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summary A derivation of the bias of the least squares estimate of the coefficients of a multivariate autoregressive model is presented. The explicit expression obtained is of use when, in the case of small samples, one needs to make a bias correction prior to bootstrapping.

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This paper is concerned with autoregressive models in which the coefficients are assumed to be not constant but subject to random perturbations so that we are considering a class of random coefficient autoregressive models. By means of a two stage regression procedure estimates of the unknown parameters of these models are obtained. The estimates are shown to be strongly consistent and to satisfy a central limit theorem. A number of Monte Carlo experiments was carried out to illustrate the estimation procedure and their results are reported.

Spectral methods are used to construct an estimate of the variance of the prediction error for a normal, stationary process. The estimate obtained is shown to be strongly consistent and asymptotically normally distributed. Some aspects of the computations with respect to the fast Fourier transform are considered. The latter half of the article consists of a number of simulations, based on both generated and real data, which illustrate the results obtained. The relation between the estimate and that obtained from a high order autoregression is discussed.