Asymptotic theory for cointegration analysis when the cointegration rank is deficient

Bernstein, David H.; Nielsen, Bent Article Asymptotic theory for cointegration analysis when the cointegration rank is deficient Econometrics Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Bernstein, David H.; Nielsen, Bent (2019) : Asymptotic theory for cointegration analysis when the cointegration rank is deficient, Econometrics, ISSN 2225-1146, MDPI, Basel, Vol. 7, Iss. 1, pp. 1-24, https://doi.org/10.3390/econometrics7010006 This Version is available at: https://hdl.handle.net/10419/247506 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ econometrics Article Asymptotic Theory for Cointegration Analysis When the Cointegration Rank Is Deficient David H. Bernstein 1and Bent Nielsen 2,∗ 1Department of Economics, University of Miami, Coral Gables, FL 33146, USA; [email protected] 2 Department of Economics & Nuffield College & Programme on Economic Modelling, University of Oxford, Oxford OX1 1NF, UK *Correspondence: [email protected] Received: 3 May 2018; Accepted: 8 January 2019; Published: 18 January 2019 Abstract: We consider cointegration tests in the situation where the cointegration rank is deficient. This situation is of interest in finite sample analysis and in relation to recent work on identification robust cointegration inference. We derive asymptotic theory for tests for cointegration rank and for hypotheses on the cointegrating vectors. The limiting distributions are tabulated. An application to US treasury yields series is given. Keywords: cointegration; rank deficiency; weak identification JEL Classification: C32 1. Introduction Determination of the cointegration rank is an important part of analyzing the cointegrated vector autoregressive model in the framework of Johansen (1988,1991,1995), Johansen and Juselius (1990), and Juselius (2006). We consider the rank deficient case where the cointegration rank of the data generating process is smaller than the rank used in the statistical analysis. In that case, the data generating process has more unit roots than the number of unit roots imposed in the statistical analysis and the usual asymptotic theory fails. We provide asymptotic theory for cointegration rank tests and tests on cointegration vectors along with simulated tables of the asymptotic distributions. Cointegration analysis is conducted in three steps. First, the specification of the model is checked. Second, the rank is determined using a sequential procedure using Dickey-Fuller type distributions. Third, the cointegrating vectors are estimated and restrictions can be tested using standard inference. Asymptotic theory shows that estimated rank is consistent in the sense that the probability that the estimated rank is not equal to the true rank equals the size of tests, whereas the probability that the estimated rank is too small vanishes, see Johansen (1992,1995) and Paruolo (2001). Hence, the rank deficiency problem does not arise in the asymptotic analysis. In practice, rank deficiency matters in two ways. The asymptotic theory often suffers from considerable finite sample distortion. Further, if an investigator wants to focus on the inference on the cointegrating relations then problems can arise if the rank is taken as known when in fact it is deficient. These problems mirror those of instrumental variable estimation with weak instruments, see Mavroeidis et al. (2014). When conducting inference on the cointegrating vector under near rank deficiency the parameters are weakly identified. At the extreme when testing on the cointegrating vector in the case of a deficient rank the model is mis-specified. This problem arises in cointegration as well as in instrumental variable estimation. In both cases maximum likelihood is conducted using reduced rank regression. The weak identification problem has attracted considerable attention in the instrumental variable literature, see for instance Mavroeidis et al. (2014). Khalaf and Urga (2014) discussed the weak Econometrics 2019,7, 6; doi:10.3390/econometrics7010006 www.mdpi.com/journal/econometrics Econometrics 2019,7, 6 2 of 24 identification problem for cointegration, that is when testing for a known cointegrating vector in the nearly rank deficient situation. These authors investigate various methods to adjust the asymptotic distribution in the weak identification case. This includes a bounds-based critical value suggested by Dufour (1997). This method requires knowledge of the asymptotic theory for the rank deficient case, which we provide here. The practical problem of ignoring rank deficiency is illustrated using yield curve data. The expectation hypothesis is often interpreted as follows. Interest rates at different maturities are integrated series, but cointegrate so that spreads are stationary. Spreads are often found to be non-stationary. Thus, it is quite possible that a pair of interest rates do not cointegrate. An investigator may proceed by assuming cointegration when there is none, so that the rank is deficient, and conduct inference on the coefficients on the alleged cointegrating vector using standard inference. Our theory shows that the inference is then severely distorted. When the rank is deficient or nearly deficient it is incorrect to use standard inference on the cointegrating vectors. Nonetheless, applying standard inference in the particular example leads to marginal rejection of the hypothesis. Applying the bounds test of Khalaf and Urga (2014) shifts the distribution to the right and there is not much power to reject a hypothesis. If the rank is deficient, which is possible in the example, the alleged cointegrating vector cannot be cointegrating. Rank deficiency also matters when the rank is determined empirically. Different asymptotic distributions arise in the standard case and when the rank is deficient. The asymptotic distribution tends to give a very good approximation to the finite sample distribution when the rank is far from being deficient, see for instance Nielsen (1997,2004) When the parameters are in the vicinity of rank deficiency the finite sample distribution tends to be a combination of the two asymptotic distributions. When the parameters are not too close to the rank deficient case a Bartlett correction using a fixed parameter second-order asymptotic expansion works very well, see Johansen (2000,2002) Bootstrap solutions have been discussed in simulation studies by Fachin (2000); Gredenhoff and Jacobson (2001); Swensen (2004); Cavaliere et al. (2012). When the parameters are closer to rank deficient a local-to-unity asymptotic expansion gives an improvement, see Nielsen (2004) for the cointegration case and Nielsen (1999,2001) for the corresponding instrumental variable case. A starting point for the finite sample analysis is knowledge of the fixed-parameter first-order asymptotic theory across the parameter space, including rank deficient cases. We discuss the asymptotic theory for models without and with deterministic terms in Sections 2 and 3, respectively. The implications for finite sample analysis and the weakly identified case are discussed in Section 4along with an application to US treasury zero coupon yields. Section 5concludes. Proofs are given in an Appendix A. 2. The Model without Deterministic Terms We consider the Gaussian cointegrated vector autoregressive model in the case with no deterministic terms. The asymptotic theory for tests for reduced cointegration rank and for a known cointegrating vector is derived when the rank is deficient. Finally, we analyze the case of near rank deficiency. 2.1. Model and Hypotheses Consider a p -dimensional time series Xt for t= 1 −k , . . . , 0, 1, . . . T . The unrestricted vector autoregressive model can be written as ∆Xt=ΠXt−1+ k−1 ∑ i=1 Γi∆Xt−i+εtfor t=1, . . . , T, (1) where the innovations εt are independent normal N p( 0, Ω) -distributed. The parameters Π , Γi , Ω are freely varying p-dimensional square matrices so that Ωis symmetric, positive definite. Econometrics 2019,7, 6 3 of 24 The hypothesis of reduced cointegration rank is formulated as Hz(r): rank Π≤r, (2) for some 0 ≤r≤p . The interpretation of the hypotheses follows from the Granger-Johansen representation presented in Section 2.2 below. The subscript z indicates that the model has a zero deterministic component. The rank hypotheses are nested so that Hz(0)⊂ · · · ⊂ Hz(r)⊂ · · · ⊂ Hz(p). (3) The rank deficiency problem arises when testing the hypothesis H z(r) when in fact the sub-hypothesis H z(r− 1 ) is satisfied. The rank is determined to be r if the hypothesis H z(r) cannot be rejected while the sub-hypothesis H z(r− 1 ) is rejected. As a short-hand we write H ◦ z(r) = H z(r)\ H z(r− 1 ) for this situation. The rank can be determined along the procedure outlined in Johansen (1992,1995) [Section 12.1] and Paruolo (2001). In practice, these decisions are often marginal, hence the need to study the asymptotic theory of test statistics in the rank deficient case. The rank hypothesis can equivalently be written as Hz(r):Π=αβ0, (4) where α and β are p×r matrices. The advantage of this formulation is that α and β vary in vector spaces. The formulation does, however, allow rank deficiency where the rank of Π is smaller than r . We follow Johansen (1991, Equation (2.2)) and refer to β as the cointegrating vectors. We find the terminology useful, although it is ambiguous. Indeed, for a particular data generating process where Π has rank less than r then the identity Π=αβ0 can be satisfied while columns of β may not be row-eigenvectors of Π in which case β0Xt cannot be stationary. Even when Π has rank r then β0Xt is only (approximately) stationary under the I(1) condition introduced below. However, from a statistical viewpoint, the estimator of Π under the restriction of rank r will in a finite sample have rank r with probability one. In practice our only knowledge of the rank arises from inference. Johansen’s terminology appears to be focused on the statistical viewpoint which we will follow even when studying the rank deficient cases. The hypothesis of known cointegration vectors is Hz,β(r):Π=αb0, (5) for some unknown matrix α and a known matrix b , both of dimension p×r , so that b has full column rank. The standard analysis is concerned with the situation where α has full column rank, but in the rank deficient case, it has reduced column rank, so that the hypothesis H z(r− 1 ) is satisfied. When referring to b as the cointegrating vectors, we, once again, follow the terminology of Johansen (1991, Equation (3.1)) even though b0Xtcannot be stationary under rank deficiency. 2.2. Granger-Johansen Representation The Granger-Johansen representation provides an interpretation of the cointegration model that is useful in the asymptotic analysis. We work with the result stated by Johansen (1995, Theorem 4.2). The theorem requires the following assumption. I(1) Condition .Suppose rank Π=s where s≤p . Consider the characteristic roots satisfying 0 =det{A(z)} where A(z) = ( 1 −z)Ip−Πz−∑k−1 i=1Γizi( 1 −z) . Suppose there are p−s unit roots, and that the remaining roots are stationary roots, so satisfying |z|>1. Econometrics 2019,7, 6 4 of 24 The Granger-Johansen theorem assumes that a process satisfying the model (1) so that rank Π=r and we can write Π=αβ0 while the I(1) condition holds with s=r . The process then has the representation Xt=C t ∑ i=1 εi+St+τ, (6) where the impact matrix C for the random walk has rank p−r and satisfies β0C= 0 and Cα= 0, the process St can be given a zero mean stationary initial distribution and τ depends on the initial observations in such a way that β0τ= 0. In other words, the process Xt behaves like a random walk with cointegrating relations β0Xtthat can be given a stationary initial distribution. 