The joint moment generating function of quadratic forms in multivariate autoregressive series - The case with deterministic components

Let {X-t} follow a discrete Gaussian vector autoregression with determinist ic components. We derive the exact finite-sample joint moment generating fu nction (MCF) of the quadratic forms that form the basis for the sufficient statistic. The formula is then specialized to the limiting MGF of functiona ls involving multivariate and univariate Ornstein-Uhlenbeck processes, drif ts, and time trends. Such processes arise asymptotically from more general non-Gaussian processes and also from the Gaussian {X-t} and have also been used in areas other than time series, such as the "goodness of fit" literat ure.