A Jackknife Correction to a Test for Cointegration Rank

Transcription

1 Econoetrics 205, 3, ; doi:0.3390/econoetrics OPEN ACCESS econoetrics ISSN Article A Jackknife Correction to a Test for Cointegration Rank Marcus J. Chabers Departent of Econoics, University of Essex, Wivenhoe Park, Colchester, Essex CO4 3SQ, UK; chab@essex.ac.uk; Tel.: ; Fax: Acadeic Editor: Kerry Patterson Received: 24 July 204 / Accepted: 5 May 205 / Published: 20 May 205 Abstract: This paper investigates the perforance of a jackknife correction to a test for cointegration rank in a vector autoregressive syste. The liiting distributions of the jackknife-corrected statistics are derived and the critical values of these distributions are tabulated. Based on these critical values the finite saple size and power properties of the jackknife-corrected tests are copared with the usual rank test statistic as well as statistics involving a sall saple correction and a Bartlett correction, in addition to a bootstrap ethod. The siulations reveal that all of the corrected tests can provide finite saple size iproveents, while aintaining power, although the bootstrap procedure is the ost robust across the siulation designs considered. Keywords: jackknife correction; bias reduction; cointegration rank test JEL classifications: C2; C32. Introduction The concept of cointegration has assued a proinent role in the analysis of econoic and financial tie series since the pioneering work of Engle and Granger [], and tests for the cointegration rank of a vector of tie series have becoe an essential part of the applied econoetrician s toolkit. The ost popular test for cointegration rank is the trace statistic proposed by Johansen [2,3] which exploits the reduced rank regression techniques of Anderson [4] in the context of a vector autoregressive (VAR) odel. The liiting distribution of the test statistic can be expressed as a functional of a vector Brownian otion process, the diension of which depends upon the difference between the nuber of variables under consideration and the cointegration rank under the null hypothesis. Percentage points of the

2 Econoetrics 205, liiting distribution have been tabulated by siulation and can be found, for exaple, in Johansen [5], Doornik [6] and MacKinnon, Haug and Michelis [7]. The accuracy of the liiting distribution as a description of the finite saple distribution has been exained in a nuber of studies. Toda [8,9] found that the perforance of the tests is dependent on the value of the stationary roots of the process, and that a saple of 00 observations is insufficient to detect the true cointegrating rank when the stationary root is close to one (0.8 or above). Doornik [6] proposed Gaa distribution approxiations to the liiting distributions, finding that they are ore accurate than previously published tables of critical values, while Nielsen [0] used local asyptotic theory to iprove the ability of the liiting distribution to act as an approxiation in finite saples. In view of the experiental evidence reported above, there have been a nuber of further attepts to iprove inference in finite saples when using the asyptotic critical values for the trace test of cointegration rank. Johansen [] deonstrated how Bartlett corrections can be ade to the statistic; these rely on various asyptotic expansions of the statistic s expectation and result in coplicated functions of the odel s paraeters that can be estiated fro the saple data. A sipler sall saple correction factor was suggested by Reinsel and Ahn [2] and involves a degrees-of-freedo type of adjustent to the saple size when calculating the value of the test statistic. This sall saple correction was shown to work well by Reiers [3] although, as Johansen [5] (p. 99) notes, the theoretical justification for this result presents a very difficult atheatical proble. More coputationally intensive bootstrap procedures have recently been advocated by, inter alia, Swensen [4] and Cavaliere, Rahbek and Taylor [5]. The ai of this paper is to analyse the properties of a siple jackknife-corrected test statistic for cointegration rank. The approach is far less deanding, coputationally, than bootstrap ethods and does not require an explicit analytical derivation of an asyptotic expansion for the saple oent as in the Bartlett approach, erely relying on its existence. The idea behind the jackknife is to cobine the statistic based on the full saple of observations with statistics based on a set of sub-saples, the weights used in the linear cobination being chosen so that the leading ter in the bias expansion is eliinated. Although intended priarily as a ethod of bias reduction in paraeter estiation following the work of Quenouille [6] and Tukey [7], the jackknife can equally be applied to test statistics in order to achieve the sae type of outcoe as the Bartlett correction. In the case of stationary autoregressive tie series Chabers [8] has shown that jackknife ethods are capable of producing substantial bias reductions as well as reductions in root ean square errors copared to other ethods in the estiation of odel paraeters; the jackknife results were also shown to be robust to departures fro norality and conditional heteroskedasticity as well as other types of isspecification. However, soe care has to be taken when applying these techniques with non-stationary data, as pointed out by Chabers and Kyriacou [9,20], who propose ethods that can be used to ensure that the jackknife procedure achieves the bias reduction as intended in the case of non-stationarity. The paper is organised as follows. Section 2 begins by defining the odel as well as the test statistic of interest. Three variants of the odel are considered, corresponding to different specifications of the deterinistic linear trend, although ost attention is given to the two variants of greatest epirical interest. The jackknife-corrected version of the statistic is defined and the appropriate liiting distributions are derived and presented in Theore. Section 3 is devoted to siulation results and