2.3. Test Statistics The likelihood ratio test statistic for the reduced rank hypothesis H z(r) against the unrestricted model H z(p) is found by reduced rank regression, see Johansen (1995, Section 6). It can be described as a two-step procedure. First, the differences ∆Xt and the lagged levels Xt−1 are regressed on the lagged differences ∆Xt−i , i= 1, . . . , k− 1 giving residuals R0,t , R1,t . Secondly, the squared sample correlations, 1 ≥b λ1≥ · · · ≥ b λp≥ 0 say, of R0,t and R1,t are found, by computing product moments Sij =T−1∑T t=1Ri,tR0 j,t and solving the eigenvalue problem 0 =det(λS11 −S10S−1 00 S01) . The log likelihood ratio test statistic for the rank hypothesis is then LR{Hz(r)|Hz(p)}=−T p ∑ j=r+1 log(1−b λj). (7) Under the hypothesis of known cointegration vectors, the likelihood is maximised by least squares regression. The log likelihood ratio test statistic against the unrestricted model H z(p) is therefore given by LR{Hz,β(r)|Hz(p)}=−Tlog det{S00 −S01S−1 11 S10} det{S00 −S01b(b0S11b)−1b0S10}. (8) The log likelihood ratio statistic for the hypothesis of known cointegrating vector against the rank hypothesis is found by combining the statistics in (7) and (8), that is LR{Hz,β(r)|Hz(r)}=LR{Hz,β(r)|Hz(p)} − LR{Hz(r)|Hz(p)}. (9) The relationship will be useful in the asymptotic theory. For instance, Theorems 1and 2give the asymptotic distributions of LR{ H z(r)| H z(p)} and LR{ H z,β(r)| H z(p)} , respectively. From this we can derive an expression for the distribution of LR{ H z,β(r)| H z(r)} . When it comes to tabulation we will need to simulate all three distributions. This would be the case even if the former two statistics were independent. 2.4. Asymptotic Theory for the Rank Test In the asymptotic analysis it is possible to relax the assumption to the innovations. While the likelihood is derived under the assumption of independent, identically Gaussian distributed innovations less is needed for the asymptotic theory. Johansen (1995) assumes the innovations are independent, identically distributed with mean zero and finite variance and uses linear process results from Phillips and Solo (1992). This could be relaxed further to, for instance, a martingale difference assumption. However, for expositional simplicity we follow Johansen’s argument and assumptions. Econometrics 2019,7, 6 5 of 24 Theorem 1. Consider the rank hypothesis H z(r): rank Π≤r . Suppose H ◦ z(s) = H z(s)\ H z(s− 1 ) holds for some s≤r and that the I(1) condition holds for that s . Let Fu=Bu be a p−s -dimensional standard Brownian motion on [0, 1]. Let 1≥ρ1≥ · · · ≥ ρp−s≥0be the eigenvalues of the eigenvalue problem 0=det ρZ1 0FuF0 udu −Z1 0Fu(dBu)0Z1 0(dBu)F0 u(10) Then, for T →∞, LR{Hz(r)|Hz(p)}=−T p ∑ j=r+1 log(1−b λj)D → p−s ∑ j=r−s+1 ρj. (11) In the standard non-deficient situation where r=s the result reduces to the result of Johansen (1995, Theorem 6.1). The rank deficient case was also discussed by Johansen (1995, p. 158) and Nielsen (2004, Theorem 6.1). Table 1reports the asymptotic distribution of the rank test reported in Theorem 1. The simulation were done using Ox (Doornik 2007). The simulation design follows that of Johansen (1995, Section 15). That is, the stochastic integrals in (10) were descretized with T= 1000 and zero initial observations with one million repetitions. The table reports simulated quantiles and moments for r−s= 0, 1, 2 and p−r= 1, 2, 3, 4. However, the case of p−r= 1 and r−s= 0 are analytic values from Nielsen (1997) and where the quantiles were provided by Karim Abadir using his results in Abadir (1995). Bernstein (2014) reports values for higher dimensions. The 85% quantile has not been computed analytically in this case. Table 1. Quantiles, mean, and variance of LR{ H z(r)| H z(p)} , where the data generating process has rank s=rank Π≤r. r−s p −r50% 80% 85% 90% 95% 97.5% 99% Mean Var 0 1 0.60 1.88 — 2.98 4.13 5.32 6.94 1.14 2.22 2 5.48 8.48 9.31 10.44 12.30 14.07 16.34 6.09 10.61 3 14.39 18.94 20.13 21.70 24.22 26.54 29.37 15.02 25.13 4 27.29 33.35 34.88 36.91 40.04 42.93 46.45 27.93 45.66 1 1 0.36 1.13 1.38 1.74 2.35 2.98 3.81 0.67 0.70 2 4.27 6.25 6.78 7.50 8.65 9.76 11.14 4.61 4.66 3 11.92 15.20 16.04 17.14 18.88 20.50 22.48 12.31 13.22 4 23.47 28.09 29.25 30.76 33.10 35.21 37.83 23.89 26.96 2 1 0.30 0.97 1.18 1.48 1.98 2.47 3.11 0.56 0.48 2 3.93 5.57 6.01 6.59 7.51 8.38 9.46 4.18 3.24 3 11.04 13.82 14.53 15.46 16.91 18.24 19.87 11.34 9.63 4 21.84 25.83 26.82 28.11 30.09 31.91 34.13 22.18 20.21 The first panel of Table 1reports the distribution for the standard case where s=r . This corresponds to Table 15.1 of Johansen (1995). The second and third panel of Table 1report the distribution for the rank deficient case where s=r− 1 so r−s= 1 and where s=r− 2 so r−s= 2. The first entry in panel 2 for s=r− 1 and p−r= 1, so r−s= 1, corresponds to Table 6 of Nielsen (2004). It is seen that as the rank becomes more deficient the distribution shifts to the left. It should be noted that if the rank is non deficient, but the I(1) condition is not satisfied then the distribution would tend to shift to the right, see Nielsen (2004) for a discussion. The simulations reported in Table 8 of that paper indicates that the distribution is between these extremes if the rank is deficient and the I(1) condition fails. The rank test statistic in (7) has been analyzed analytically for the canonical correlation problem in cross-sectional models in Nielsen (1999,2001) This test also corresponds to the test for relevance in the instrument variable problem. In that case, analytic expressions are available when p= 2, r= 1 Econometrics 2019,7, 6 6 of 24 and s= 0, 1. When s= 1 we have a χ2 -distribution with mean 1 and variance 2. When s= 0 the mean is 0.429 and the variance is 0.575 −( 0.429 )2= 0.391, see Nielsen (1999). Thus, the impact of rank deficiency is similar to what is seen in Table 1for cointegration rank testing. 