Likelihood Ratio Tests on Cointegrating Vectors, Their Disequilibrium Adjustment Vectors, and Their Orthogonal Complements EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 3, No. 3, 2010, 541-571 ISSN 1307-5543 – www.ejpam.com Email address: norman.j.morin@frb.gov *The opinions expressed are those of the author and not necessarily those of the Board of Governors of the Federal Reserve System or its staff. http://www.ejpam.com 541 © 2010 EJPAM All rights reserved. SPECIAL ISSUE ON GRANGER ECONOMETRICS AND STATISTICAL MODELING DEDICATED TO THE MEMORY OF PROF. SIR CLIVE W.J. GRANGER Likelihood Ratio Tests on Cointegrating Vectors, Disequilibrium Adjustment Vectors, and Their Orthogonal Complements Norman Morin* Division of Research and Statistics, Federal Reserve Board, Washington, DC, USA Abstract. Cointegration theory provides a flexible class of statistical models that combine long-run (cointegrating) relationships and short-run dynamics. This paper presents three likelihood ratio (LR) tests for simultaneously testing restrictions on cointegrating relationships and on how quickly each variable in the system reacts to the deviation from equilibrium implied by the cointegrating relationships. Both the orthogonal complements of the cointegrating vectors and of the vectors of adjustment speeds have been used to define the common stochastic trends of a nonstationary system. The restrictions implicitly placed on the orthogonal complements of the cointegrating vectors and of the adjustment speeds are identified for a class of LR tests, including those developed in this paper. It is shown how these tests can be interpreted as tests for restrictions on the orthogonal complements of the cointegrating relationships and of their adjustment vectors, which allow one to combine and test for economically meaningful restrictions on cointegrating relationships and on common stochastic trends. 2000 Mathematics Subject Classifications: 62H12, 62H15 Key Words and Phrases: Cointegration, common stochastic trend, likelihood ratio tests 1. Introduction Since its introduction by Granger [14,15] cointegration has become a widely investigated and extensively used tool in multivariate time series analysis. Cointegrated N. Morin / Eur. J. Pure Appl. Math 542 models combine short-run dynamics and long-run relationships in a framework that lends itself to investigating these features in economic data. The relationship between cointegrated systems, their vector autoregressive (VAR) and vector moving-average representations, and vector error-correction models (VECM) were developed by Granger in [14,15] and by Engle and Granger in [7]. In a cointegrated system of time series, the cointegrating vectors can be interpreted as the long-run equilibrium relationships among the variables towards which the system will tend to be drawn. Economic theories and economic models may imply long-run relationships among variables. Certain ratios or spreads between nonstationary variables are expected to be stationary, that is, these variables are cointegrated with given cointegrating vectors. For example, neoclassical growth models imply “balanced growth” among income, consumption, and investment (for example [29, 41]), implying that their ratios are mean- reverting. Other theories, rather than implying given ratios or spreads are cointegrated, may imply that some linear combinations of the variables are stationary, that is, the variables are cointegrated without specifying the cointegrating relationships (for example [25]). Johansen’s maximum likelihood approach to cointegrated models [19] provides an efficient procedure for the estimation of cointegrated systems and provides a useful framework in which to test restrictions of the sorts mentioned above. For example, Johansen [19, 21] and Johansen and Juselius [25, 26] derive likelihood ratio tests for various structural hypotheses concerning the cointegrating relationships and the speed of adjustment to the disequilibrium implied by the cointegrating relationships (or weights); Konishi and Granger [30] use this approach to derive and test for separation cointegration, and Gonzalo and Granger [12] use this framework for estimation of and testing for their multivariate version of Quah’s [37] permanent and transitory (P-T) decomposition. Further, building on the univariate work of Beveridge and Nelson [1] and the multivariate