Elsevier

Journal of Econometrics

Volume 88, Issue 2, February 1999, Pages 301-339
Journal of Econometrics

Likelihood analysis of seasonal cointegration

https://doi.org/10.1016/S0304-4076(98)00035-9Get rights and content

Abstract

The error correction model for seasonal cointegration is analyzed. Conditions are found under which the process is integrated of order 1 and cointegrated at seasonal frequency, and a representation theorem is given. The likelihood function is analyzed and the numerical calculation of the maximum likelihood estimators is discussed. The asymptotic distribution of the likelihood ratio test for cointegrating rank is given. It is shown that the estimated cointegrating vectors are asymptotically mixed Gaussian. The results resemble the results for cointegration at zero frequency when expressed in terms of a complex Brownian motion. Tables are provided for asymptotic inference.

Introduction

This paper contains a systematic treatment of the statistical analysis of seasonal cointegration in the vector autoregressive model. The theory started with the paper by Hylleberg et al. (1990)which gave the main results on the representation and the single-equation tests for cointegration at complex frequencies. An analysis of seasonal cointegration of Japanese consumption was given in Engle et al. (1993). Seasonal cointegration analysis is prompted by the empirical finding that the vector autoregressive model often describes macro-data quite well. The occurrence of unit roots in the fitted process implies that it is non-stationary, and the occurrence of unit roots at seasonal frequency implies a non-stationary seasonal variation. This again implies the possibility of seasonal cointegration, and the phenomenon that the seasonality drifts, such that ‘summer becomes winter’.

The paper on maximum likelihood inference by Lee (1992)sets the stage for the analysis of multivariate systems. Unfortunately, it does not treat all aspects of asymptotic inference, and the test for cointegration rank is only partially correct. The two papers by Gregoir 1993a, Gregoir 1993b) deal with a very general situation of unit roots allowing for processes to be integrated of order greater than 1 at any frequency, but do not treat likelihood inference.

The purpose of this paper is therefore to improve on the previous analysis and discuss maximum likelihood estimation, calculation of test statistics, and derivation of asymptotic distributions in the context of the vector autoregressive model. In the process of doing so it is natural to give the mathematical theory of the Granger representation, the error correction model, the role of the constant term, and seasonal dummies. The basic new trick is the introduction of the complex Brownian motion, which makes many calculations more natural and greatly simplifies formulae for limit distributions. We focus mainly on complex roots, since the case with roots at 1 is well known from the literature, see Johansen (1996), and the situation with a root at −1 can be dealt with using the same methods, see Lee (1992).

We consider the autoregressive model defined for an n-dimensional process Xt by the equations:Xt=j=1lΠjXt−j+ΦDtt,where we assume that the initial values X0,…,X−l+1 are fixed. When deriving estimators and test statistics we also assume that εt are i.i.d. Nn(0,Ω), while the asymptotic results are proved under the assumption that the errors are i.i.d. with finite variance and mean zero. This assumption can be further relaxed, see Chan and Wei (1988), and the comments in Section 4. The deterministic terms Dt may contain a constant, a linear term, or seasonal dummies. Various models defined by restrictions on the deterministic terms will be considered. The properties of the process generated by Eq. (1)are as usual expressed in terms of the characteristic polynomialA(z)=Inj=1lΠjzjwith determinant |A(z)|, and where In is an n×n identity matrix.

The paper is organized as follows: in Section 2the error correction model for seasonal cointegration of processes that are integrated of order 1 at seasonal frequency is discussed. The equations are solved in the form of a Granger representation theorem, applying a general result about inversion of matrix polynomials. This is applied to analyze the role of constant, linear term, and seasonal dummies, see Franses and Kunst (1995). In Section 3the Gaussian likelihood analysis and calculation of maximum likelihood estimators in the model with unrestricted deterministic terms, as well as in some models, defined by restrictions of deterministic terms, is discussed. In Section 4some technical asymptotic results on the behavior of the process and product moments are given. Section 5contains asymptotic results for the maximum likelihood estimator of the cointegrating vectors, and the likelihood ratio test for cointegration rank at seasonal frequency.

In Appendix A a brief description of the (real) matrix representation of complex matrices is given along with proofs of the technical results in Section 4. Finally, Appendix B contains tables of limit distributions of the likelihood ratio tests for cointegrating rank for various models defined by restrictions on the deterministic terms.

Section snippets

The representation theorem and the error correction model

This section contains the necessary analytic results from the theory of real polynomials A(z) with values in the set of n×n matrices. Theorem 1 gives Lagrange’s expansion for a polynomial around arbitrary points and it is shown in Corollary 2 how this contains the formulation of an error correction model. The basic result, however, is Theorem 3, which gives a necessary and sufficient condition for the inverse matrix polynomial to have poles of order 1. In Theorem 4 this result is interpreted as

The models for seasonal cointegration and their statistical analysis

In this section the statistical model for autoregressive processes integrated of order 1 at seasonal frequency which allows for seasonal cointegration is defined. Various models defined by restrictions on the deterministic terms are given. We discuss Gaussian maximum likelihood estimation and the formulation of some hypotheses on the cointegrating ranks, the cointegrating vectors, and the adjustment coefficients.

Asymptotic results

This section deals with some technical results on asymptotic behavior of various processes and product moments. The proofs are given in Appendix A. We assume throughout that the processes are generated by autoregressive equations without deterministic terms and that the ε are i.i.d. with mean zero and variance Ω. We start with the sums St(m) and then find the limiting behavior of Xt(m) and finally investigate S11,S10,S00, and Sε1 which are based on residuals from the regression (15).

The limit

Asymptotic inference on rank and cointegrating relations

The main result about the estimator β̂ is that it is asymptotically mixed Gaussian such that asymptotic inference on its coefficients can be conducted in the χ2 distribution. The test statistic for hypotheses on the rank at seasonal frequency has a limit distribution, which is similar to the usual one at frequency zero, when expressed in terms of the complex Brownian motion.

Acknowledgements

The authors would like to thank David Hendry for discussions of the paper and the Danish Social Sciences Research Council for their continuing support.

References (14)

There are more references available in the full text version of this article.

Cited by (0)

View full text