2.5. Asymptotic Theory for the Test on the Cointegrating Vectors In the analysis of the test for known cointegrating vectors, we focus on the situation where the data generating process has rank s= 0. In this situation the asymptotic distribution is relatively simple to describe, because it does not depend on the value of the hypothesized cointegrating vectors b . This is adequate for a discussion of aspects of situations considered in Khalaf and Urga (2014). If the rank is non-zero but deficient so 0 <s<r , then the data generating process will have cointegrating vectors β0 of dimension p×s and the asymptotic theory will depend on β0 and b . In practice, it is rare to test for simple hypotheses when there is more than one hypothesized cointegrating vector, so we do not pursue this complication. The analysis of the test for known cointegrating vectors is somewhat different from the analysis in Johansen (1995). His analysis is aimed at the situation where different restrictions are imposed on the cointegrating vectors. The argument then involves an intriguing consistency proof for the estimated cointegrating vectors. However, when testing the hypothesis of known cointegrating vectors the likelihood is maximized by the least squares method and the consistency argument is not needed. The asymptotic theory can then be described by the following result. Theorem 2. Consider the hypothesis H z,β(r):Π=αb0 , where α , b have dimension p×r and where α is unknown and b is known with full column rank. Suppose Hz(0)is satisfied, so that α=0and s =0, and that the I(1) condition is satisfied with s= 0. Let Bu be a p -dimensional standard Brownian motion on [ 0, 1 ] with components B1,uand B2,uof dimension r and p −r, respectively. Then, for T →∞, LR{Hz,β(r)|Hz(p)}D →tr{Z1 0dBuB0 u(Z1 0BuB0 udu)−1Z1 0Bu(dBu)0 −Z1 0dBuB0 1,u(Z1 0B1,uB0 1,udu)−1Z1 0B1,u(dBu)0}. (12) The convergence of the test statistic LR{Hz,β(r)|Hz(p)}holds jointly with the convergence for the rank test statistic LR{ H z(r)| H z(p)} , for s= 0, in Theorem 1. Thus, when s= 0the formula (9) implies that the limit distribution of the test statistic for known β within the model with rank of at most r can be found as the difference of the two limiting variables. Table 2reports the asymptotic distribution of the test for known cointegrating vector in the model where the rank is at most r . When s=r the asymptotic distribution is χ2 with r(p−r) degrees of freedom, see Johansen (1995, Theorem 7.2.1). When s= 0 the asymptotic distribution reported in Theorem 2applies. The simulation design is as before. It is seen that in the rank deficient case the distribution is shifted to the right. This matches the finite sample simulations reported by Johansen (2000, Table 2). Econometrics 2019,7, 6 7 of 24 Table 2. Quantiles, mean, and variance of LR{ H z,β(r)| H z(r)} , where the data generating process has rank s=rank Π≤r. p r s 50% 80% 85% 90% 95% 97.5% 99% Mean Var 2 1 1 0.45 1.64 2.07 2.71 3.84 5.02 6.63 1 2 0 2.62 5.44 6.22 7.30 9.05 10.75 12.96 3.31 8.71 3 2 2 1.39 3.22 3.79 4.61 5.99 7.38 9.21 2 4 0 5.80 9.42 10.40 11.71 13.82 15.77 18.27 6.42 15.53 3 1 1 1.39 3.22 3.79 4.61 5.99 7.38 9.21 2 4 0 6.79 10.58 11.57 12.89 15.02 17.00 19.49 7.33 17.52 Table 3reports the simulated asymptotic distribution of the test for known cointegrating vector in the model where the rank is unrestricted. The distribution is shifted to the right in the rank deficient case. Note, that the table reports the distribution of the convolution of the statistics simulated in Tables 1and 2, see (9). Thus, up to a simulation error the expectations reported in Tables 1and 2add up to the expectation reported in Table 3. In the full rank case r=s the statistics in Tables 1and 2are independent, as proved below, so also the variances are additive. Theorem 3. Consider the hypothesis H ◦ z,β(r) . Suppose H ◦ z(r) = H ◦ z(r)/ H ◦ z(r− 1 ) is satisfied and that the I( 1 ) condition holds with s=r . Then the rank test statistic LR{ H ◦ z(r)| H ◦ z(p)} and the statistic LR{ H ◦ z,β(r)| H ◦ z(r)} for testing a simple hypothesis on the cointegrating vector are asymptotically independent. The asymptotic distribution of the rank statistic LR{ H ◦ z(r)| H ◦ z(p)} is given in Theorem 1, while the statistic for the cointegrating vector LR{H◦ z,β(r)|H◦ z(r)}is asymptotically χ2{r(p−r)}. Table 3. Quantiles, mean, and variance of LR{ H z,β(r)| H z(p)} , where the data generating process has rank s=rank Π≤r. p r s 50% 80% 85% 90% 95% 97.5% 99% Mean Var 2 1 1 1.54 3.43 4.01 4.83 6.22 7.62 9.47 2.15 4.23 0 3.35 6.11 6.89 7.95 9.70 11.38 13.57 3.98 8.82 3 2 2 2.52 4.85 5.53 6.48 8.07 9.60 11.62 3.15 6.26 0 6.36 9.96 10.92 12.22 14.32 16.29 18.79 6.98 15.35 3 1 1 7.50 11.03 11.98 13.27 15.34 17.30 19.81 8.13 14.73 0 11.33 15.73 16.88 18.41 20.83 23.09 25.91 11.96 23.31 2.6. The Case of Nearly Deficient Rank With the above results we have two extremes. First, the full rank case where standard results apply, that is Johansen’s Dickey-Fuller type distribution for rank testing and χ2inferences for testing constraints on the cointegrating vectors. Second, the rank deficient case where new Dickey-Fuller type distributions apply both for rank testing and for testing constraints on the cointegrating vectors. In between these extremes we have the nearly rank deficient case corresponding to weak identification in the instrumental variable literature. These nearly deficient cases can be analyzed using local-to-unity parametrization. However, a full theory is notationally complicated as there will be many nuisance parameters. We therefore consider a simple special case inspired by the power analysis of Johansen (1995, Section 14) and distribution analysis of Nielsen (2004). The main finding is that the appropriate local rate is T−1 as in power analysis for unit tests and cointegration rank tests as opposed to T−1/2 for stationary models as in Andrews and Cheng (2012). Consider a bivariate, first order, local-to-unity vector autoregressive model where ∆Xt=1 T b1b2 0 0 !Xt−1+εtfor t=1, . . . , T, (13) Econometrics 2019,7, 6 8 of 24 where the innovations εtare independent normal N2(0, I2)-distributed where b16=0. We now have the following variant of the result for the rank test in Theorem 1. Theorem 4 (Nielsen 2004, Theorem 6.2) . Consider the data generating process (13). Let Bu be a bivariate standard Brownian motion on [0, 1]and let Jube the bivariate Ornstein-Uhlenbeck process given by Ju= b1b2 0 0 !Zu 0Jsds +Bu. Let 1≤ρ1≤ρ2≤0be the eigenvalues of the eigenvalue problem 0=det ρZ1 0JuJ0 udu −Z1 0Ju(dBu)0Z1 0(dBu)J0 u Then, for T →∞, LR{Hz(1)|Hz(2)}D →ρ2. The limit distribution is tabulated in Nielsen (2004, Table 8). We now consider the test for known cointegrating vector, b= (b1 , b2)0 . The result in Theorem 2is modified as follows. Theorem 5. Consider the data generating process (13). Let Bu , Ju be defined as in Theorem 4and let J1,u= b0Ju. Then LR{Hz,β(1)|Hz(2)}D →tr {Z1 0dBuJ0 u(Z1 0JuJ0 udu)−1Z1 0Ju(dBu)0 Z1 0dBuJ0 1,u(Z1 0J1,uJ0 1,udu)−1Z1 0J1,u(dBu)0}. 