generalization by Stock and Watson [42], cointegration analysis may be used to decompose a system of variables into permanent components (based on the variables’ common stochastic trends) and temporary (or cyclical) components. Several methods have been proposed to separate cointegrated systems into their permanent and temporary components (for example, [12, 21, and 27]). In each case, the permanent component is based either on the orthogonal complements of the cointegrating relationships or on the orthogonal complements of the disequilibrium adjustments to the cointegrating relationships. In this paper, new hypothesis tests are presented in Johansen’s maximum likelihood framework that allow one to combine restrictions on the cointegrating relationships and on their disequilibrium adjustments. These tests possess closed-form solutions and do not require iterative methods to estimate the restricted parameters under the null hypothesis. Secondly, both for Johansen’s likelihood ratio tests for coefficient restrictions and for the new tests presented below, the restrictions implicitly placed on the orthogonal complements of the cointegrating relationships and on the orthogonal complements of the adjustment speeds are presented. Johansen’s tests and the tests developed in this paper can be interpreted as tests of restrictions on the various definitions of common stochastic trends, since these definitions depend on the orthogonal complements either of the cointegrating relationships or of the disequilibrium adjustments. Thus, one has great flexibility in formulating and testing hypotheses of economic interest simultaneously on the cointegrating relationships and on the common stochastic trends—the long-run relationships among the variables in the system and the variables driving the trending behavior the system, respectively. N. Morin / Eur. J. Pure Appl. Math 543 The organization of this paper is as follows: In section 2, the basic model and notation are introduced, and maximum likelihood estimation of the unrestricted model is briefly described. In section 3, likelihood ratio tests for restrictions on cointegrating relationships and on their weights are briefly described, and three new tests in this framework are presented. In section 4, the implications for the orthogonal complements of the cointegrating vectors and of the adjustment vectors are developed for the tests described in section 3. It is shown how these tests can be used for testing restrictions on the orthogonal complements of cointegrating vectors and on the orthogonal complements of the disequilibrium adjustment vectors—thus allowing for combinations of tests on cointegrating relationships and on the different definitions of common stochastic trends. Section 5 concludes, and the appendix contains the mathematical proofs. 2. The Unrestricted Cointegrated Model Let ( )I d denote a time series that is integrated of order d, that is, d applications of the differencing filter, 1 L∆ = − , yield a stationary process. Let tX be a p×1 vector of possibly I(1) time series defined by the kth-order vector autoregression (VAR), 1 k t i t i t t i X X D ε− = = Π + Φ +∑ , 1, ,t T=  , (1) and generated by initial values 0, ,kX X−  , by p-dimensional normally-distributed zero- mean random variables { } 0 T t t ε = with variance matrix Ω and by a vector of deterministic components tD (possibly constants, linear trends, and seasonal and other dummy variables). Using the lag polynomial expression for (1), ( ) t t tL X D εΠ = Φ + , (2) where ( ) 1 k i i i L I L = Π = − Π∑ , the VAR in the levels in (1) can be rewritten in first differences as 1 1 1 k t t i t i t t i X X X D ε − − − = ∆ = Π + Γ ∆ + Φ +∑ , (3) where ( ) 1 1 k i i I =   Π = −Π = − − Π    ∑ and 1 , 1, , 1 k i j j i i k = + Γ = − Π = −∑  . The long-run behavior of the system depends on the rank of the p×p matrix Π. If the matrix has rank 0 (that is, Π = 0) then there are p unit roots in the system, and (3) is simply a traditional VAR in differences. If Π has full rank p, then tX is an I(0) process, that is, tX is stationary in its levels. If the rank of Π is r with 0 r p< < , then tX is said to be cointegrated of order r. This implies that there are r

