Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T08:08:51.825Z Has data issue: false hasContentIssue false
  • English
  • Français

OPTIMALITY OF GLS FOR ONE-STEP-AHEAD FORECASTING WITH REGARIMA AND RELATED MODELS WHEN THE REGRESSION IS MISSPECIFIED

Published online by Cambridge University Press:  06 September 2007

David F. Findley
David F. Findley
Affiliation:
U.S. Census Bureau
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the modeling of a time series described by a linear regression component whose regressor sequence satisfies the generalized asymptotic sample second moment stationarity conditions of Grenander (1954, Annals of Mathematical Statistics 25, 252–272). The associated disturbance process is only assumed to have sample second moments that converge with increasing series length, perhaps after a differencing operation. The model's regression component, which can be stochastic, is taken to be underspecified, perhaps as a result of simplifications, approximations, or parsimony. Also, the autoregressive moving average (ARMA) or autoregressive integrated moving average (ARIMA) model used for the disturbances need not be correct. Both ordinary least squares (OLS) and generalized least squares (GLS) estimates of the mean function are considered. An optimality property of GLS relative to OLS is obtained for one-step-ahead forecasting. Asymptotic bias characteristics of the regression estimates are shown to distinguish the forecasting performance. The results provide theoretical support for a procedure used by Statistics Netherlands to impute the values of late reporters in some economic surveys.The author thanks two referees and the co-editor for comments and suggestions that led to substantial improvements in the exposition and also thanks John Aston and Tucker McElroy for helpful comments on an earlier draft. Any views expressed are the author's and not necessarily those of the U.S. Census Bureau.

Type
Research Article
Information
Econometric Theory , Volume 23 , Issue 6 , December 2007 , pp. 1083 - 1107
Copyright
© 2007 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Aelen, F. (2004) Improving Timeliness of Industrial Short-Term Statistics Using Time Series Analysis. Statistics Netherlands, Division of Technology and Facilities, Methods and Informatics Discussion paper 04005. Statistics Netherlands. http://www.cbs.nl/NR/rdonlyres/EC77EDFA-1579-4AF6-9763-8EFEEB9F4CC5/0/discussionpaper04005.pdf.
Amemiya, T. (1973) Generalized least squares with an estimated autocovariance matrix. Econometrica 41, 723–732.Google Scholar
Anderson, T.W. (1971) The Statistical Analysis of Time Series. Wiley.
Bell, W.R. & S.C. Hillmer (1983) Modelling time series with calendar variation. Journal of the American Statistical Association 78, 526–534.Google Scholar
Box, G.E.P. & G.M. Jenkins (1976) Time Series Analysis: Forecasting and Control, rev. ed. Holden-Day.
Box, G.E.P. & G.C. Tiao (1975) Intervention analysis with applications to economic and environmental problems. Journal of the American Statistical Association 70, 70–79.Google Scholar
Brillinger, D.R. (1975) Time Series. Holt, Rinehart and Winston.
Findley, D.F. (2003) Convergence of Estimates of Misspecified regARIMA Models and Generalizations and the Optimality of Estimated GLS Regression Coefficients for One-Step-Ahead Forecasting. Statistical Research Division Research Report Statistics 2003-03. U.S. Census Bureau. http://www.census.gov/srd/papers/pdf/ rrs2003-06.pdf.
Findley, D.F. (2005) Asymptotic stationarity properties of out-of-sample forecast errors of misspecified regARIMA models and the optimality of GLS for one-step-ahead forecasting. Statistica Sinica 15, 447–476.Google Scholar
Findley, D.F., B.C. Monsell, W.R. Bell, M.C. Otto, & B.C. Chen (1998) New capabilities and methods of the X-12-ARIMA seasonal adjustment program. Journal of Business & Economic Statistics 16, 127–177 (with discussion).Google Scholar
Findley, D.F., B.M. Pötscher, & C.Z. Wei (2001) Uniform convergence of sample second moments of time series arrays. Annals of Statistics 29, 815–838.Google Scholar
Findley, D.F., B.M. Pötscher, & C.Z. Wei (2004) Modeling of time series arrays by multistep prediction or likelihood methods. Journal of Econometrics 118, 151–187.Google Scholar
Findley, D.F. & R.J. Soukup (2000) Modeling and model selection for moving holidays. In 2000 Proceedings of the Business and Economic Statistics Section of American Statistical Association, pp. 102–107. American Statistical Association. Also http://www.census.gov/ts/papers/asa00_eas.pdf.
Grenander, U. (1954) On the estimation of regression coefficients in the case of an autocorrelated disturbance. Annals of Mathematical Statistics 25, 252–272.Google Scholar
Grenander, U. & M. Rosenblatt (1984) Time Series, 2nd ed. Chelsea.
Hannan, E.J. (1970) Multiple Time Series. Wiley.
Hannan, E.J. (1982) Testing for autocorrelation and Akaike's criterion. In J. Gani & E.J. Hannan (eds.), Essays in Statistical Science, Papers in Honour of P.A.P. Moran, pp. 403–412. Applied Probability Trust.
Koreisha, S.G. & Y. Fang (2001) Generalized least squares with misspecified correlation structure. Journal of the Royal Statistical Society, Series B 63, 515–532.Google Scholar
Newton, H.J. & M. Pagano (1983) The finite memory prediction of covariance stationary time series. SIAM Journal of Scientific and Statistical Computation 4, 330–339.Google Scholar
Peña, D., G.E. Tiao, & R.S. Tsay (2001) A Course in Time Series Analysis. Wiley.
Pierce, D.A. (1971) Least squares estimation in the regression model with autoregressive-moving average errors. Biometrika 58, 299–312.Google Scholar
Pötscher, B.M. (1987) Convergence results for maximum likelihood type estimators in multivariate ARMA models. Journal of Multivariate Analysis 21, 29–52.Google Scholar
Pötscher, B.M. (1991) Noninvertibility and pseudo-maximum likelihood estimation of misspecified ARMA models. Econometric Theory 7, 435–449. Corrections: Econometric Theory 10, 811.Google Scholar
Pötscher, B.M. & I.R. Prucha (1997) Dynamic Nonlinear Econometric Models: Asymptotic Theory. Springer-Verlag.
Stock, J.H. & M.W. Watson (2002) Introduction to Econometrics. Addison-Wesley.
Thursby, J.G. (1987) OLS or GLS in the presence of misspecification error? Journal of Econometrics 35, 359–374.Google Scholar
West, K.D. (1996) Asymptotic inference about predictive ability. Econometrics 64, 1067–1084.Google Scholar