3. The Model with a Constant We now consider the model augmented with a constant. In the cointegrated model the constant is restricted to the cointegrating space. Thus, the cointegrating vectors consist of vectors relating the dynamic variable extended by a further coordinate for the constant. There are now two rank conditions; one related to the dynamic part of these extended cointegrating vectors and one relating to the deterministic part of the cointegrating vectors. The condition to the cointegration rank in the standard theory can therefore fail in two ways. 3.1. Model and Hypotheses The unrestricted vector autoregressive model is ∆Xt=ΠXt−1+µ+ k−1 ∑ i=1 Γi∆Xt−i+εtfor t=1, . . . , T, (14) where the innovations εt are independent normal N p( 0, Ω) -distributed. The parameters are the p -dimensional square matrices Π , Γi , Ω and the p -vector µ . They vary freely so that Ω is symmetric, positive definite. For the model with a constant there are two types of cointegration rank hypotheses: Hc`(r): rank Π≤r, (15) Hc(r): rank (Π,µ)≤r. (16) Econometrics 2019,7, 6 15 of 24 The notation in Khalaf and Urga (2014) differs slightly from the present notation. The hypothesis of known cointegration vectors is stated as β0= (Ir , b0 0)0 for some known b0 , corresponding to the present hypotheses Hz,β(r)and Hc`,β(r). The test statistics are LR(b0) = LR{Hm,β(r)|Hm(p)}, (25) LRC(b0) = LR{Hm,β(r)|Hm(r)}, (26) for m=z , c` . Moreover they consider the hypothesis H m,Π(r) , say, of a known impact matrix Π of rank r. This is tested through the statistic LR∗=LR{Hm,Π(r)|Hm(p)}. (27) When the rank is not deficient the test statistic LRC(b0) is asymptotically χ2 r(p−r) , see Johansen (1995, Section 7). The test statistic LR(b0) has a Dickey-Fuller type distribution as derived in Theorem 2for the case without deterministic terms, contradicting the χ2 asymptotics suggested by Khalaf and Urga (2014, Section 4). Table 2indicates that this distribution is close to, but different from, a χ2 p(p−r) -distribution when p= 2, 3 and p−r= 1. When p= 3 and r= 1, the limiting distribution is further from a χ2 p(p−r) -distribution. Likewise, the statistic LR∗ converges to a Dickey-Fuller-type distribution. This can be proved through a modification of the proof of Theorem 2. Khalaf and Urga’s Theorem 1 is concerned with bounding the distribution of the likelihood ratio statistic for the hypothesis Π=ab0 , where a , b are known p×r -matrices so that b has rank r , against the alternative where Π is unrestricted. The idea of their Theorem is to come up with a bound to the critical value when a , b may have deficient rank s≤r . Unfortunately, their theorem evolves around the incorrect χ2 distribution although unit root testing is implicitly involved. We therefore reformulate the result in terms of the limiting distributions derived herein. We consider the test statistic LR(b0) = LR{ H z,β( 1 )| H z( 1 )} when the rank of Π is nearly deficient. Suppose the rank is nearly deficient in the sense that Π≈T−1M for some matrix M along the lines of the theory in Section 2.6. Then, intuitively, the limiting distribution will be a combination of those arising when the true rank is 0 and when it is 1. The asymptotic theory developed here gives the relevant bounds. In the case of the zero level model the Theorems 1and 2imply the following pointwise result. Theorem 11. Let θ denote the parameters of the model (1). Consider the parameter space Θz where the hypothesis H z,β( 1 ):Π=αb0 holds. Here α , b are both of dimension p× 1. Here α is unknown, while b is known and has full column rank. Suppose the data generating process satisfies the I( 1 ) condition with s≤ 1. Let qz,s be the asymptotic ( 1 −ψ) quantile of LR{ H z,β( 1 )| H z( 1 )} when the data generating process satisfies H◦ z,β(s)for s =0, 1. Let qz,∗=maxs=0,1 qz,s. Then it holds for all θ∈Θzthat lim T→∞ P[LR{Hz,β(1)|Hz(1)} ≥ qz,∗]≤ψ. (28) The simulated values in Table 2show that for ψ=5% then qz,∗=max(qz,0,qz,1) = (max(9.05, 3.84) = 9.05 for p=2, max(13.82, 5.99) = 13.82 for p=3. (29) The interpretation is as follows. Suppose the hypothesis H z( 1 ) has not been rejected, but it is unclear whether the rank could be nearly deficient. Then the hypothesis of a known β0 is rejected if the statistic LR{Hz,β(1)|Hz(1)}is larger than qz,∗. Econometrics 2019,7, 6 16 of 24 The bound for qz,∗ seems very extreme. Khalaf and Urga therefore suggest to use the alternative statistic LR{ H z,β( 1 )| H z(p)} . Theorem 11 could be modified to cover this statistic. The simulations in Table 3indicate that we would then use bounds ˜ qz,∗=max(˜ qz,0,˜ qz,1) = (max(9.70, 6.22) = 9.70 for p=2, max(20.83, 15.34) = 20.83 for p=3. (30) We can establish a similar result for the constant level model using Theorems 8and 9. However, it is necessary to exclude the possibility of a linear trends in the rank deficient model as this would give a very complicated result. Theorem 12. Let θ denote the parameters of the model (14). Consider the parameter space Θc where the hypothesis H c,β( 1 ):(Π , µ) = α(b0 , b0 c) holds. Here α , b are both of dimension p× 1, while bc is a scalar. Further b , bc are known and b6= 0. Suppose the data generating process satisfies the I( 1 ) condition with s= 0 or s= 1. Let qc,s be the asymptotic ( 1 −ψ) quantile of LR{ H c,β( 1 )| H c( 1 )} when the data generating process satisfies H◦ c,β(s)for s =0, 1. Let qc,∗=maxs=0,1 qc,s. Then it holds for all θ∈Θ1that lim T→∞ P[LR{Hc,β(1)|Hc(1)} ≥ qc,∗]≤ψ. (31) The simulated values in Table 7show that for ψ=5% then qc,∗=max(qz,0,qz,1) = (max(14.05, 5.99) = 14.05 for p=2, max(19.66, 7.82) = 19.66 for p=3. (32) If the alternative is taken as Hc(p)instead of Hc(1)the bounds are modified as ˜ qc,∗=max(˜ qc,0,˜ qc,1) = (max(18.18, 12.38) = 18.18 for p=2, max(32.04, 24.39) = 32.04 for p=3. (33) The bounds (32), (33) for the constant level model appear further apart than the corresponding bounds (29), (30) for the zero level model. So in the constant level case there is perhaps less reason to use the test against the unrestricted model. 