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 . The second‐order asymptotic approximation of 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 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.     

Bias‐corrected bootstrap prediction intervals for autoregressive model: new alternatives with applications to tourism forecasting

This paper proposes the use of the bias‐corrected bootstrap for interval forecasting of an autoregressive time series with an arbitrary number of deterministic components. We use the bias‐corrected bootstrap based on two alternative bias‐correction methods: the bootstrap and an analytic formula based on asymptotic expansion. We also propose a new stationarity‐correction method, based on stable spectral factorization, as an alternative to Kilian's method exclusively used in past studies. A Monte Carlo experiment is conducted to compare small‐sample properties of prediction intervals. The results show that the bias‐corrected bootstrap prediction intervals proposed in this paper exhibit desirable small‐sample properties. It is also found that the bootstrap bias‐corrected prediction intervals based on stable spectral factorization are tighter and more stable than those based on Kilian's stationarity‐correction. The proposed methods are applied to interval forecasting for the number of tourist arrivals in Hong Kong. Copyright © 2010 John Wiley & Sons, Ltd.     

A non‐Gaussian generalization of the Airline model for robust seasonal adjustment

In their seminal book Time Series Analysis: Forecasting and Control, Box and Jenkins (1976) introduce the Airline model, which is still routinely used for the modelling of economic seasonal time series. The Airline model is for a differenced time series (in levels and seasons) and constitutes a linear moving average of lagged Gaussian disturbances which depends on two coefficients and a fixed variance. In this paper a novel approach to seasonal adjustment is developed that is based on the Airline model and that accounts for outliers and breaks in time series. For this purpose we consider the canonical representation of the Airline model. It takes the model as a sum of trend, seasonal and irregular (unobserved) components which are uniquely identified as a result of the canonical decomposition. The resulting unobserved components time series model is extended by components that allow for outliers and breaks. When all components depend on Gaussian disturbances, the model can be cast in state space form and the Kalman filter can compute the exact log‐likelihood function. Related filtering and smoothing algorithms can be used to compute minimum mean squared error estimates of the unobserved components. However, the outlier and break components typically rely on heavy‐tailed densities such as the t or the mixture of normals. For this class of non‐Gaussian models, Monte Carlo simulation techniques will be used for estimation, signal extraction and seasonal adjustment. This robust approach to seasonal adjustment allows outliers to be accounted for, while keeping the underlying structures that are currently used to aid reporting of economic time series data. Copyright © 2006 John Wiley & Sons, Ltd.