Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Assessing prediction error in autoregressive models


Authors: Ping Zhang and Paul Shaman
Journal: Trans. Amer. Math. Soc. 347 (1995), 627-637
MSC: Primary 62M10; Secondary 62M20
DOI: https://doi.org/10.1090/S0002-9947-1995-1277143-9
MathSciNet review: 1277143
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Assessing prediction error is a problem which arises in time series analysis. The distinction between the conditional prediction error $ e$ and the unconditional prediction error $ E(e)$ has not received much attention in the literature. Although one can argue that the conditional version is more practical, we show in this article that assessing $ e$ is nearly impossible. In particular, we use the correlation coefficient $ \operatorname{corr} (\hat e,e)$, where $ \hat e$ is an estimate of $ e$, as a measure of performance and show that $ {\lim _{T \to \infty }}\sqrt T \operatorname{corr} (\hat e,e) = C$ where $ T$ is the sample size and $ C > 0$ is some constant. Furthermore, the value of $ C$ is large only when the process is extremely non-Gaussian or nearly nonstationary.


References [Enhancements On Off] (What's this?)

  • [1] H. Akaike, Statistical predictor identification, Ann. Inst. Statist. Math. 22 (1970), 203-217. MR 0286233 (44:3447)
  • [2] -, Information theory and an extension of the maximum likelihood principle, Proc. 2nd International Symposium on Information Theory (B. N. Petrov and Csáki, eds.), Akademiai Kiado, Budapest, 1973, pp. 267-281. MR 0483125 (58:3144)
  • [3] T. W. Anderson and R. P. Mentz, On the structure of the likelihood function of autoregressive and moving average models, J. Time Ser. Anal. 1 (1980), 83-94. MR 622137 (83b:62175)
  • [4] D. R. Brillinger, Some history of the study of higher-order moments and spectra, Statistica Sinica 1 (1991), 465-476. MR 1130128 (92j:62003)
  • [5] W. A. Fuller and D. P. Hasza, Properties of predictors for autoregressive time series, J. Amer. Statist. Assoc. 76 (1981), 155-161. MR 608187 (82c:62132)
  • [6] I. Johnstone, On inadmissibility of some unbiased estimates of loss, Statistical Decision Theory and Related Topics IV, vol. 1 (S. S. Gupta and J. O. Berger, eds.), Springer-Verlag, New York, 1988, pp. 361-379. MR 927112 (89c:62017)
  • [7] H. Linhart and W. Zucchini, Model selection, Wiley, New York, 1986. MR 866144 (88a:62002)
  • [8] P. Shaman, Properties of estimates of the mean square error of prediction in autoregressive models, Studies in Econometrics, Time Series, and Multivariate Statistics (S. Karlin, T. Amemiya, and L. A. Goodman, eds.), Academic Press, New York, 1983, pp. 331-342. MR 738660 (85h:62126)
  • [9] P. Shaman and R. A. Stine, The bias of autoregressive coefficient estimators, J. Amer. Statist. Assoc. 83 (1988), 842-848. MR 963814
  • [10] R. Shibata, Selection of the order of an autoregressive model by Akaike's information criterion, Biometrika 63 (1976), 117-126. MR 0403130 (53:6943)
  • [11] T. P. Speed, What is an analysis of variance (with discussion)?, Ann. Statist. 15 (1987), 885-941. MR 902237 (88k:62126)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 62M10, 62M20

Retrieve articles in all journals with MSC: 62M10, 62M20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1277143-9
Keywords: Conditional prediction error, correlation, cumulant, higher-order spectrum, non-Gaussian model
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society