4.3. Empirical Illustration The identification robust inference can be illustrated using a series of monthly US treasury zero-coupon yields over the period 1987:8 to 2000:12. The data are taken from Giese (2008) and runs from the start of Alan Greenspan’s chairmanship of the Fed and finishes before the burst of the dotcom bubble. Giese considers 5 maturities (1, 3, 18, 48, 120 months), but here we only consider 2 maturities (12, 24 months). The empirical analysis uses OxMetrics, see Doornik and Hendry (2013). Figure 1shows the data in levels and differences along with the spread. The spread does not appear to have much of a mean reverting behaviour. It is not crossing the long-run average for periods of up to 4 years. This point towards a random walk behaviour which contradicts the expectations hypothesis in line with Giese’s analysis. She finds two common trends among five maturities. The two common trends can be interpreted as short-run and long-run forces driving the yield curve. The cointegrating relations match an extended expectations hypothesis where spreads are not cointegrated but two spreads cointegrate. This is sometimes called butterfly spreads and gives a more flexible match to the yield curve. This is in line with earlier empirical work. Hall et al. (1992), among others, found only one common trend when looking at short-term maturities, while Shea (1992); Zhang (1993) and Carstensen (2003) found more than one common trend when including longer maturities. Econometrics 2019,7, 6 17 of 24 b12 b24 1990 1995 2000 5.0 7.5 10.0 (a) zero coupon yields in levels b12 b24 ∆b12 ∆b24 1990 1995 2000 -1 0 1(b) zero coupon yields in differences ∆b12 ∆b24 s 1990 1995 2000 0.0 0.5 1.0 (c) yield spread s Figure 1. Zero coupon yields in (a) levels; (b) differences; and (c) spread. A vector autoregression of the form (14) with an intercept, k= 4 lags as well as a dummy variable for 1987:10 was fitted to the data. This has the form ∆Xt=ΠXt−1+µ+ 3 ∑ i=1 Γi∆Xt−i+Φ1(t=1987:10)+εtfor t=1, . . . , T, where Xt is the bivariate vector of the 12 and 24 month zero-coupon yields and periods t= 1 and t=Tcorrespond to 1987:8 and 2000:12 giving T=161. Table 9reports specification test statistics with p -values in square brackets. The tests do not provide evidence against the initial model. They are the autocorrelation test of Godfrey (1978) the cumulant based normality test, see Doornik and Hansen (2008), and the ARCH test of Engle (1982). For the validity of applying the autoreregressive and normality tests for non-stationarity autoregressions, see Engler and Nielsen (2009), Kilian and Demiroglu (2000), and Nielsen (2006). The dummy variable matches the policy intervention after the stock market crash on 19 October 1987. Empirically, the dummy variable can be justified in two ways. First, the plot of yield differences in Figure 1b indicate a sharp drop in yields at that point. Secondly, the robustified least squares algorithm analyzed in Johansen and Nielsen (2016) could be employed for each of the two equations in the model. The algorithm uses a cut-off for outliers in the residuals that is controlled in terms of the gauge, which is the frequency of falsely detected outliers that can be tolerated. The gauge is chosen small in line with recommendations of Hendry and Doornik (2014, Section 7.6), see also Johansen and Nielsen (2016). Thus, we choose a cut-off of 3.02 corresponding to a gauge of 0.25%. When running the autoregressive distributed lag models without outliers, only 1987:10 has an absolute residual exceeding the cut-off. Next, when re-running the model including a dummy for 1987:10, no further residuals exceed the cut-off. This is a fixed point for the algorithm. The detection of outliers may have some impact on specification tests, estimation, and inference. Johansen and Nielsen (2009,2016) analyze the impact on estimation when the data generating process has no outliers. They find that outlier detection only gives a modest efficiency loss compared to standard least squares when the cut-off is as large as chosen here. Berenguer-Rico and Nielsen (2017) find a considerable impact on the normality test employed above. At present, there is no theory for these algorithms for data generating processes with Econometrics 2019,7, 6 18 of 24 outliers, albeit some results are available for cointegration analysis with known break date, including the broken trend analysis of Johansen et al. (2000) and the structural change model of Hansen (2003). Table 9. Specification tests for the unrestricted vector autoregression. Test b12,tb24,tTest System χ2 normality (2)3.8 [0.15]4.1 [0.13]χ2 normality (4)4.3 [0.36] Far,1−7(7, 144)1.7 [0.11]1.0 [0.45]Far,1−7(28, 272)1.2 [0.24] Farch,1−7(7, 147)1.8 [0.09]1.0 [0.41] Table 10 reports cointegration rank tests. The fifth column shows conventional p -values based on Tables 4and 6for s=r corresponding to Johansen (1995, Tables 15.2, 15.3). The sixth column shows p -values based on Tables 5and 6assuming data have been generating by a model satisfying H c( 0 ) = H z( 0 ) . In both cases the p -values are approximated by fitting a Gamma distribution to the reported mean and variance, see Nielsen (1997); Doornik (1998) for details. As expected, the latter p -values tend to be higher than the former. Overall this provide overwhelming evidence in favour of a pure random walk model in line with Giese (2008). Table 10. Cointegration rank tests. Hypothesis rLikelihood LR p-Value s=rHc(0) Hc`(2) = Hc(2)2 134.63 Hc`(1)1 