Prediction Error of Multivariate Time Series With Mis‐specified Models

We consider the situation in which an incorrectly specified autoregressive moving‐average model is used to predict future values of a stationary multivariate time series. The use of an incorrect model for prediction results in an increase in mean‐square prediction error over that of the optimal predictor, and an expression for this increase is first given for fixed values of the parameters in the incorrect model. For the case in which the incorrect model is an autoregression, we also take into account parameter estimation error by first deriving the asymptotic distribution and limiting moment properties of the least‐squares estimator of the parameters in the mis‐specified model. An asymptotic approximation to the increase in mean‐square prediction error is then obtained. Numerical examples are provided to demonstrate the accuracy of the asymptotic approximation in finite samples. Our results are consistent with those obtained in the univariate case, indicating that fitted autoregressions of high order can yield substantially sub‐optimal forecasts.