Misspecified prediction for time series

Let {X-t} be a stationary process with spectral density g(lambda). It is often that the true structure g(lambda) is not completely specified. This paper discusses the problem of misspecified prediction when a conjectured spectral density f(theta)(lambda), theta is an element of Theta, is fitted to g(lambda). Then, constructing the best linear predictor based on f(theta)(lambda), we can evaluate the prediction error M(theta). Since theta is unknown we estimate it by a quasi-MLE <(<theta>)over cap>(Q). The second-order asymptotic approximation of M(<(<theta>)over cap>(Q)) is given. This result is extended to the case when X, 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(<(<theta>)over cap>(Q)) for a time series regression model having a long-memory residual process with the true spectral density g(lambda). 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 (C) 2001 John Wiley & Sons, Ltd.