133.71 1.8 0.18 0.39 Hc(1)1 133.71 1.8 0.80 0.75 Hc`(0)0 129.70 9.8 0.30 0.46 Hc(0)0 129.21 10.8 0.57 0.57 If we have a strong belief in the expectation hypothesis we would, perhaps, ignore the rank tests and seek to test the expectations hypothesis directly. If we maintain the model H c( 1 ) , we could have to contemplate that the cointegration vectors could be nearly unidentified. A mild form of the expectation hypothesis is that the spread is zero mean stationary. Thus, we test the restriction b∗= ( 1, − 1, 0 ) . The likelihood ratio statistic is 4.0. Assuming the data generating process satisfies either H ◦ c( 0 ) or H ◦ c( 1 ) , but not by H ◦ c`( 0 ) , we can apply the Khalaf-Urga (2014)-type bound test established in Theorem 12. The 95% bound in (32) is 14.05 so the hypothesis cannot be rejected based on this test. This contrasts with the above rank tests which gave strong evidence against the expectations hypothesis. The results reconcile if the bounds test does not have much power in the weakly identified case. Indeed, this seems to be the case when looking at Table 3, ρ= 0.99-panels in Khalaf and Urga (2014), corresponding to near rank deficiency or weak identification. Thus, assuming the rank is one when in fact the data generating process appears to be nearly rank deficient seems to reduce power for tests on the cointegrating vector. That is, when the alleged cointegrating vector is not cointegrating it would be useful to be able to falsify the economic hypothesis. The above mentioned simulations indicate that this is not the case. 5. Conclusions We have derived asymptotic theory for cointegration rank tests and tests on cointegrating vectors in the rank deficient case. The asymptotic distributions have been simulated and tabulated. The results shed some light on the finite sample theory for cointegration analysis. They can be used to improve the theory on identification robust inference developed by Khalaf and Urga (2014). This was applied to two US treasury yield series. Econometrics 2019,7, 6 19 of 24 It appears that large distortions arise when applying standard cointegration inference in the situation where the rank is deficient or nearly deficient. The rank hypothesis gives an inequality for the rank, that is rank Π≤r . This includes cases where the rank is r and where it is less than r . Thus, the parameter space for the model where rank Π≤r therefore has a lower dimensional subset where the rank is deficient. Inferential procedures for rank determination are consistent but do leave a positive probability of deciding for a deficient rank in finite samples. In practice, it is therefore possible to end up in a situation of rank deficiency or near deficiency. When proceeding to testing restrictions on the cointegrating vectors, the model is therefore mis-specified or nearly mis-specified. The asymptotic analysis of the test distributions gives the following results. When testing for cointegration rank, the distribution shifts to the left when the rank is deficient. When testing for restrictions on the cointegrating vector, the distribution shifts to the right when the rank is deficient. When the rank is nearly deficient the distribution will tend to shift in similar directions. As a consequence, a test for cointegration restrictions using conventional critical has a size control problem previously observed by Johansen (2000). One can instead apply identification robust tests as suggested by Khalaf and Urga (2014), but our impression is that while these tests are better behaved in terms of size, they have modest power to reject incorrect restrictions. Our recommendation is to test for rank before testing restrictions on cointegrating vectors in line with Johansen’s framework. If the conclusion from the rank determination is ambiguous it is best to proceed with caution and possibly explore different choices for rank. This is a common theme in the applied work of Juselius. Author Contributions: The authors made equal contributions. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest. Appendix A. Proofs Processes are considered on the space of right continuous processes with left limits, D[ 0, 1 ] . A discrete time process Xt for t= 1, . . . , T is embedded in D[ 0, 1 ] through Xinteger(Tu) for 0 ≤u≤ 1. For processes Yt , Zt for t= 1, . . . , T the residuals from regressing Yt on Zt are denoted (Yt|Zt) = Yt−∑T s=1YsZ0 s(∑T s=1ZsZ0 s)−1Zt. Proof of Theorem 1 . This follows the outline of the proof in Johansen (1995, §10, 11). Let Π=α0β0 0 for p×s -matrices α0 , β0 with full column rank. Let Γ=Ip−∑k−1 i=1Γi . Under the I(1) condition the Granger-Johansen representation (6) holds with rank s and Johansen’s Lemma 10.1 stands with r replaced by s. His Lemmas 10.2, 10.3 hold with BT=β0⊥(β0 0⊥β0⊥)−1so that, on D[0, 1], T−1/2B0 TXinteger(Tu)=B0 TCT−1/2 integer(Tu) ∑ t=1 εt+oP(1). (A1) For later use we will note that the Brownian motion B can be chosen as follows. For any orthogonal square matrix ˜ M so ˜ M0˜ M=Ip−s choose the ( p−s )-dimensional standard Brownian motion B so that T−1/2 ˜ M0(α0 0⊥Ωα0⊥)−1/2α0 0⊥Γβ0⊥(β0 0⊥β0⊥)−1β0 0⊥X[Tu] D →Bu(A2) on D[0, 1]. Econometrics 2019,7, 6 20 of 24 Proof of Theorem 2 . Introduce the notation b ΩU=S00 −S01S−1 11 S10 for the unrestricted variance estimator and b ΩR=S00 −S01b(b0S11b)−1b0S10 for the restricted variance estimator. Then the likelihood ratio test statistic satisfies LR{Hz,β(r)|Hz(p)}=−Tlog det(b ΩU) det(b ΩR)=Tlog det{Ip+b Ω−1 U(b ΩR−b ΩU)}. If it is shown that b ΩUis consistent and T(b ΩR−b ΩU)converges in distribution then LR{Hz,β(r)|Hz(p)}=tr{Ω−1T(b ΩR−b ΩU)}+oP(1), (A3) following Johansen (1995, p. 224). The consistency of the unrestricted variance estimator b ΩU follows from Johansen (1995, Lemma 10.3) used with r=s=0 and BT=Ip. Consider T(b ΩR−b ΩU) . Note first that the data generating process has cointegration rank s= 0. Thus α0 , β0 are empty matrices so that their complements can be chosen as the identity matrix. The I(1) condition then implies that Γ=Ip−∑k−1 i=1Γi is invertible. The asymptotic convergence