenvaue tests in her specia setup when the actua process order is unknown and may be infinite. The same resut was aso found for three-dimensiona DGPs when H(0) : r = 0 or H(1) : r = 1 are tested (see Tabe 4). Generay, the sizes of the trace variants of the L R + and L R PC tests exceed the sizes of the corresponding maximum eigenvaue versions a bit for ψ 1 = 0.9 and ψ 1 = in the case of arge innovation correation ( = (, )). For = (, ) we aso observe that the trace variants of L R + and L R PC reject sighty too often. Both the trace and the maximum eigenvaue versions of L R i0 and L R are affected by this probem as we. In contrast, in the absence of innovation correation ( = (0, 0)) the tests are quite conservative, especiay the maximum eigenvaue variants. However, the L R SL tests have reasonabe size properties for both kinds of innovation correation and therefore outperform the other tests in this respect.

17 Maximum eigenvaue vs. trace tests 303 In the absence of innovation correation ( = (0, 0, 0)) we observe simiar size properties of the tests for four-dimensiona DGPs with r = 1 as for three-dimensiona ones. However, in the case of high innovation correation ( = (,, )) the probem of an excessive size distortion is much more severe, as shown in Tabe 5. A tests, except L R GLS, reject far too often. Gonzao and Pitarakis (1999) have pointed out that this kind of size distortion is a typica probem of LR-type tests that seem to be emerging in arge systems. However, the excessive size distortion is ess pronounced for the maximum eigenvaue tests than for the trace tests. So the former have a sight advantage in this respect. The four-dimensiona case aso aows us to compare the size of the test variants for processes with two cointegration reations (r = 2). As the reative performance of the trace and maximum eigenvaue tests is amost identica, we do not show the resuts here. Nevertheess, we mention that a tests are very conservative in this case. Generay, the sizes of the tests do not exceed the vaue of 2 and are often beow 1. Ony the L R SL tests are a bit ess conservative. To sum up, the size properties of the trace and the maximum eigenvaue variants are in genera rather simiar for a test proposas. However, in specific situations the trace tests suffer more from an excessive size distortion than the maximum eigenvaue tests. The sma-sampe power resuts of the tests are depicted in the Figures 5 9. Figure 5 shows that the trace and maximum eigenvaue tests perform quite simiary in the bivariate case with r = 0 or 1 when H(0) : r = 0 is tested. In accordance with resuts by Toda (1994) for tests which aow for a time trend, the trace tests are sighty better for aternatives cose to the nu hypothesis and the maximum eigenvaue tests have an advantage in situations further away from the nu hypothesis. Interestingy, this pattern is observed independenty of the innovation correation and the tests assumption regarding the deterministic terms. Obviousy, the differences are quite sma and may be due to samping variabiity in our simuations, athough we use repications. To further investigate the significance of the differences in our resuts we aso show the reative powers of the maximum eigenvaue tests divided by the powers of the corresponding trace tests in Figure 6 where statisticay significant differences at the 5% eve are indicated by cooured circes, boxes, etc. The significance is checked using the statistic {m/[ ˆp tr (1 ˆp tr ) + ˆp max (1 ˆp max ) 2 ˆτ]} 1/2 ( ˆp tr ˆp max ) which is asymptoticay standard norma if the underying rejection probabiities of the two tests are equa. Here m denotes the number of repications in our Monte Caro experiment, as before (i.e. m = ), ˆp tr and ˆp max are the observed reative rejection frequencies of the trace and maximum eigenvaue tests, respectivey, and ˆτ = ˆγ ( ˆp max ˆp tr ), where ˆγ is the proportion of joint rejections of both tests. The statistic was aso used by Paruoo (2001) in comparing his oca power resuts. More detais can be found in his study. Obviousy, Figure 6 shows that most of the power differences are statisticay significant at the 5% eve. On the other hand, the differences between corresponding maximum eigenvaue and trace tests are so sma (usuay ess than 10%) that they are hardy of practica importance. Because most curves cross the ine at one, Figure 6 aso shows that in most cases power advantages are not uniform over the fu range of parameter vaues. It is worth noting that the tests assuming no inear trend have higher sma-sampe power than those aowing for a trend (see Figure 5). Hence, it pays to specify the deterministic terms propery. On the other hand, comparing the different test versions within the respective groups (µ 1 = 0 or µ 1 0), a cear winner for a situations is not found. Pesavento (2000) compares the sma-sampe power of maximum eigenvaue tests for the case of testing H(0) : rk( ) = 0 in a

