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PREDICTION ERRORS IN NONSTATIONARYAUTOREGRESSIONS OF INFINITE ORDER

Published online by Cambridge University Press:  26 October 2009

Ching-Kang Ing,
Chor-yiu Sin and
Shu-Hui Yu
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Abstract

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Assume that observations are generated fromnonstationary autoregressive (AR) processes ofinfinite order. We adopt a finite-orderapproximation model to predict future observationsand obtain an asymptotic expression for themean-squared prediction error (MSPE) of the leastsquares predictor. This expression provides thefirst exact assessment of the impacts ofnonstationarity, model complexity, and modelmisspecification on the corresponding MSPE. It notonly provides a deeper understanding of the leastsquares predictors in nonstationary time series, butalso forms the theoretical foundation for acompanion paper by the same authors, which obtainsasymptotically efficient order selection innonstationary AR processes of possibly infiniteorder.

Information

Type
Research Article
Information
Econometric Theory , Volume 26 , Issue 3 , June 2010, pp. 774 - 803
Copyright
Copyright © Cambridge University Press 2009

Footnotes

The authors are deeply grateful to a co-editorand two referees for their helpful suggestions andcomments. The research of the first and thirdauthors as partially supported by the NationalScience Council of Taiwan under grants NSC94-2118-M-001-013 and NSC 94-2416-H-260-019,respectively.

References

REFERENCES

Akaike, H. (1974) A new look at the statistical model identification. IEEE Transactions on Automatic Control 19, 716–723.CrossRefGoogle Scholar
Barria, J. & Halmos, P.R. (1982) Asymptotic toeplitz operators. Transactions of the American Mathematical Society 273, 621–630.CrossRefGoogle Scholar
Berk, K.N. (1974) Consistent autoregressive spectral estimates. Annals of Statistics 2, 489–502.10.1214/aos/1176342709CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measure. Wiley.Google Scholar
Chan, N.H. & Wei, C.Z. (1988) Limiting distribution of least squares estimates of unstable autoregressive processes. Annals of Statistics 19, 367–401.Google Scholar
Choi, M.-D. (1983) Tricks or treats with the Hilbert matrix. American Mathematical Monthly 90, 301–312.CrossRefGoogle Scholar
Chow, Y.S. & Teicher, H. (1997) Probability Theory: Independence, Interchangeability, Martingale, 3rd ed. Springer-Verlag.CrossRefGoogle Scholar
Findley, D.F. & Wei, C.Z. (2002) AIC, overfitting principles, and the boundedness of moments of inverse matrices for vector autoregressions and related models. Journal of Multivariate Analysis 83, 415–450.CrossRefGoogle Scholar
Fuller, W.A. & Hasza, D.P. (1981) Properties of predictors for autoregressive time series. Journal of the American Statistical Association 76, 155–161.CrossRefGoogle Scholar
Gerencsér, L. (1992) AR (∞) estimation and nonparametric stochastic complexity. IEEE Transactions on Information Theory 38, 1768–1778.CrossRefGoogle Scholar
Ing, C.-K. (2003) Multistep prediction in autoregressive processes. Econometric Theory 19, 254–279.CrossRefGoogle Scholar
Ing, C.-K. (2007) Accumulated prediction errors, information criteria and optimal forecasting for autoregressive time series. Annals of Statistics 35, 1238–1277.CrossRefGoogle Scholar
Ing, C.-K. & Sin, C.-y. (2006) On prediction errors in regression models with nonstationary regressors. In Ho, H.C., Ing, C.-K., and Lai, T.L. (eds.), Time Series and Related Topics: In Memory of Ching-Zong Wei, pp. 60–71. IMS Lecture Notes–Monograph Series 52. Institute of Mathematical Statistics.CrossRefGoogle Scholar
Ing, C.-K., Sin, C.-y., & Yu, S.-H. (2007) Using Information Criteria to Select an Autoregressive Model: A Unified Approach without Knowing the Order of Integratedness. Technical report, Academia Sinica.Google Scholar
Ing, C.-K. & Wei, C.Z. (2003) On same-realization prediction in an infinite-order autoregressive process. Journal of Multivariate Analysis 85, 130–155.CrossRefGoogle Scholar
Ing, C.-K. & Wei, C.Z. (2005) Order selection for same-realization predictions in autoregressive processes. Annals of Statistics 33, 2423–2474.CrossRefGoogle Scholar
Kunitomo, N. & Yamamoto, T. (1985) Properties of predictors in misspecified autoregressive time series. Journal of the American Statistical Association 80, 941–950.CrossRefGoogle Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277–301.CrossRefGoogle Scholar
Shibata, R. (1980) Asymptotic efficient selection of the order of the model for estimating parameters of a linear process. Annals of Statistics 8, 147–164.10.1214/aos/1176344897CrossRefGoogle Scholar
Wei, C.Z. (1987) Adaptive prediction by least squares predictors in stochastic regression models with applications to time series. Annals of Statistics 15, 1667–1682.CrossRefGoogle Scholar