3 Econoetrics 205, is divided into two subsections. The first sub-section coputes (asyptotic) critical values of the liiting distributions of the jackknife-corrected statistics. The second sub-section is concerned with the finite saple size and power properties of the jackknife-corrected statistics, which are copared with the unadjusted statistic as well as the sall-saple adjustent of Reinsel and Ahn [2], the Bartlett correction of Johansen [] and the bootstrap approach of Cavaliere, Rahbek and Taylor [5]. Section 4 concludes and discusses soe directions for future research, while an Appendix provides a proof of Theore and soe details on the siulation ethod used to derive the critical values in Section 3.. The following notation will be used. For a p r atrix C of rank r < p there exists a full rank p (p r) atrix C satisfying C C = 0; P C = C(C C) C denotes the projection atrix of C; and C = ( p r i= j= c2 ij) /2 denotes the Euclidean nor of C. In addition, for a square atrix A, det[a] denotes the deterinant, and I p denotes the p p identity atrix. The sybol = d denotes d p equality in distribution; denotes convergence in distribution; denotes convergence in probability; denotes weak convergence of the relevant probability easures; L denotes the lag operator such that L j y t = y t j for soe integer j and variable y t ; and B(s) denotes a standard vector Brownian otion process. Functionals of B(s), such as 0 B(s)B(s) ds, are denoted 0 BB for notational convenience. 2. The Model and Tests for Cointegration Rank Following Johansen [5] the odel under consideration is the following VAR(k) syste in the p vector y t : k y t = Π i y t i + ΦD t + ɛ t, t =,..., T () i= where ɛ,..., ɛ T are independent and identically distributed p rando vectors with ean vector zero and positive definite covariance atrix Ω, D t is a q vector of deterinistic ters, Π,..., Π k are p p atrices of autoregressive coefficients, Φ is a p q atrix of coefficients on the deterinistic ters, and the initial values, y k+,..., y 0, are assued to be fixed. It is convenient, in the analysis of cointegration, to write Equation () in the vector error correction odel (VECM) for k y t = Πy t + Γ i y t i + ΦD t + ɛ t, t =,..., T (2) i= where Π = k i= Π i I p and Γ i = k j=i+ Π j (i =,..., k ). The following assuption will be ade. Assuption. (i) Let A(z) = ( z)i p Πz k i= Γ i( z)z i. Then det[a(z)] = 0 iplies that z > or z = ; (ii) The atrix Π has rank r < p and has the representation Π = αβ where α and β are p r with rank r; (iii) The atrix α Γβ is nonsingular, where Γ = I p k i= Γ i. This assuption is coon in the cointegration literature and iplies (see Theore 4.2 of Johansen [5], for exaple) that y t has the representation t y t = C (ɛ i + ΦD i ) + C(L) (ɛ t + ΦD t ) + P β y 0, t =,..., T (3) i= where C(z) = ( z)a(z) = i= C iz i and C = C() = β (α Γβ ) α. This representation is convenient because it decoposes y t into a stochastic trend coponent, a stationary coponent, and a

4 Econoetrics 205, ter that depends on the initial condition y 0. Assuption also ensures that, although the vector y t is I(), both y t E( y t ) and β y t E(β y t ) can be given initial distributions such that they are I(0). A proof of Equation (3) and a discussion of its iplications can be found in Johansen [5] (pp ). The specification of the deterinistic coponent ΦD t in Equation (2) has iplications not only for the interpretation of the error correction odel but also for the level process y t itself. The following three leading cases have received ost attention in the literature: Case : no deterinistic coponents. In this case ΦD t = 0 and so, fro Equations (2) and (3), k y t = αβ y t + Γ i y t i + ɛ t y t = C i= t ɛ i + C(L)ɛ t + P β y 0 Case 2: restricted intercept. Here ΦD t = αρ 0, where ρ 0 is r, and hence i= k y t = α (β y t + ρ 0 ) + Γ i y t i + ɛ t y t = C where τ 0 is a vector of intercepts. i= t ɛ i + C(L)ɛ t + τ 0 + P β y 0 i= Case 3: restricted linear trend. In this specification ΦD t = µ 0 + αρ t, where µ 0 and ρ are p and r vectors respectively. It follows that k y t = α (β y t + ρ t) + Γ i y t i + µ 0 + ɛ t y t = C i= t ɛ i + C(L)ɛ t + τ 0 + τ t + P β y 0 i= where τ 0 and τ are vectors of constants. Hence in case there are no deterinistic ters at all, in case 2 the intercept is restricted to the cointegrating relationships, while in case 3 there is an unrestricted intercept and the tie trend only enters through the cointegrating relationships although a linear trend is present in the levels representation. The trace statistic has becoe a popular ethod of testing the null hypothesis of r < p cointegrating vectors against the aintained hypothesis that Π has full colun rank p, in which case the process y t is stationary. In what follows it is convenient to further express the VECM Equation (2) in the for Z 0t = αβ Z t + ΨZ 2t + ɛ t, t =,..., T (4) where Z 0t = y t and the reaining ters are defined with respect to the three cases concerning the specification of the deterinistic coponent ΦD t defined above: Case : β = β, Z t = y t, Z 2t = ( y t,..., y t k ), Ψ = [Γ,..., Γ k ].

