Misspecified prediction for time series

Let { X t } be a stationary process with spectral density g (λ).It is often that the true structure g (λ) is not completely specified. This paper discusses the problem of misspecified prediction when a conjectured spectral density f θ (λ), θ∈Θ, is fitted to g (λ). Then, constructing the best linear predictor based on f θ (λ), we can evaluate the prediction error M (θ). Since θ is unknown we estimate it by a quasi‐MLE $\hat{\theta}_{Q}$ . The second‐order asymptotic approximation of $M(\hat{\theta}_{Q})$ is given. This result is extended to the case when X t contains some trend, i.e. a time series regression model. These results are very general. Furthermore we evaluate the second‐order asymptotic approximation of $M(\hat{\theta}_{Q})$ for a time series regression model having a long‐memory residual process with the true spectral density g (λ). Since the general formulae of the approximated prediction error are complicated, we provide some numerical examples. Then we illuminate unexpected effects from the misspecification of spectra. Copyright © 2001 John Wiley & Sons, Ltd.