18 304 Hemut Lütkepoh et a. bivariate setting to singe-equation tests which are appicabe if it is known that both variabes are I (1) and there is at most one cointegrating reation. If this knowedge is utiized, considerabe power gains may be possibe by using singe-equation cointegration tests. Compared with the foregoing bivariate setup, a tests have a ower sma-sampe power when the corresponding three- and four-dimensiona DGPs with r = 0 or 1 are considered. Nevertheess, the reative characteristics of the test variants remain unchanged, so we do not present the reevant graphs here. We have aso simuated the sma-sampe power for three- and four-dimensiona processes with two cointegration reations when testing H(0) and H(1). The quaitative resuts are argey the same for both dimensions. Therefore we just refer to the three-dimensiona case. In Figure 7, the power for the case of testing H(0) is depicted. In this case, for ψ 1 < 1, there are two more cointegrating reations than are specified in the nu hypothesis (i.e. r r 0 = 2). Hence, for the maximum eigenvaue tests the nu and aternative hypotheses are incorrect and, thus, the trace tests may have an advantage. This is ceary refected in the power curves. In the absence of innovation correation ( = (0, 0) ) the sma-sampe power of the trace tests exceeds the power of the maximum eigenvaue tests by up to about 20 percentage points. In some cases, the power curves of the trace tests are ceary steeper than those of the maximum eigenvaue tests so that the superior performance of the former is ony party attributabe to their arger size. When testing H(1), there is again just one extra cointegration reation under the aternative (r r 0 = 1) and the power of a tests is rather ow for ψ 2 = 0.9, as can be seen in Figure 8. Obviousy, in some situations it is not very ikey that a cointegrating rank of two wi be found. Setting the second autoregressive eigenvaue to ψ 2 = 0.7 increases the sma-sampe power of the tests for = (0, 0) remarkaby (see Figure 9). In the case of testing H(0) so that r r 0 = 2 (panes A and B), the advantage of the trace tests is even more pronounced than for the DGPs with ψ 2 = 0.9. When the nu hypothesis is H(1) and, hence, r r 0 = 1, both test versions are once more rather simiar (panes C and D). We have aso extended our simuations in the different directions. First, we have generated processes with fat-taied, non-norma residuas. More precisey, we have used t-distributions with 5 degrees of freedom. The resuts are very simiar to those with norma residuas and are therefore not shown. 2 A second extension considered is processes with a component cose to I (2). The resuting test sizes were simiar to the previousy discussed I (1) cases. The power of the tests appied to the near I (2) processes is not necessariy worse than that in the I (1) cases because some of the power resuts for those cases are aso very poor. Hence, there is itte that can be earnt in additionay from the near I (2) case. Therefore the resuts are aso not shown. 3 In summary, we can concude that with respect to sma-sampe power, both the trace and the maximum eigenvaue tests have simiar properties, in ine with the oca power resuts. However, in some cases, the trace tests are ceary superior to the corresponding maximum eigenvaue versions in terms of power. This happens in particuar if the actua rank r exceeds the rank specified in the nu hypothesis, r 0, by more than one, r r 0 > 1. In those cases where the maximum eigenvaue tests dominate, their power advantage is ony minor. On the other hand, the atter tests seem to have smaer size distortions than the trace tests. Sti, our overa recommendation is to use the trace tests if one wants to appy just one test version. Of course, there is nothing wrong with the common practice of using both versions simutaneousy. 2 Detaied resuts are posted on the internet at 3 Detais are aso posted on the internet.