5 Econoetrics 205, Case 2: β = (β, ρ 0 ), Z t = (y t, ), and Ψ and Z 2t are defined as in case. Case 3: β = (β, ρ ), Z t = (y t, t), Z 2t = ( y t,..., y t k, ), Ψ = [Γ,..., Γ k, µ 0 ]. Based on Equation (4) it is possible to define the atrices M ij = T T t= Z it Z jt, (i, j = 0,, 2), S ij = M ij M i2 M 22 M 2j, (i, j = 0, ) The trace statistic is then obtained as S r = T p i=r+ log( ˆλ i ) (5) where the (ordered) eigenvalues > ˆλ >... > ˆλ p > 0 solve the deterinantal equation det[λs S 0 S00 S 0 ] = 0. Let B(s) denote a (p r)-diensional standard Brownian otion process. Then, as T, { ( ) } S r trace dbf F F F db (6) where the stochastic process F (s) is defined for each case as follows: Case : F (s) = B(s); Case 2: F (s) = (B(s), ) ; Case 3: F (s) = [ ( B(s) 0 B ) ] (7), s 2 A proof of Equation (6) can be found in Theore. of Johansen [5]. The asyptotic distribution given in Equation (6) has been found to provide a poor approxiation to the finite saple distribution of S r in a nuber of cases leading to alternative approaches being developed. Recent work has suggested bootstrap techniques as a way of iproving inference concerning the cointegration rank in finite saples, for exaple Swensen [4] and Cavaliere, Rahbek and Taylor [5], while Bartlett corrections have been proposed by Johansen []. The idea behind Bartlett correction is to adjust the statistic S r so that its finite saple distribution is closer to the liiting distribution. To see how this works, suppose it is possible to expand the expectation of S r as E(S r ) = a 0 + a T + O(T 2 ) (8) where a 0 denotes the liit of the expectation as T and a is either a known constant (typically a function of the odel paraeters) or can be estiated consistently fro the saple data. Then the adjusted statistic S B r = ( + a ) S r (9) a 0 T can be shown to satisfy E(S B r ) = a 0 + O(T 2 ), thereby iproving the accuracy of the liiting distribution as an approxiation to the finite saple distribution (at least in ters of the ean of the

6 Econoetrics 205, distribution). Johansen [] derives expressions for the Bartlett correction factor which depend on the paraeters of the odel in a coplicated way and, hence, ust be estiated fro the saple data. Siulations reveal that the adjusted statistic iproves the size of the cointegration rank test in a large part of the paraeter space. A procedure related to the Bartlett approach is the jackknife. It achieves the sae order of reduction in the bias of the statistic but does not require a foral derivation of the precise ters in the expansion in Equation (8), erely relying on the existence of such a representation. The idea is to cobine the statistic S r in a linear cobination with the ean of a set of statistics obtained fro sub-saples, the weights being chosen to eliinate the first order bias ter a /T. Suppose, then, that the full saple of observations is divided into non-overlapping sub-saples, each sub-saple containing l = T/ observations. If S rj denotes the statistic coputed fro sub-saple j then the jackknife statistic is defined by S J r, = S r S rj (0) Provided that E(S rj ) = a 0 + a l + O(l 2 ) for j =,..., and l = O(T ), it is straightforward to show that E(Sr,) J = a 0 + O(T 2 ) by substitution of the relevant expressions. Hence both statistics, Sr B and Sr,, J achieve the sae order of bias reduction but by different eans. Although valid in cases 2 and 3, the above arguent concerning the jackknife statistic Sr, J falters in case on the assuption that the sub-saple statistics, S rj (j =,..., ), all share the sae expansion in Equation (8) as the full-saple statistic S r. It has been shown by Chabers and Kyriacou [9,20] that, in a univariate setting with a unit root, the sub-saple statistics (for j ) have different properties to the full saple statistic in the liit due to the stochastic order of agnitude of the pre-sub-saple value, and the sae phenoenon also arises here in case. The iplication is that the liiting distributions of the sub-saple statistics S rj (at least for j ) will differ fro that of S r ; the expansions of E(S rj ) will differ fro that of E(S r ); and, hence, the jackknife statistic (as defined above) will not fully eliinate the O(T ) ter in the bias. These probles do not arise in cases 2 and 3 because the presence of the intercept and/or tie trend ensures that the distributions are invariant to the initial (pre-sub-saple) conditions. Although it is possible to overcoe these probles in case by siply subtracting y (j )l fro the observations in sub-saple j an idea proposed in the univariate unit root setting by Chabers and Kyriacou [9] we do not pursue this avenue any further in view of the liited applicability of case in practice. Instead, we focus on the application of the jackknife correction in the ore epirically relevant cases 2 and 3. In order to econoise on notation it is convenient to define the functional { ( ) } Q(U, V, δ) = trace duv V V V du δ where U(s) and V (s) are vector stochastic processes defined on s δ, and to define the intervals δ 0 = [0, ] and δ j, = [(j )/, j/] (j =,..., ). With this notation the liiting distribution in Equation (6), for exaple, can be represented as S r Q(B, F, δ 0 ) δ j= δ