> > and corresponding eigenvectors ( )1 ˆ ˆ ˆ, , pV v v=  normalized by 11 ˆ ˆ pV S V I′ = . Thus the maximum likelihood estimate for the cointegrating vectors β is ( )1 ˆ ˆ ˆ, , rv vβ =  , (13) and the normalization implies that the estimate of the weights in (8) is 01 ˆˆ Sα β= . (14) Then, apart from a constant, the maximized likelihood can be written as ( )2 max 00 1 ˆ1 r T i i L S λ− = = −∏ . (15) Likelihood ratio tests of the hypothesis of r unrestricted cointegrating relationships in the unrestricted VAR model and for r unrestricted cointegrating relationships against the alternative of r+1 unrestricted cointegrating relationships—the trace and maximum eigenvalue tests—are derived in [19]. The asymptotic distribution of the trace and maximum eigenvalue tests for different deterministic components may be found in [19] and [25], and the tabulated critical values for various values of r and for different deterministic components may be found in [20, 23, 34]; small-sample adjustments to the critical values that are based on response surface regressions may be found in [2] and [31]. The unrestricted orthogonal complements of β and α, β⊥ and α⊥ , can be estimated three ways: One may use the eigenvectors associated with the zero eigenvalues of ββ ′ and αα′ [12] (given a p×r matrix of full column rank A, one can quickly construct A⊥ as the ordered eigenvectors corresponding to the p-r zero-eigenvalues of AA′ ), and one may estimate α⊥ as the eigenvectors corresponding to the p-r smallest eigenvalues that solve the dual of the eigenvalue problem in (12), 1 00 01 11 10 0S S S Sλ −− = , normalized such that N. Morin / Eur. J. Pure Appl. Math 546 00ˆ ˆ p rS Iα α⊥ ⊥ −′ = , and by setting 10 ˆ ˆSβ α⊥ ⊥= . Johansen [23] shows one may estimate them from (12) by ( )11 1, ,r pS v v+  and ( )1 00 01 1, ,r pS S v v− +  , respectively. 3. Testing Restrictions on β and α Economic theory may suggest that certain ratios or spreads between variables will be cointegrating relationships. For example, some neoclassical growth models with a stochastic productivity shock imply “balanced growth” among income, consumption, and investment (that is, the ratios are cointegrated), and certain one-factor models of the term structure of the interest rates imply that the spreads between the different interest rate maturities will be cointegrated. One might also be interested in testing for the absence of certain variables in the system from any of the cointegrating relationships. Complicated restrictions on β or α may be formulated, for example, neutrality hypotheses in Mosconi and Giannini [32] and separation cointegration in Konishi and Granger [30]. Based on their maximum likelihood framework, Johansen [19, 21] and Johansen and Juselius [25, 26] formulate a series of likelihood ratio tests for linear restrictions on β or α and tests for a subset of known vectors in β or α. After briefly summarizing this set of five tests, three new tests for combining linear restrictions and known vectors will be derived. The tests for restrictions on the cointegrating relationships and disequilibrium adjustment vectors described below are asymptotically chi-squared distributed. The finite sample properties of some of the tests have been studied (see, for example [18]) and are shown to have significant size distortions in small samples, though they generally perform well with larger samples. Johansen [24] introduces a Bartlett-type correction for tests (1) and (2) below that depend on the size of the system, the number of cointegrating vectors, the lag length in the VECM, the number of deterministic terms (restricted versus unrestricted), the parameter values, and the sample size under the null hypothesis. Haug [18] demonstrates that the Bartlett correction is successful in moving the empirical size of the test close to the nominal size of the test and also demonstrates that the power of the tests for restrictions on β depend on the speed of adjustment to the long-run equilibrium relationships in the system, with slower adjustment speeds leading to tests with lower power. The tests below are all based on the reduced rank regression representation of the VECM in (4), 0 1 , 1, ,t t tR R t Tαβ ε′= + =  , (16) the same equation that is the starting point for the maximum likelihood estimates of the parameters of the VECM. The estimators and test statistics are all calculated in terms of the residual product moment matrices , , 0,1ijS i j = and by their eigenvalues. The parameter estimates under the restrictions and the maximized likelihood functions can be explicitly