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MULTISTEP PREDICTION IN AUTOREGRESSIVE PROCESSES

Published online by Cambridge University Press:  31 January 2003

CHING-KANG ING
CHING-KANG ING
Affiliation:
National Taipei University
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Abstract

In this paper, two competing types of multistep predictors, i.e., plug-in and direct predictors, are considered in autoregressive (AR) processes. When a working model AR(k) is used for the h-step prediction with h > 1, the plug-in predictor is obtained from repeatedly using the fitted (by least squares) AR(k) model with an unknown future value replaced by their own forecasts, and the direct predictor is obtained by estimating the h-step prediction model's coefficients directly by linear least squares. Under rather mild conditions, asymptotic expressions for the mean-squared prediction errors (MSPEs) of these two predictors are obtained in stationary cases. In addition, we also extend these results to models with deterministic time trends. Based on these expressions, performances of the plug-in and direct predictors are compared. Finally, two examples are given to illustrate that some stationary case results on these MSPEs can not be generalized to the nonstationary case.The author is deeply grateful to the co-editor Pentti Saikkonen and two referees for their helpful suggestions and comments on a previous version of this paper.

Type
Research Article
Information
Econometric Theory , Volume 19 , Issue 2 , April 2003 , pp. 254 - 279
Copyright
© 2003 Cambridge University Press

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References

REFERENCES

Bhansali, R.J. (1996) Asymptotically efficient autoregressive model selection for multistep prediction. Annals of the Institute of Statistical Mathematics 48, 577–602.CrossRefGoogle Scholar
Bhansali, R.J. (1997) Direct autoregressive predictors for multistep prediction: order selection and performance relative to the plug in predictors. Statistica Sinica 7, 425–449.Google Scholar
Bhansali, R.J. & F. Papangelou (1991) Convergence of moments of least squares estimators for the coefficients of an autoregressive process of unknown order. Annals of Statistics 19, 1155–1162.CrossRefGoogle Scholar
Brockwell, P.J. & R.A. Davis (1987) Time Series: Theory and Methods. New York: Springer-Verlag.
Findley, D.F. (1984) On some ambiguities associated with the fitting of ARMA models to time series. Journal of Time Series Analysis 5, 213–225.CrossRefGoogle Scholar
Findley, D.F. & C.Z. Wei (2002) AIC, overfitting principles, and boundedness of moments of inverse matrices for vector autoregressive models fit to weakly stationary time series. Journal of Multivariate Analysis, forthcoming.Google Scholar
Fuller, W.A. & D.P. Hasza (1981) Properties of predictors for autoregressive time series. Journal of American Statistical Association 76, 155–161.CrossRefGoogle Scholar
Hamilton, J.D. (1994) Time Series Analysis. Princeton: Princeton University Press.
Hemerly, E.M. & M.H.A. Davis (1989) Strong consistency of the PLS criterion for order determination of autoregressive processes. Annals of Statistics 17, 941–946.CrossRefGoogle Scholar
Ing, C.K. (2001) A note on mean-squared prediction errors of the least squares predictors in random walk models. Journal of Time Series Analysis 22, 711–724.CrossRefGoogle Scholar
Ing, C.K. (2002) Selecting optimal multistep predictors in autoregressive processes. Manuscript.
Ing, C.K. & C.Z. Wei (2002) On same-realization prediction in an infinite-order autoregressive process. Journal of Multivariate Analysis, forthcoming.Google Scholar
Kunitomo, N. & T. Yamamoto (1985) Properties of predictors in misspecified autoregressive time series. Journal of the American Statistical Association 80, 941–950.CrossRefGoogle Scholar
Papangelou, F. (1994) On a distributional bound arising in autoregressive model fitting. Journal of Applied Probability 31, 401–408.CrossRefGoogle Scholar
Rissanen, J. (1986) Order estimation by accumulated prediction errors. In Essays in Time Series and Applied Processes, special volume 23A of Journal of Applied Probability, 55–61.Google Scholar
Schwarz, G. (1978) Estimating of dimension of a model. Annals of Statistics 6, 461–464.CrossRefGoogle Scholar
Shibata, R. (1980) Asymptotically efficient selection of the order of the model for estimating parameters of a linear process. Annals of Statistics 8, 147–164.CrossRefGoogle Scholar
Wei, C.Z. (1987) Adaptive prediction by least squares predictors in stochastic regression models with application to time series. Annals of Statistics 15, 1667–1682.CrossRefGoogle Scholar