7 Econoetrics 205, 3 36 The foral stateent of the ain result is as follows. Theore. Let y,..., y T be generated according to Equation (2) and let Assuption hold. Then, as T : (a) If is fixed: Case 2. S rj Q(B, F 2, δ j, ) (j =,..., ), and S J r, Q(B, F 2, δ 0 ) Q(B, F 2, δ j, ) j= where F 2 (s) = [B(s), ]. Furtherore, Q(B, F 2, δ j, ) and Q(B, F 2, δ k, ) are independent for j k. Case 3. S rj Q(B, F j,, δ j, ) (j =,..., ), and S J r, Q(B, F 3, δ 0 ) Q(B, F j,, δ j, ) j= where and F j, (s) = [( F 3 (s) = [ ( B(s) B(s) 0 B ) ], s 2 B(s)ds), s j 2 ], j =,..., Furtherore, Q(B, F j,, δ j, ) and Q(B, F k,, δ k, ) are independent for j k. (b) If + T 0: Case 2. S J r, Q(B, F 2, δ 0 ). Case 3. S J r, Q(B, F 3, δ 0 ). Theore (a) shows that, when the nuber of sub-saples is fixed, the liiting distribution of the jackknife-corrected statistic is the sae linear cobination of the liiting distributions of the full- and sub-saple statistics. Note that the length of each sub-saple, l = T/, increases with T for fixed. However, when is allowed to increase with T but at a slower rate, Theore (b) shows that the liiting distribution is equivalent to that of the full-saple statistic alone. In this case note that, if T 0 as T, then l = T. In order to use these distributions for inference it is necessary to obtain the appropriate critical values; these are provided in Section 3 for a range of values of and p r. Further analysis of the finite saple properties of the jackknife-corrected statistics is provided in the next section by eans of Monte Carlo siulations.

8 Econoetrics 205, Siulation Results 3.. Critical Values of Liiting Distributions The liiting distributions of the jackknife statistic S J r, presented in Theore for fixed values of the jackknife paraeter are nonstandard and, in order to be useful in practice, it is necessary to obtain the appropriate critical values. Tables and 2 provide the 90%, 95% and 99% points of the distributions for values of p r ranging fro to 2 for the epirically relevant cases 2 and 3, respectively, and for values of {2, 3, 4, 5, 6, 8, 0, 2, 6, 20}. A total of 00,000 replications were carried out with a saple size of T = ax{200, 00 } spanning the interval [0, ] ensuring that when 2 each sub-saple (of length l = T/) contains 00 points. The ethod described in Chapter 5 of Johansen [5] was eployed, with suitable odifications to allow for the sub-saple statistics used in constructing the jackknife corrections; details are provided in the Appendix. It can be seen that, in all cases, the critical values for S J r, are larger than those for S r that are reported in, for exaple, Johansen [5], Doornik [6] and MacKinnon, Haug and Michelis [7]. The critical values are also seen to decrease as increases for a given value of p r. Table. Percentage points of liiting distributions of S J r,: case 2. : p r 90% p r 95%

9 Econoetrics 205, Table. Cont. : p r 99% Table 2. Percentage points of liiting distributions of S J r,: case 3. : p r 90% p r 95%

10 Econoetrics 205, Table 2. Cont. : p r 99% Finite Saple Properties The finite saple size and power properties of the jackknife-corrected test statistics were investigated using the siulation odel adopted by Cavaliere, Rahbek and Taylor [5] (denoted CRT2) for the purpose of evaluating their bootstrap procedure. The odel takes p = 4 and is given by y t = αβ y t + Γ y t + ɛ t, t =,..., T where ɛ t is a vector of norally distributed independent rando variables with covariance atrix I 4, the saple size T {50, 00, 200}, and the initial condition is y 0 = y 0 = 0. The short-run adjustent atrices are defined as α = (a, 0, 0, 0) and Γ = γ δ 0 0 δ γ γ γ

