SOME EXACT DISTRIBUTION-THEORY FOR MAXIMUM-LIKELIHOOD ESTIMATORS OF COINTEGRATING COEFFICIENTS IN ERROR-CORRECTION MODELS

This paper derives some exact finite sample distributions and characte rizes the tail behavior of maximum likelihood estimators of the cointe grating coefficients in error correction models. It is shown that the reduced rank regression estimator has a distribution with Cauchy-like tails and no finite moments of integer order. The maximum likelihood e stimator of the coefficients in a particular triangular system represe ntation is studied and shown to have matrix t-distribution tails with finite integer moments to order T - n + r where T is the sample size, n is the total number of variables in the system, and r is the dimensi on of the cointegration space. These results help to explain some rece nt simulation studies where extreme outliers are found to occur more f requently for the reduced rank regression estimator than for alternati ve asymptotically efficient procedures that are based on the triangula r representation. In a simple triangular system, the Wald statistic fo r testing linear hypotheses about the columns of the cointegrating mat rix is shown to have an F distribution, analogous to Hotelling's T-2 d istribution in multivariate linear regression.