19 Maximum eigenvaue vs. trace tests CONCLUSIONS In this study we have compared maximum eigenvaue and trace tests for the cointegrating rank of a VAR process. The comparison is performed for test variants suitabe for different types of deterministic terms. More precisey, a coupe of tests aowing for a nonzero mean and a group of tests aowing in addition for a deterministic inear trend are considered. The asymptotic distributions under oca aternatives are given and a oca power comparison is presented. In that comparison no major differences between corresponding maximum eigenvaue and trace tests are detected. In a sma-sampe simuation comparison it is found, however, that in some situations trace tests tend to have more heaviy distorted sizes whereas their power performance is superior to that of the maximum eigenvaue competitors. In particuar, the trace tests are advantageous if there are at east two more cointegrating reations in the process than are specified under the nu hypothesis. Based on our simuations we have a preference for the trace tests. This resut justifies the common practice in empirica work of using either both types of tests simutaneousy or appying the trace tests excusivey. In accordance with other authors (see, for exampe, Hubrich et a. (2001)) we aso found that the aternative LR-type test versions for a specific deterministic term sometimes differ more substantiay than the corresponding maximum eigenvaue and trace tests. Therefore, based on our simuation resuts, it appears to be more sensibe to appy different versions with respect to the treatment of deterministic terms rather than the maximum eigenvaue and the trace variant of one specific test version. The choice of the deterministic term may be based, for exampe, on the tests presented in Johansen (1995, Ch. 11) for this purpose. In specia cases it may aso be possibe to increase the power by using other tests designed for specific situations. For exampe, Pesavento (2000) shows that, in a bivariate system with two I (1) variabes, testing for a cointegrating rank of zero is perhaps better based on singe-equation tests. Aso, if some cointegration reation is known, power improvements may be possibe as in Horvath and Watson (1995) by using this prior knowedge. ACKNOWLEDGEMENTS We thank Raf Brüggemann, two anonymous referees and Karim Abadir for hepfu comments and we are gratefu to the Deutsche Forschungsgemeinschaft, SFB 373, the Yrjö Jahnsson Foundation and the European Commission under the Training and Mobiity of Researchers Programme (contract No. ERBFMRXCT980213), for financia support. The second author aso thanks the Aexander von Humbodt Foundation for financia support under a Humbodt research award. Part of this research was done whie he was visiting the Humbodt University in Berin. REFERENCES Gonzao, J. and J. -Y. Pitarakis (1999). Dimensionaity effect in cointegration anaysis. In R. Enge and H. White (eds), Cointegration, Causaity, and Forecasting. A Festschrift in Honour of Cive W. J. Granger, pp Oxford: Oxford University Press. Hansen, P. and S. Johansen (1998). Workbook on Cointegration. Oxford: Oxford University Press. Horvath, M. T. K. and M. W. Watson (1995). Testing for cointegration when some of the cointegrating vectors are prespecified. Econometric Theory 11,

20 306 Hemut Lütkepoh et a. Hubrich, K., H. Lütkepoh and P. Saikkonen (2001). A review of systems cointegration tests. Econometric Reviews 20, Johansen, S. (1988). Statistica anaysis of cointegration vectors. Journa of Economic Dynamics and Contro 12, Johansen, S. (1991). Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive modes. Econometrica 59, Johansen, S. (1992). Determination of cointegration rank in the presence of a inear trend. Oxford Buetin of Economics and Statistics 54, Johansen, S. (1994). The roe of the constant and inear terms in cointegration anaysis of nonstationary time series. Econometric Reviews 13, Johansen, S. (1995). Likeihood Based Inference in Cointegrated Vector Autoregressive Modes. Oxford: Oxford University Press. Johansen, S. and K. Juseius (1990). Maximum ikeihood estimation and inference on cointegration with appications to the demand for money. Oxford Buetin of Economics and Statistics 52, Lütkepoh, H. and P. Saikkonen (2000). Testing for the cointegrating rank of a VAR process with a time trend. Journa of Econometrics 95, Paruoo, P. (2001). The power of ambda max. Oxford Buetin of Economics and Statistics 63, Perron, P. and J. Y. Campbe (1993). A note on Johansen s cointegration procedure when trends are present. Empirica Economics 18, Pesavento, E. (2000). Anaytica evauation of the power of tests for the absence of cointegration. Discussion Paper San Diego: University of Caifornia. Saikkonen, P. and H. Lütkepoh (1999). Loca power of ikeihood ratio tests for the cointegrating rank of a VAR process. Econometric Theory 15, Saikkonen, P. and H. Lütkepoh (2000a). Trend adjustment prior to testing for the cointegrating rank of a vector autoregressive process. Journa of Time Series Anaysis 21, Saikkonen, P. and H. Lütkepoh (2000b). Testing for the cointegrating rank of a VAR process with an intercept. Econometric Theory 16, Saikkonen, P. and R. Luukkonen (1997). Testing cointegration in infinite order vector autoregressive processes. Journa of Econometrics 81, Toda, H. Y. (1994). Finite sampe properties of ikeihood ratio tests for cointegrating ranks when inear trends are present. Review of Economics and Statistics 76, Toda, H. Y. (1995). Finite sampe performance of ikeihood ratio tests for cointegrating ranks in vector autoregressions. Econometric Theory 11, APPENDIX. DERIVATION OF ASYMPTOTIC DISTRIBUTIONS OF TEST STATISTICS As mentioned in Section 3, some of the imiting distributions of the test statistics under consideration have been given by Johansen (1995), Hansen and Johansen (1998) and Paruoo (2001). In particuar, the case when no deterministic term is present has been treated in detai by these authors. Moreover, Saikkonen and Lütkepoh (1999) discuss the imiting distributions of the trace test versions when deterministic terms are present. To derive the imiting distributions of the maximum eigenvaue tests, we show that the genera framework of the atter paper can be used. This resut does not foow automaticay from Saikkonen and Lütkepoh (1999) because