11 Econoetrics 205, where a, γ and δ are scalar paraeters defined below for each of the three data generation processes (DGPs) considered: DGP: a = 0.4, β = (, 0, 0, 0), γ = 0.8, δ {0, 0.2}. DGP2: a = 0.4, β = (, 0, 0, 0), γ = 0.5, δ {0, 0.2}. DGP3: a = 0, δ = 0, γ {0, 0.5, 0.8, 0.9}. In DGPs and 2 there is a single cointegrating vector while in DGP3 there is no cointegration and y t is an I() VAR(2) process (or, equivalently, y t is an I(0) VAR() process). The for of cointegration in DGPs and 2 was considered in CRT2 and iplies that y t is I(0). The value of δ that appears in the atrix Γ was used by CRT2 because it is related to soe auxiliary conditions relevant for the bootstrap procedure of Swensen [2]. These conditions are satisfied for δ = 0 but not for δ = 0.2. CRT2 also included two additional values of δ (equal to 0. and 0.3) but, as will be seen, the value of δ does not have a ajor ipact on the test procedures under consideration and so we restrict attention to just two of the four values used in CRT2. Note that, in DGP3, δ = 0 and the atrix Γ is diagonal with the scalar γ foring the diagonal eleents. It is also necessary for the DGPs to satisfy the three parts of Assuption. The first requires the roots of the equation det[a(z)] = 0 to have odulus greater than or equal to one, where in this case A(z) = ( z)i 4 αβ z Γ ( z)z. In DGPs and 2 there are three unit roots and in DGP3 there are four; the oduli of the non-unit roots are reported in Table 3, where it can be seen that Assuption (i) is satisfied in all cases. Coparing DGP with DGP2, the effect of reducing γ fro 0.8 to 0.5 is to increase the odulus of each of the non-unit roots. In DGP3, increasing the paraeter γ reduces the non-unit roots towards unity, and in the extree case of γ =, y t becoes an I(2) process. It is well known that the rank test perfors poorly as this extree case is approached; see, for exaple, the siulation evidence in Johansen (2002). Note that, when γ = 0, there are no roots in addition to the four unit roots because, in this case, A(z) = ( z)i 4 and hence det[a(z)] = ( z) 4. Assuption (ii) is obviously satisfied, while it can be shown that det[α Γβ ] = ( γ) 3 and hence Assuption (iii) is satisfied provided γ. A total of seven test statistics for cointegration rank were considered. The first is the standard (unadjusted) trace statistic S r defined in Equation (5). The second uses the sall saple correction proposed by Reinsel and Ahn [2]; the resulting statistic, denoted Sr RA, is defined by S RA r = (T pk) p i=r+ log( ˆλ i ) = (T pk) S r () T The third statistic is the Bartlett-corrected statistic defined in Equation (9); full details concerning coputation of the correction factors can be found in Johansen []. The fourth ethod is based on the bootstrap procedures of CRT2. The bootstrap saples are obtained by estiating the VECM under the null hypothesis, checking that the roots of the estiated atrix polynoial equation det[a(z)] = 0 satisfy Assuption (i), and then generating a total of N BS saples recursively using an appropriate

12 Econoetrics 205, ethod. In the siulations reported here a wild bootstrap was eployed in which, if ˆɛ it denotes eleent i of the residual vector ˆɛ t, then the residuals used for the bootstrap saples were of the for ( ) T ˆɛ BS it = u it ˆɛ it T ˆɛ it, i =,..., 4, t =,..., T t= where the u it are independent noral variates. The statistic S r is coputed in each bootstrap saple and the critical value is obtained fro the distribution of S r over the N BS boostrap saples. We denote this test by Sr BS but ephasise that the test statistic is actually S r which is copared with the critical value fro the finite saple bootstrap distribution rather than the critical value fro the asyptotic distribution. Full details of the procedure can be found in CRT2. Table 3. Moduli of non-unit roots in siulations. δ/γ Moduli DGP DGP DGP In addition to the above statistics, three versions of the jackknife statistic are considered. The first is Sr, J defined in Equation (0). This was coputed for a range of values of where practicable, although we report ainly the results for = 2; details of how the tests perfor for other values of are also provided. The reaining two jackknife statistics are based on sall saple adjustents to either the full saple statistic S r and/or the sub-saple statistics S rj upon which the jackknife is based. In particular the two additional jackknife statistics are defined by S J r, = S J2 r, = SRA r SRA r in which the sall saple adjusted sub-saple statistics are defined analagously to Equation () by S RA rj = j= j= S rj S RA rj (l pk) S rj, j =,...,. l The first of these statistics uses the sall saple adjustent purely on S r while the second also uses it on the sub-saple statistics.

13 Econoetrics 205, A total of R = 0, 000 replications were perfored for each cobination of paraeter values for each DGP and, as in CRT2, the VAR odel was fitted with a restricted intercept (case 2). The bootstrap procedure is the ost coputationally intensive coponent in the siulations, requiring a sufficiently large nuber (N BS ) of bootstrap saples in each replication in order to copute the critical value fro the bootstrap distribution; CRT2, for exaple, set N BS = 399. The bootstrap coputations are therefore O(RN BS ) copared to O(R) for the other statistics. While using a large nuber of bootstrap saples poses no proble in a single epirical application, it is ore of a coputational (and tie) burden when coputing a large nuber of bootstrap saples for each one of a large nuber of Monte Carlo replications. We therefore eployed the approach of Davidson and MacKinnon [22] and Giacoini, Politis and White [23] and used only one bootstrap saple per replication, i.e., N BS =. Instead of using a large nuber of bootstrap saples to deterine the critical value fro the bootstrap distribution in each replication, the critical value is obtained fro the bootstrap distribution across the R Monte Carlo replications. This warp-speed ethod reduces substantially the nuber of bootstrap coputations fro O(RN BS ) to O(R) in accordance with the other non-bootstrap statistics. The siulation results are suarised in Tables 4 7; in all cases the tests are based on a noinal size of 5%. Table 4 contains the epirical size of each of the seven test statistics in the case of the VAR with a single cointegrating vector. The value of δ has a relatively sall ipact on the perforance of the tests but the reduction in γ fro 0.8 to 0.5 has a uch larger ipact. It is apparent that, in all DGPs, the unadjusted Johansen statistic S has large size distortions, with the epirical size being as large as 45% in DGP with δ = 0.2 and T = 50. The sall saple adjustent that results in S RA reduces the size closer to its noinal level in all cases, particularly in DGP 2 (where γ = 0.5) but less so in DGP (γ = 0.8) where size distortions reain. The Bartlett correction has a tendency to over-copensate, leading to epirical sizes below 5% (and around 2% for T = 50) in ost cases. The bootstrap produces sizes around 5% in DGP but in DGP 2 the epirical size tends to be slightly lower than the noinal size. The jackknife statistic S,2 J anages to reduce the size towards the 5% level copared to the unadjusted statistic S with epirical sizes around 6% 7% in DGP 2 and a bit higher in DGP. The sall saple adjustent in S,2 J reduces the epirical size in all cases, copared to S,2, J while S,2 J2 produces sizes close to the noinal level in DGP 2 but shows little iproveent (if any) over S,2 J in DGP. The power perforance of the tests in the cointegrated VAR is suarised in Table 5 in which the probability of rejecting the null hypothesis that r = 0 is reported. Beginning with the unadjusted statistic S 0, the high power at the saller saple sizes in DGP is a reflection of the large size distortions reported in Table 4. All of the adjusted statistics are less powerful than S 0 but it should be reebered that they do have better size properties. The statistic S0,2 J has particularly low power for T = 50 in DGP 2. The size properties of the tests in a non-cointegrated VAR are reported in Table 6. As the value of γ increases fro 0 to 0.9 the unadjusted statistic S 0 suffers fro huge size distortions, rising to 92% for T = 50 when γ = 0.9. The size properties of the adjusted statistcs are all better than S 0 with the bootstrap test controlling size best over this range of paraeters. The Bartlett adjustent again tends to reduce epirical size to below its noinal level as γ increases while, for the jackknife statistics, S0,2 J2