calculated; other tests not discussed here may be solved using iterative methods (see [6] and [22]). Denote the unrestricted model of at most r cointegrating relationships in the VECM (4) as ( )H r . For any rectangular matrix with full column rank, A, define the notation ( ) 1A A A A −′≡ , which implies rA A A A I′ ′= = . Five tests for restrictions on β and α from Johansen [19, 20] and Johansen and Juselius [25] are briefly described before turning to three new tests for restrictions on β and α. N. Morin / Eur. J. Pure Appl. Math 547 (1) 0 :H Hβ φ= (Johansen [19]), (17) where H p×r is known and φ s×r is unknown, r≤s = = =      . Then the restricted estimators are ( )2 1 ˆ , , r mv vφ −=    (40) 2 2 ˆ ˆHβ φ= (41) 01 2̂ˆ A S Hψ φ⊥′= (42) ( ) 1 1 11. 10. ˆ k kH S H H S Aφ −′ ′= (43) ( ) 1 1 2 11. 10. 2 ˆ ˆ ˆ ˆ, ,k kH H S H H S A Hβ β β φ−   ′ ′= =    (44) [ ] ( ) 1 01 2 ˆˆ ˆˆ, , ,A A A A A A A A Sα τ ψ β− ⊥ ⊥ ⊥ ⊥ ⊥  ′ ′ = = =    , (45) where 1 . , , 0,1ij k ij ik kk kjS S S S S i j−= − = is calculated from (37) evaluated at 2̂ ˆ,φ ψ . The maximized likelihood function, apart from a constant, is ( ) ( )00. 002 max 1 1 1 1 r m m kT i i i i A S A A S A L A A A A λ ρ − ⊥ ⊥− = =⊥ ⊥ ′ ′ = − − ′ ′ ∏ ∏  . (46) The proof of Theorem 3 is in the Appendix. Theorem 4. The likelihood ratio test statistic of the hypothesis [ ]0 : , ,H H Aβ φ α τ= = verses ( )H r is expressed as: N. Morin / Eur. J. Pure Appl. Math 551 ( )( ) ( ) ( ) ( ) 0 00. 00 00 1 1 1 | ln ... ˆln ln 1 ln 1 ln 1 k r m m r i i j i i j LR H H r A S A A S A T A A A A S λ ρ λ ⊥ ⊥ ⊥ ⊥ − = = = =  ′ ′  −  ′ ′     + − + − − −   ∑ ∑ ∑  , (47) where { } 1,î i r λ = are from the unrestricted maximized likelihood in (15), and is asymptotically distributed as 2χ with m(p-r)+r(p-s)degrees of freedom. The proof of Theorem 4 is in the Appendix. Next, a hypothesis test on αβ ′Π = of the form 1 2Π = Π + Π is presented in which 1 AH ′Π = is known. This test, which combines tests (2) and (4), implies one is testing that both a subset of the cointegrating vectors and the associated adjustment vectors are known. It might seem too optimistic or restrictive to believe one might not only know certain cointegrating vectors but also know the adjustments to them. A test of this sort, however, might be useful as the end of a general-to-simple strategy for testing structural hypotheses or for testing very specific theoretical implications. More usefully, one might estimate the cointegrating relationships and adjustment vectors from a subset of a system of variables and then desire to test whether these estimated relationships hold in the full system of variables. (8) [ ] [ ]0 : , , ,H H Aβ θ α τ= = where both ,H A are known p×s matrices with s = = =     solve the eigenvalue problem ( ) 1 11. 10. 00. 01. 0k k k kH S H H S A A S A A S Hρ −′ ′ ′ ′− = . (134) The variance-covariance matrix is then estimated by ˆ ˆ ˆ ˆ ˆ AA AA A A A A A A A A⊥ ⊥ ⊥ ⊥ ⊥ ⊥  Ω Ω ′    Ω =     Ω Ω  , (135) where the estimators of A A⊥ ⊥ Ω , 1 AA A Aω ⊥ ⊥ ⊥ −= Ω Ω , and 1 .AA A AA AA A A A A⊥ ⊥ ⊥ ⊥ ⊥ −Ω = Ω − Ω Ω Ω are used to recover ˆ A A⊥ ⊥ Ω , ˆ ˆˆAA A Aω ⊥ ⊥ ⊥ Ω = Ω , ˆ ˆ A A AA⊥ ⊥ ′Ω = Ω , and . ˆ ˆ ˆˆAA AA A A Aω ⊥ ⊥ Ω = Ω + Ω . Finally, the maximized likelihood is ( ) ( )00 00.2 max 1 1 1 1 r m m kT i i i i A S A A S A L A A A A λ ρ − ⊥ ⊥− = =⊥ ⊥ ′ ′ = − − ′ ′ ∏ ∏  .  (136) Proof of Theorem 4. The likelihood ratio test for 0H in H(r) is ( )( ) ( ) ( )( )( )0 02lnLR H H r L H L H r= − . (137) The constant terms in both cancel, and one can write the likelihood ratio test statistic from (15) and (136), ( )( ) ( ) ( ) ( ) 0 00. 00 00 1 1 1 | ln ln ... ˆln 1 ln 1 ln 1 k r m m r i i j i i j LR H H r A S A A S A T S A A A A λ ρ λ ⊥ ⊥ ⊥ ⊥ − = = = =  ′ ′  −  ′ ′     + − + − − −   ∑ ∑ ∑  . (138) The number of free parameters in the unrestricted model for r cointegrating relationships, from Theorem 2, is 2pr-r 2 . In the restricted model, 1 2A H A Hφ ψφ⊥′ ′ ′ ′Π = + (see [23, Lemma 7.1]) has ms+(p-m+s-(r-m))(r-m) free parameters. The degrees of freedom for the likelihood ratio tests is the difference in free parameters between the unrestricted and restricted models, m(p-r)+r(p-s). So, the likelihood ratio test is asymptotically distributed as 2χ with m(p-r)+r(p-s) degrees of freedom.  REFERENCES 568 Proof of Theorem 5. Under the hypothesis 0 : , , ,H H H A Aβ φ α ψ⊥ ⊥  = =    where ,H A are known p×s matrices and φ and ψ (p-s)×(r-s) are unknown, s≤r