Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T18:17:54.940Z Has data issue: false hasContentIssue false
  • English
  • Français

Unbiasedness of Predictions from Etimated Vector Autoregressions

Published online by Cambridge University Press:  18 October 2010

Jean-Marie Dufour
Jean-Marie Dufour*
Affiliation:
Université de Montréal
Get access Rights & Permissions [Opens in a new window]

Abstract

Forecasts from a univariate autoregressive model estimated by OLS are unbiased, irrespective of whether the model fitted has the correct order; this property only requires symmetry of the distribution of the innovations. In this paper, this result is generalized to vector autoregressions and a wide class of multivariate stochastic processes (which include Gaussian stationary multivariate stochastic processes) is described for which unbiasedness of predictions holds: specifically, if a vector autoregression of arbitrary finite order is fitted to a sample from any process in this class, the fitted model will produce unbiased forecasts, in the sense that the prediction errors have distributions symmetric about zero. Different numbers of lags may be used for each variable in each autoregression and variables may even be missing, without unbiasedness being affected. This property is exact in finite samples. Similarly, the residuals from the same autoregressions have distributions symmetric about zero.

Type
Research Article
Information
Econometric Theory , Volume 1 , Issue 3 , December 1985 , pp. 387 - 402
Copyright
Copyright © Cambridge University Press 1985 