14 Econoetrics 205, perfors best for saller values of γ while S J 0,2 produces the best perforance of the three for larger values of γ. Table 4. Epirical size: cointegrated VAR. δ T S S RA S B S BS S J,2 S J,2 S J2,2 DGP DGP Table 5. Epirical power: cointegrated VAR. δ T S 0 S RA 0 S B 0 S BS 0 S J 0,2 S J 0,2 S J2 0,2 DGP DGP

15 Econoetrics 205, Table 6. Epirical size: non-cointegrated VAR (DGP3). γ T S 0 S RA 0 S B 0 S BS 0 S J 0,2 S J 0,2 S J2 0, Table 7. Epirical size of S J, for varying. δ/γ T DGP DGP DGP The results for the jackknife tests in Tables 4 6 are based on = 2 sub-saples, but it is of interest to ascertain how the perforance of the tests is affected using different values of. For T = 50 there is little scope to increase uch further; with = 2 each sub-saple has only l = 25 observations,

16 Econoetrics 205, so increasing soon akes sub-saple estiation infeasible. However, for larger saple sizes soe experientation is possible, and so Table 7 reports the epirical size of S J, for {2, 4, 5} when T = 00 and {2, 4, 5, 8, 0} when T = 200; in each case, for the largest value of, the sub-saples contain just l = 20 observations. Table 7 shows that the epirical size of S J, is rearkably robust to the value of, with the exception of DGP3 when γ = 0.9. To suarise the siulation results, it appears that rank test statistics based on soe for of correction factor can provide size iproveents over the unadjusted Johansen statistic while still aintaining good power properties, although a bootstrap approach offers the ost consistent perforance over the range of DGPs considered. It should be stressed, however, that the corrected statistics and the bootstrap operate in rather different ways. All of the corrected statistics whether the correction is a siple sall saple adjustent, a (paraetric) Bartlett correction or a (nonparaetric) jackknife correction ai to adjust the raw statistic in such a way that the distribution of the corrected statistic atches better the asyptotic distribution, the critical values of which the corrected statistic is copared with. The bootstrap, on the other hand, uses as the test statistic the unadjusted statistic itself, but by generating bootstrap saples whose size is equal to the given finite nuber of observations, uses critical values fro the finite saple bootstrap distribution against which to copare the statistic. The evidence obtained here suggests that the latter approach is the ost robust in practice. 4. Conclusions This paper has investigated the asyptotic properties and finite saple perforance of jackknife-corrected test statistics for cointegration rank in a VAR syste. In particular, the liiting distributions of jackknife-corrected test statistics have been derived for the two trend specifications of ost epirical relevance; the asyptotic critical values for these cases have been tabulated; and the finite saple size and power properties of the jackknife-corrected tests have been copared with the usual (unadjusted) rank test statistic as well as statistics using various sall saple corrections and a bootstrap approach. The siulations reveal that all the corrected statistics, including the jackknife variants, can provide size iproveents over the unadjusted statistic while still aintaining good power properties, although a bootstrap approach offers the ost consistent perforance over the range of DGPs considered. There are a nuber of ways in which the analysis of this paper can be built upon. In practice the precise for of the VAR (i.e., the specification of the deterinistic trend function and the nuber of lags) is unknown, and various pre-tests are often conducted, including the use of inforation criteria to deterine the VAR order. This has an ipact on the perforance of the rank tests, and it would be of interest to ascertain how well the jackknife ethods perfor relative to other tests in such a scenario. Another potentially fruitful area of investigation concerns the use of jackknife ethods in estiating the cointegrating paraeters theselves. Additionally, bootstrap ethods have been shown to be adaptable to situations where heteroskedasticity (both conditional and unconditional) is present as well as breaks in variance and correlations; see, for exaple, Cavaliere, Rahbek and Taylor [24,25]. Jackknife ethods have also been found to be robust to conditional heteroskedasticity in stationary autoregressions by

