An explicit expression for the Fisher information matrix of a multiple time series process

The principal result in this paper is concerned with the derivative of a vector with respect to a block vector or matrix. This is applied to the asymptotic Fisher information matrix (FIM) of a stationary vector autoregressive and moving average time series process (VARMA). Representations which can be used for computing the components of the FIM are then obtained. In a related paper [1], the derivative is taken with respect to a vector. This is obtained by vectorizing the appropriate matrix products whereas in this paper the corresponding matrix products are left unchanged. AMS classification: 15A69, 62H12