Asymptotic Likelihood Based Inference for Co‐integrated Homogenous Gaussian Diffusions

Abstract

In this paper we consider inference for a multivariate Gaussian homogenous diffusion which is co‐integrated, i.e. admits a representation in terms of stable relations (ergodic diffusions) plus Brownian motions. We show that inference on co‐integration rank (the number of stable relations) in continuous time can be based on existing asymptotic distributions from discrete time co‐integration analysis. Likewise the asymptotic distributions of the co‐integration parameters are shown to be mixed Gaussian. Special attention is given to the parametrization of the drift terms. It is shown that the asymptotic distribution of the co‐integration rank test statistic does not depend on the level of the process as a result of the chosen parametrization.