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

1. Akaike, H. Statistical predictor identification. Annals of the Institute of Statistical Mathematics 22 (1970): 203–217.10.1007/BF02506337Google Scholar
2. Baillie, R. T. Asymptotic prediction mean squared error for vector autoregressive models. Biometrika 66 (1979a): 675–678.Google Scholar
3. Baillie, R. T. The asymptotic mean squared error of multistep prediction from the regression model with autoregressive errors. Journal of the American Statistical Association 74 (1979b): 175–184.Google Scholar
4. Baillie, R. T. Prediction from ARMAX models. Journal of Econometrics 12 (1980): 365–374.10.1016/0304-4076(80)90062-7Google Scholar
5. Baillie, R. T. Prediction from the dynamic simultaneous equation model with vector auto-regressive errors. Econometrica 49 (1981): 1331–1337.10.2307/1912757Google Scholar
6. Barone, P. & Roy, R.. On a predictive test for stability of multivariate autoregressive-moving average models. Publication 457, Département d'informatique et de recherche opérationnelle, Université de Montréal, 1983.Google Scholar
7. Bhansali, R. J. Asymptotic mean-square error of predicting more than one-step ahead using the regression method. Applied Statistics 2 (1974): 35–42.10.2307/2347051Google Scholar
8. Bhansali, R. J. Linear prediction by autoregressive model fitting in the time domain. The Annals of Statistics 6 (1978): 224–231.Google Scholar
9. Bhansali, R. J. Effects of not knowing the order of an autoregressive process on the mean squared error of prediction—I. Journal of the American Statistical Association 76 (1981): 588–597.Google Scholar
10. Bloomfield, P. On the error of prediction of a time series. Biometrika 59 (1972): 501–507.10.1093/biomet/59.3.501Google Scholar
11. Box, G. E. P. & Jenkins, G.. Time series analysis, forecasting and control. San Francisco: Holden-Oay, 1970.Google Scholar
12. Cleveland, W. S. Fitting time series models for prediction. Technometrics 13 (1971): 713–723.10.1080/00401706.1971.10488843Google Scholar
13. Cogger, K. O. Specification analysis. Journal of the American Statistical Association 68 (1973): 899–905.10.1080/01621459.1973.10481443Google Scholar
14. Davies, N. & Newbold, P.. Forecasting with misspecified models. Applied Statistics 29 (1980): 87–92.10.2307/2346415Google Scholar
15. Davisson, I. D. The prediction error of stationary Gaussian time series of unknown covari-ance. IEEE Transactions on Information Theory IT-11 (1965): 527–532.10.1109/TIT.1965.1053829Google Scholar
16. Doan, T., Litterman, R. & Sims, C.. Forecasting and conditional projection using realistic prior distributions. Econometric Reviews 3 (1984): 1–100.Google Scholar
17. Don, F. J. M. & Magnus, J. R.. On the unbiasedness of iterated GLS estimators. Communications in Statistics—Theory and Methods A9 (1980): 519–527.Google Scholar
18. Dufour, J. M. Unbiasedness of predictions from estimated autoregressions when the true order is unknown. Econometrica 52 (1984): 209–215.10.2307/1911469Google Scholar
19. Fotopoulos, S. B. & Ray, W. D.. Components of prediction errors for a stationary process with estimated parameters. Journal of Time Series Analysis 4 (1983): 1–8.10.1111/j.1467-9892.1983.tb00352.xGoogle Scholar
20. Fuller, W. A. & Battese, G. E.. Transformations for estimation of linear models with nested-error structure. Journal of the American Statistical Association 68 (1973): 626–632.10.1080/01621459.1973.10481396Google Scholar
21. Fuller, W. & Hasza, D.P.. Predictors for the first-order autoregressive process. Journal of Econometrics 13 (1980): 139–157.Google Scholar
22. Fuller, W. & Hasza, D.P.. Properties of predictors for autoregressive time series. Journal of the American Statistical Association 76 (1981): 155–161.10.1080/01621459.1981.10477622Google Scholar
23. Guilkey, D. K. & Schmidt, P.. Estimation of seemingly unrelated regressions with vector autoregressive errors. Journal of the American Statistical Association 68 (1973): 642–647.10.1080/01621459.1973.10481399Google Scholar
24. Hartley, H. O. & Jayatillake, K. S. E.. Estimation for linear models with unequal variances. Journal of the American Statistical Association 68 (1973): 189–192.10.1080/01621459.1973.10481361Google Scholar
25. Harvey, A. C. On the unbiasedness of robust regression estimates. Communications in Statistics—Theory and Methods A7 (1978): 779–783.10.1080/03610927808827668Google Scholar
26. Hurwicz, L. Least squares bias in time series. In Koopmans, T. C. (ed.), Statistical Inference in Dynamic Economic Models. New York: Wiley, 1950.Google Scholar
27. Jenkins, G. M. & Alavi, A. S.. Some aspects of modelling and forecasting multivariate time series. Journal of Time Series Analysis 2 (1981): 1–47.10.1111/j.1467-9892.1981.tb00309.xGoogle Scholar
28. Kackar, R. N. & Harville, D. A.. Unbiasedness of two-stage estimation and prediction pro -cedures for mixed linear models. Communications in Statistics—Theory and Methods A10 (1981): 1249–1261.10.1080/03610928108828108Google Scholar
29. Kakwani, N. C. The unbiasedness of Zellner's seemingly unrelated regression equations estimators. Journal of the American Statistical Association 62 (1967): 141–142.10.1080/01621459.1967.10482895Google Scholar
30. Kakwani, N. C. Note on the unbiasedness of a mixed regression estimator. Econometrica 36 (1968): 610–611.Google Scholar
31. Khatri, C. G. & Shah, K. R.. On the unbiased estimation of fixed effects in a mixed model for growth curves. Communications in Statistics—Theory and Methods A10 (1981): 401–406.10.1080/03610928108828046Google Scholar
32. Loève, M. Probability Theory I, Fourth Edition. New York: Springer, 1977.Google Scholar
33. Magnus, J. R. Maximum likelihood estimation of the GLS model with unknow n parameters i n the disturbance covariance matrix. Journal of Econometrics 1 (1978): 281–312.10.1016/0304-4076(78)90056-8Google Scholar
34. Malinvaud, E. M'ethodes statistiques de l'économétrie, 2nd ed. Paris: Dunod, 1969.Google Scholar
35. Phillips, P. C. B. The sampling distribution of forecasts from a first-order autoregression. Journal of Econometrics 9 (1979): 241–261.10.1016/0304-4076(79)90073-3Google Scholar
36. Reinsel, G. Asymptotic properties of prediction errors for the multivariate autoregressive model using estimated parameters. Journal of the Royal Statistical Society, Series B 76 (1980): 802–816.Google Scholar
37. Sawa, T. The exact moments of the least squares estimator for the autoregressive model. Journal of Econometrics 8 (1978): 159–172.10.1016/0304-4076(78)90025-8Google Scholar
38. Schmidt, P. The asymptotic distribution of forecasts in the dynamic simulation of an econometric model. Econometrica 42 (1974): 303–309.10.2307/1911980Google Scholar
39. Seely, J. & Hogg, R. V.. Symmetrically distributed and unbiased estimators in linear models. Communications in Statistics—Theory and Methods All (1982): 721–729.10.1080/03610928208828266Google Scholar
40. Shenton, R. L. & Johnson, W. L.. Moments of a serial correlation coefficient. Journal of the Royal Statistical Society, Series B 27 (1965): 308–320.Google Scholar
41. Sims, C. A. Macroeconomics and reality. Econometrica 48 (1980): 1–48.10.2307/1912017Google Scholar
42. Spitzer, I. J. & Baillie, R. T.. Small-sample properties of predictions from the regression model with autoregressive errors. Journal of the American Statistical Association 78 (1983): 258–263.10.1080/01621459.1983.10477958Google Scholar
43. Tanaka, K. & Maekawa, K.. The sampling distribution of the predictor for an autoregressive model under misspecifications. Journal of Econometrics 25 (1984): 327–351.10.1016/0304-4076(84)90005-8Google Scholar
44. Tiao, G. C. & Box, G. E. P.. Modeling multiple time series with applications. Journal of the American Statistical Association 76 (1981): 802–816.Google Scholar
45. Toyooka, Y. Prediction error in a linear model with estimated parameters. Biometrika 69 (1982): 453–459.10.1093/biomet/69.2.453Google Scholar
46. White, J. S. Asymptotic expansions for the mean and variance of the serial correlation coefficient. Biometrika 48 (1961): 85–94.10.1093/biomet/48.1-2.85Google Scholar
47. Yajima, Y. On prediction of integrated moving average processes. Annals of the Institute of Statistical Mathematics 32 (1980): 81–94.10.1007/BF02480313Google Scholar
48. Yamamoto, T. Asymptotic mean square prediction error for an autoregressive model with estimated coefficients. Applied Statistics 25 (1976): 123–127.10.2307/2346680Google Scholar
49. Yamamoto, T. Predictions of multivariat e autoregressive-moving average models. Biomet-rika 65 (1981): 485–492.10.1093/biomet/68.2.485Google Scholar