Matrix differential calculus applied to multiple stationary time series and an extended Whittle formula for information matrices

The purpose of this paper is to set forth easily implementable expressions for the Fisher information matrix (FIM) of a Gaussian stationary vector autoregressive and moving average process with exogenous or input variables, a vector ARMAX or VARMAX process. The entries of the FIM are represented as circular integral expressions and can be computed by applying Cauchy’s residue theorem. An extension of the Whittle formula for the FIM of multiple time series processes is developed for VARMAX processes. It will be shown that the extended Whittle formula yields the FIM when a bivariate structure, consisting of the VARMAX process and the exogenous-input process, is considered. Consequently, the equivalence between a frequency and time domain representation of the FIM of VARMAX processes is established. In order to obtain the results presented in this paper, the differentiation techniques developed and used in [A. Klein, P. Spreij, An explicit expression for the Fisher information matrix of a multiple time series process, Linear Algebra Appl. 417 (2006) 140–149] are applied.