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 ε 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 θ_Q. The second-order asymptotic approximation of M(θ̂_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(θ̂_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.