in (A2) then reduces to T−1/2 ˜ M0Ω−1/2ΓXinteger(Tu)=T−1/2 ˜ M0Ω−1/2 integer(Tu) ∑ t=1 εt+oP(1)D →Bu, (A4) where B is a standard Brownian motion of dimension p and for any orthonormal ˜ M so that ˜ M0˜ M=Ip . In particular, we will choose ˜ Mso ˜ M="{b0Γ−1Ω(Γ0)−1b}−1/2b0Γ−1Ω1/2 (b0 ⊥Γ0Ω−1Γb⊥)−1/2b0 ⊥Γ0Ω−1/2 #. (A5) Let B1,u,B2,ube the first rand the last p−rcoordinates of Bu, respectively. Then we get {b0Γ−1Ω(Γ0)−1b}−1/2b0Xinteger(Tu) D →B1,u. The variance estimators are b ΩR=Sεε −Sε1b(b0S11b)−1b0S1ε and b ΩU=Sεε −Sε1S−1 11 S1ε . In particular, the difference of the variance estimators is T(b ΩR−b ΩU) = T{Sε1M(M0S11 M)−1M0S1ε−Sε1bm(m0b0S11bm)−1mb0S1ε}, (A6) for any invertible matrices M , m and in particular for M0=˜ M0Ω−1/2Γ and m={b0Γ−1Ω(Γ0)−1b}−1/2 . In light of the identity ˜ M0˜ M=Ip , the random walk convergence in (A4), the rules for the trace and the notation v=mb write tr{Ω−1T(b ΩR−b ΩU)}=tr{˜ M0Ω−1/2T(b ΩR−b ΩU)Ω−1/2 ˜ M} =tr [˜ M0Ω−1/2T{Sε1M(M0S11M)−1M0S1ε−Sε1v(v0S11v)−1v0S1ε}Ω−1/2 ˜ M]. Then the product moment convergence results in Johansen (1995, Lemma 10.3) imply tr{Ω−1T(b ΩR−b ΩU)}D →tr{Z1 0dBuB0 u(Z1 0BuB0 udu)−1Z1 0Bu(dBu)0 −Z1 0dBuB0 1,u(Z1 0B1,uB0 1,udu)−1Z1 0B1,u(dBu)0}. Econometrics 2019,7, 6 21 of 24 This is also the limit of the likelihood ratio test statistic due to (A3). The convergence holds jointly with the convergence of the likelihood ratio test statistic for rank in Theorem 1since the orthogonal matrix ˜ Min (A2) can be chosen freely. Proof of Theorem 3 . We need a number of results from Johansen (1995). Let B , V be independent standard Brownian motions. His Theorem 11.1 shows LR{Hz(r)|Hz(p)}D →tr {Z1 0dBuB0 u(Z1 0BuB0 udu)−1Z1 0BudB0 u}, (A7) while his Lemma 13.8 shows LR{Hz,β(r)|Hz(r)}D →tr {Z1 0dVuB0 u(Z1 0BuB0 udu)−1Z1 0BudV0 u}. (A8) Johansen does not explicitly argue that the convergence results hold jointly. This can be done by going into the proofs of the results, find the asymptotic expansions of the test statistic, and express them in terms of random walks that converge to the processes B , V when normalized by T1/2 . The asymptotic distribution in (A8) is mixed Gaussian since B , V are independent. Thus, by conditioning on B we see that LR{ H z,β(r)| H z(r)} is asymptotically χ2 and hence independent of B . In turn the two test statistics are asymptotically independent. Proof of Theorem 5 . We follow Stockmarr and Jacobsen (1994) or Johansen (1995, Theorem 14.1, Lemma 14.3) and find that T−1/2Xinteger(Tu) converges to Ju as a process on D[ 0, 1 ] while (S00,S1ε,S11/T)converges in distribution to (I2,R1 0JudB0 u,R1 0JuJ0 udu). Now, proceed as in the proof of Theorem 2. It has to be argued that b ΩU converges in probability to I2 and that T(b ΩR−b ΩU) has the limit distribution postulated in the Theorem. The convergence of the b ΩU follows from the listed properties of the product moment matrices. For T(b ΩR−b ΩU) we have as in Equation (A6) that T(b ΩR−b ΩU) = T{Sε1(S11)−1S1ε−Sε1b(b0S11b)−1b0S1ε}. Again, we can apply the listed properties of the product moment matrices. Proof of Theorem 6 . Similar to the proof of Theorem 1, the relevant Granger-Johansen representation is (22) with rank s . Use Johansen’s Lemmas 10.2, 10.3 with BT={γ(γ0γ)−1 , T−1/2τ`(τ0 `τ`)−1} , where τ`=Cµ, while γ∈span(β0⊥)so that γ0τ`=0 and the expansion (A1) is replaced by T−1/2B0 TXinteger(Tu)=((γ0γ)−1γ0CT−1/2 ∑integer(Tu) t=1εt u)+oP(1)(A9) on D[0, 1]. Thus, ∆Xthas a non-zero level, but this is eliminated by regression on the intercept. Proof of Theorem 7 . Similar to the proof of Theorem 1. Use the Granger-Johansen representation (22) with rank s and τ`=Cµ= 0, and Johansen’s Lemmas 10.2, 10.3 with BT=β0⊥(β0 0⊥β0⊥)−1 so that T−1/2B0 TXinteger(Tu)has expansion (A1). Proof of Theorem 8 . Similar to the proof of Theorem 1. Use the Granger-Johansen representation (22) with rank s , and τ` . Use Johansen’s Lemmas 10.2, 10.3 with Xt , BT and the expansion (A1) replaced by, respectively, X∗ t= (X0 t , 1 )0 , the block diagonal matrix B∗ T=diag (BT , T1/2) where BT=β0⊥(β0 0⊥β0⊥)−1, and T−1/2B∗0 TX∗ integer(Tu)= B0 TCT−1/2 ∑integer(Tu) t=1εt 1!+oP(1)(A10) on D[0, 1]. Econometrics 2019,7, 6 22 of 24 Proof of Theorem 9 . The proof of Theorem 2is modified noting that R1,t is the (p+ 1 ) -vector (Xt−1 , 1 )0 corrected for lagged differences instead of Xt−1 corrected for lagged differences. Choose ˜ M as in (A5). Replace (A4) by T−1/2 ˜ M0Ω−1/2Γ0 0 1 ! Xinteger(Tu) 1!D →Fu. (A11) The difference of variance estimators in (A6) is now T(b ΩR−b ΩU) = T{Sε1M(M0S11 M)−1M0S1ε−Sε1b∗(b∗0S11b∗)−1b∗0S1ε}, (A12) where the invertible (p+1)-dimensional matrix Mnow is chosen as M=     b0Γ−1Ω(Γ0)−1b0 0 0b0 ⊥Γ0Ω−1Γb⊥0 0 0 1      −1/2    b0b0 c b0 ⊥Γ0Ω−1Γ0 0 1   (A13) Viewed as a ( 3 × 2 ) -block matrix, the two upper left equals the previous M . Since the random walk dominates a constant it holds that T−1/2Ip0 0 1 !M Xinteger(Tu) 1!D →Fu. (A14) Moreover, the first r coordinates of MR1,t are proportional to b∗0R1,t . Thus the argument can be completed as in the proof of Theorem 2. Proof of Theorem 10 . The proof of Theorem 3has to be modified to allow for a constant term in the cointegrating vector. The arguments leading to asymptotic results for the test statistics are sketched in Johansen and Juselius (1990) and, with more details, in Johansen et al. (2000, Theorem 3.1, Lemma A.5). Proof of Theorem 11. Write LR{Hz,β(1)|Hz(1)}=LR{Hz(1)|Hz(p)} − LR{Hz,β(1)|Hz(p)}. (A15) When s= 0 Theorems 1and 2give expansions for the right hand expressions of (A15) and in turn for the desired test statistic on the left hand of (A15). This implies an asymptotic distribution with asymptotic ( 1 −ψ) quantile qz,0 , say. When s= 1 Theorem 3in a similar way gives an asymptotic ( 1 −ψ) quantile qz,1 . Thus, with qz,∗=maxs=0,1 qz,s we get limT→∞ P [LR{ H z,β( 1 )| H z( 1 )} ≥ qz,∗]≤ψ , both with s=0 and when s=1. Proof of Theorem 12 . Similar to the proof of Theorem 11, applying Theorems 8–10 instead Theorems 1–3. References Abadir, Karim M. 1995. 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