Some Exact Distribution Theory for Maximum Likelihood Estimators of Cointegrating Coefficients in Error Correction Models
Peter C. B. Phillips
This paper derives some exact finite sample distributions and characterizes the tail behavior of maximum likelihood estimators of the cointegrating 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 estimator of the coefficients in a particular triangular system representation 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 dimension of the cointegration space. These results help to explain some recent simulation studies where extreme outliers are found to occur more frequently for the reduced rank regression estimator than for alternative asymptotically efficient procedures that are based on the triangular representation. In a simple triangular system, the Wald statistic for testing linear hypotheses about the columns of the cointegrating matrix is shown to have an $F$ distribution, analogous to Hotelling's $T^2$ distribution in multivariate linear regression.