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Misspecified prediction for time series
Authors:In‐Bong Choi  Masanobu Taniguchi
Abstract:Let {Xt} 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 fθ(λ), θ∈Θ, is fitted to g(λ). Then, constructing the best linear predictor based on fθ(λ), we can evaluate the prediction error M(θ). Since θ is unknown we estimate it by a quasi‐MLE equation image . The second‐order asymptotic approximation of equation image is given. This result is extended to the case when Xt contains some trend, i.e. a time series regression model. These results are very general. Furthermore we evaluate the second‐order asymptotic approximation of equation image 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.
Keywords:stationary process  misspecified prediction  multistep prediction  spectral density  conjectured spectral density  best linear predictor  quasi‐MLE  time series regression model  long‐memory process
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