17 Econoetrics 205, 3 37 Chabers [8], and so a further coparison with bootstrap ethods would be of interest in the context of cointegration. Such avenues are left for future work. Acknowledgents I a grateful to the Editor-in-Chief, Kerry Patterson, and three anonyous referees for helpful coents on this paper. In particular I thank one of the referees for pointing out the independence properties of the functionals Q(B, F 2, δ j, ) and Q(B, F j,, δ j, ) that appear in parts (a) and (b), respectively, of Theore. I also thank Giuseppe Cavaliere for providing Matlab code for ipleenting the bootstrap ethods of Cavaliere, Rahbek and Taylor [5], upon which I was able to base y own Gauss code, and Rob Taylor for suggesting I explore warp-speed ethods of speeding up the siulations involving the bootstrap. The initial research upon which this paper is based was funded by the Econoic and Social Research Council under grant nuber RES Appendix A.. Proof of Theore (a) Johansen [5] (Theore.) shows that S r Q(B, F, δ 0 ) as T under the stated conditions, where F (s) is defined in Equation (7) for each case. Taking each of the two relevant cases in turn: Case 2 Here, S r Q(B, F 2, δ 0 ) and the sub-saple statistics have the sae distribution defined on δ j,, i.e., S rj Q(B, F 2, δ j, ). The result for Sr, J follows straightforwardly as is fixed. To deonstrate the independence of Q(B, F 2, δ j, ) and Q(B, F 2, δ k, ) for j k, consider [ j/ ] j/ Q(B, F 2, δ j, ) = trace db(s)f 2 (s) F 2 (s)f 2 (s) ds } j/ F 2 (s)db(s) and recall that F 2 (s) = [B(s), ]. This expression is a function of B(s) for s δ j, which is clearly not independent of the process B(r) for r δ k, that enters Q(B, F 2, δ k, ). However, let A j, = I p r B(s)ds 0 p r where 0 p r denotes a (p r) vector of zeros. We can then write { Q(B, F 2, δ j, ) = trace db(s)f 2 (s) A j, [ ] j/ A j, F 2 (s)f 2 (s) A j,ds A j, F 2 (s)db(s)

18 Econoetrics 205, which is a function of the process A j, F 2 (s) = B(s) B(s)ds This shows that Q(B, F 2, δ j, ) can be represented in ters of the quasi-deeaned process B j, (s) = B(s) the process B k, (r) = B(r) B(s)ds for s δ j,, and it follows that Q(B, F 2, δ k, ) can be written in ters of k/ (k )/ B(r)dr for r δ k,. The two functionals of interest will be independent if B j, (s) and B k, (r) are independent which, due to the being Gaussian processes, only requires their covariance to be zero. The covariance of interest is ( ) j/ C j,k = E (B j, (s)b k, (r) ) = E (B(s)B(r) ) E B(s)dsB(r) E ( B(s) k/ (k )/ B(r) dr ) + 2 E ( k/ B(s)ds (k )/ B(r) dr Suppose, without loss of generality, that k > j which iplies r > s. Then we have E (B(s)B(r) ) = in(s, r)i p r = si p r, ( ) j/ E B(s)dsB(r) = in(s, r)dsi p r ) ( E B(s) k/ (k )/ ) B(r) dr = = k/ (k )/ sdsi p r = j 2 I p r, 2 in(s, r)dri p r E ( k/ B(s)ds (k )/ B(r) dr ) k/ = s dri p r = s (k )/ I p r, = k/ (k )/ in(s, r)drds Cobining these expressions we find that ( C j,k = s j 2 2 = s + 2 j 2 2 as required, where 0 k denotes a k k atrix of zeros. k/ sds dr = j 2 (k )/ 2 I p r ) I p r = 0 (p r) (p r)

19 Econoetrics 205, Case 3 For the full-saple statistic, S r Q(B, F 3, δ 0 ), while the appropriate process F j, for the sub-saples is obtained as the residual fro a continuous tie projection of B(s) and s on a constant on the interval δ j, (the process F 3 (s) is obtained in the sae way but with the projections taking place on δ 0 ). The first projection is given by B(s) B(s)ds ds = B(s) / B(s)ds = B(s) B(s)ds which provides the first p r eleents of F j, (s). The second projection residual, which is the final eleent of F j, (s), is obtained as s sds ds = s ( ) j 2 / 2 / = s j 2 The sub-saple statistics satisfy S rj Q(B, F j,, δ j, ) and the result for S J r, follows. The independence of Q(B, F j,, δ j, ) and Q(B, F k,, δ k, ) follows fro the arguents used for case 2 above, noting that F j, already contains the process B j,. (b) In both cases, when as T such that + T 0 (so that l = T ), it follows that S J r, = S r + o p () and the results are straightforward. A.2. Method of Siulation of Liiting Distributions The objective is to siulate distributions of the for { ( Q(B, F, δ) = trace dbf δ δ F F ) δ F db } where B(s) is a (p r)-diensional standard Brownian otion process and F (s) is a stochastic processes whose precise for is given in Theore and depends on the specification of the deterinistic coponent in the odel. Consider, first, Q(B, F, δ 0 ), and define the (p r)-diensional process y t = ɛ t, t =,..., T, where the eleents of ɛ t are independent standard noral rando variates and y 0 = 0. Then, following Johansen [5] (chapter 5), the distribution of Q(B, F, δ 0 ) is approxiated by T ˆQ T = trace ɛ t P t t= ( T t= for an appropriate choice of P t (t =,..., T ), as follows: Case 2: P t = ( y t, ) ; ) T P t P t P t ɛ t Case 3: P t = [ (y t ȳ), t 2 (T + )], where ȳ = T T t= y t. t=

