The Joint Distribution of Forecast Errors in the AR(1) Model

Second-order asymptotic expansion approximations to the joint distributions of dynamic forecast errors and of static forecast errors in the stationary Gaussian pure AR(1) model are derived. The approximation to the dynamic forecast errors distribution can be expressed as a multivariate normal distribution with modified mean vector and covariance matrix, thus generalizing the results of Phillips [12]. However, the approximation to the static forecast errors distribution includes skewness and kurtosis terms. Thus the class of multivariate normal distributions does not provide as good approximations (in terms of error convergence rates) to the distributions of the static forecast errors as to the distributions of the dynamic forecast errors. These results cast some doubt on the appropriateness of model validation procedures, such as Chow tests, which use the static forecast errors and implicitly assume that these have a distribution which is well approximated by a multivariate normal.