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Optimal Inference in Cointegrated Systems

P. C. B. Phillips

Abstract

This paper studies the properties of maximum likelihood estimates of cointegrated systems. Alternative formulations of such models are considered including a new triangular system error correction mechanism. It is shown that full system maximum likelihood brings the problem of inference within the family that is covered by the locally asymptotically mixed normal asymptotic theory provided that all unit roots in the system have been eliminated by specification and data transformation. This result has far reaching consequences. It means that cointegrating coefficient estimates are symmetrically distributed and median unbiased asymptotically, that an optimal asymptotic theory of inference applies, and that hypothesis tests may be conducted using standard asymptotic chi-squared tests. In short, this solves problems of specification and inference in cointegrated systems that have recently troubled many investigators. Methodological issues are also addressed and these provide the major focus of the paper. Our results favor the use of full system estimation in error correction mechanisms or subsystem methods that are asymptotically equivalent. They also point to disadvantages in the use of unrestricted VAR's that are formulated in levels and in certain single equation approaches to the estimation of error correction mechanisms. Unrestricted VAR's implicitly estimate unit roots that are present in the system and the relevant asymptotic theory for the VAR estimates of the cointegrating subspace inevitably involves unit root asymptotics. Single equation error correction mechanisms generally suffer from similar disadvantages through the neglect of additional equations in the system. Both examples point to the importance of the proper use of information in the estimation of cointegrated systems. In classical estimation theory the neglect of information typically results in a loss of statistical efficiency. In cointegrated systems deeper consequences occur. Single equation and VAR approaches sacrifice asymptotic median unbiasedness as well as optimality and they run into inferential difficulties through the presence of nuisance parameters in the limit distributions. The advantages of the use of fully specified systems techniques are shown to be all the more compelling in the light of these alternatives. Attention is also given to the information content that is necessary to achieve optimal estimation of the cointegrating coefficients. It is shown that optimal estimation of the latter does not require simultaneous estimation of the transient dynamics even when the parameters of the transient dynamics are functionally dependent on the parameters of the cointegrating relationship. All that is required is consistent estimation of the long run covariance matrix of the system residuals and this covariance matrix estimate can be utilized in regression formulae of the generalized least squares type. Thus, optimal estimation can be achieved without a detailed specification of the system's transient responses and thus, in practice, without the use of eigenvalue routines such as those employed in the Johansen (1988) procedure.