20 Econoetrics 205, In a siilar way the distributions of the sub-saple statistics are siulated using ˆQ T,j, = trace jl t=(j )l+ ɛ t P t jl t=(j )l+ P t P again subject to an appropriate choice of P t (t = (j )l +,..., jl): Case 2: P t = [ y t, ] ; t jl t=(j )l+ P t ɛ t Case 3: P t = [ (y t ȳ j ), t ( ] j 2) l, 2 where ȳj = l jl t=(j )l+ y t. The siulated values are cobined to approxiate the distribution in Theore (a) using ˆQ T j= ˆQ T,j, for a range of values of. Conflicts of Interest The author declares no conflict of interest. References. Engle, R.F.; Granger, C.W.J. Cointegration and error correction: Representation, estiation and testing. Econoetrica 987, 55, Johansen, S. Statistical analysis of cointegration vectors. J. Econ. Dyn. Control 988, 2, Johansen, S. Estiation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive odels. Econoetrica 99, 59, Anderson, T.W. Estiating linear restrictions on regression coefficients for ultivariate noral distributions. Ann. Math. Stat. 95, 22, Johansen, S. Likelihood-Based Inference in Cointegrated Vector Autoregressive Models; Oxford University Press: Oxford, UK, Doornik, J.A. Approxiations to the asyptotic distribution of cointegration tests. J. Econ. Surv. 998, 2, MacKinnon, J.G.; Haug, A.A.; Michelis, L. Nuerical distribution functions of likelihood ratio tests for cointegration. J. Appl. Econo. 999, 4, Toda, H.Y. Finite saple properties of likelihood ratio tests for cointegrating ranks when linear trends are present. Rev. Econ. Stat. 994, 76, Toda, H.Y. Finite saple perforance of likelihood ratio tests for cointegrating ranks in vector autoregressions. Econo. Theory 995,, Nielsen, B. On the distribution of likelihood ratio test statistics for cointegration rank. Econo. Rev. 2004, 23, 23.. Johansen, S. A sall saple correction for the test of cointegrating rank in the vector autoregressive odel. Econoetrica 2002, 70,

21 Econoetrics 205, Reinsel, G.C.; Ahn, S.K. Vector autoregressive odels with unit roots and reduced rank structure: Estiation, likelihood ratio test, and forecasting. J. Tie Series Anal. 992, 3, Reiers, H.E. Coparisons of tests for ultivariate cointegration. Stat. Pap. 992, 33, Swensen, A.R. Bootstrap algoriths for testing and deterining the cointegration rank in VAR odels. Econoetrica 2006, 74, Cavaliere, G.; Rahbek, A.; Taylor, A.M.R. Bootstrap deterination of the co-integration rank in VAR odels. Econoetrica 202, 80, Quenouille, M.H. Notes on bias in estiation. Bioetrika 956, 43, Tukey, J.W. Bias and confidence in not-quite large saples. In Proceedings of the Institute of Matheatical Statistics, Aes, USA, 3 5 April Chabers, M.J. Jackknife estiation of stationary autoregressive odels. J. Econo. 203, 72, Chabers, M.J.; Kyriacou, M. Jackknife Bias Reduction in the Presence of a Unit Root; Discussion Paper 685, University of Essex Departent of Econoics: Colchester, UK, Chabers, M.J.; Kyriacou, M. Jackknife estiation with a unit root. Stat. Probab. Lett. 203, 83, Swensen, A.R. Corrigendu to Bootstrap algoriths for testing and deterining the cointegration rank in VAR odels. Econoetrica 2009, 77, Davidson, R.; MacKinnon, J.G. Iproving the reliability of bootstrap tests with the fast double bootstrap. Coput. Stat. Data Anal. 2007, 5, Giacoini, R.; Politis, D.N.; White, H. A warp-speed ethod for conducting Monte Carlo experients involving bootstrap estiators. Econo. Theory 203, 29, Cavaliere, G.; Rahbek, A.; Taylor, A.M.R. Testing for co-integration in vector autoregressions with non-stationary volatility. J. Econo. 200, 58, Cavaliere, G.; Rahbek, A.; Taylor, A.M.R. Bootstrap deterination of the co-integration rank in heteroskedastic VAR odels. Econo. Rev. 204, 33, c 205 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the ters and conditions of the Creative Coons Attribution license (