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Bayesian Forecasting of Economic Time Series

Published online by Cambridge University Press:  11 February 2009

Bruce M. Hill
Bruce M. Hill
Affiliation:
University of Michigan
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Abstract

A model is suggested to forecast economic time series. This model incorporates some innovative ideas of Harrison and Stevens [20] for building into the forecasting process important external shocks to the systems. Thus the occurrence of possibly significant real-world events may cause a fundamental change in the time series in question. The Jeffreys-Savage (JS) Bayesian theory of hypothesis testing is used to test the hypothesis that a particular event has been such as to free the series from its immediate past behavior. When the event frees the series in this way, then we model the sequence of observations following such an event (until the next such event) as an exchangeable sequence. In the simplest case of 0–1 valued data, such as in recording the ups and downs of the value of a particular commodity or stock, our alternative hypothesis is a Pólya process, and the null hypothesis is a simple random walk (unit roots model) with p = .50. Any exchangeable sequence is strictly stationary, and the observations in the Polya process are positively correlated, which can give rise to “explosive” behavior of the series at isolated time points. We then use the JS theory to predict future observations by taking a weighted average of the optimal predictions for each model, with weights given by the posterior probabilities of the hypotheses. Results of simulation studies are presented which compare the predictive performance of the fully Bayesian method based upon the JS theory with those based upon the “p-value” or pre-test method. The de Finetti method for scoring predictions is used to assess their empirical performance. A theoretical methodology, which extends the “evaluation game” of Hill [28,37], is developed for comparing predictors.

Type
Articles
Information
Econometric Theory , Volume 10 , Issue 3-4 , August 1994 , pp. 483 - 513
Copyright
Copyright © Cambridge University Press 1994

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References

REFERENCES

1.Aitchison, J. & Dunsmore, I.R.. Statistical Prediction Analysis. Cambridge University Press, 1975.CrossRefGoogle Scholar
2.Arthur, B., Ermolév, Y. & Kaniovskii, Y.. A generalized urn problem and its applications. Cybernetics 19 (1983): 61–71.CrossRefGoogle Scholar
3.Bachelier, L.Théorie de la speculation. Ann. Sci. Ecole. Norm. Sup. 3 17 (1900): 21–86.CrossRefGoogle Scholar
4.Berger, J. In defense of the likelihood principle: Axiomatics and coherence. In Bernardo, J.M., DeGroot, M.H., Lindley, D.V. & Smith, A.F.M. (eds.), Bayesian Statistics 2 Amsterdam: North-Holland, 1985.Google Scholar
5.Bhat, A. Applications of Bayesian Statistics in Econometrics. Ph.D. Dissertation, University of Michigan, 1988.Google Scholar
6.Blackwell, D. & Girshick, M.A.. Theory of Games and Statistical Decisions. New York: Wiley, 1954.Google Scholar
7.Box, G. & Jenkins, G.M.. Time Series Analysis: Forecasting and Control, rev. ed. San Francisco: Holden-Day, 1976.Google Scholar
8.de Finetti, B.La prévision: ses lois logiques, ses sources subjectives. Annales de I'Institut Henri Poincaré 1 (1937): 1–68.Google Scholar
9.de Finetti, B.Theory of Probability, Vol. 1. London: Wiley, 1974.Google Scholar
10.de Finetti, B.Theory of Probability, Vol. 2. London: Wiley, 1975.Google Scholar
11.DeGroot, M.Optimal Statistical Decisions. New York: McGraw-Hill, 1970.Google Scholar
12.Diaconis, P. & Freedman, D.. Finite exchangeable sequences. Annals of Probability 8 (1980): 745–764.CrossRefGoogle Scholar
13.Doan, T.A., Litterman, R.B. & Sims, C.. Forecasting and conditional projection using realistic prior distributions. Econometric Reviews 1 (1984): 1–100.CrossRefGoogle Scholar
14.Doob, J.Stochastic Processes. New York: Wiley, 1953.Google Scholar
15.Eggenberger, F. & Pólya, G.. Über die Statistik verketteter Vorgänge. Z. Angew. Math. Mech. 3 (1923): 279–289.CrossRefGoogle Scholar
16.Feller, W.An Introduction to Probability Theory and Its Applications, Volume 1, 3rd ed., rev. New York: Wiley, 1968.Google Scholar
17.Feller, W.An Introduction to Probability Theory and Its Applications, Volume 2, 2nd rev. ed.New York: Wiley, 1971.Google Scholar
18.Ferguson, T.S.Mathematical Statistics: A Decision Theoretic Approach. New York: Acadmic Press, 1967.Google Scholar
19.Freedman, D.A.Comment on “Some subjective Bayesian considerations in the selection of models,” by B.M. Hill. Econometric Reviews 4 (1985–1986): 247–251.CrossRefGoogle Scholar
20.Harrison, P.J. & Stevens, C.F.. Bayesian forecasting. Journal of the Royal Statistical Society, B 38 (1976): 205–247.Google Scholar
21.Heath, D. & Sudderth, W.. De Finetti's theorem for exchangeable random variables. American Statistician 30 (1976): 188–189.CrossRefGoogle Scholar
22.Heath, D. & Sudderth, W.. On finitely additive priors, coherence, and extended admissibility. Annals of Statistics 6 (1978): 333–345.CrossRefGoogle Scholar
23.Hewitt, E. & Savage, L.J.. Symmetric measures on cartesian products. In The Writings of Leonard Jimmie Savage—A Memorial Selection, pp. 244–275. American Statistical Association and Institute of Mathematical Statistics, 1981.Google Scholar
24.Hill, B.M.Inference about variance components in the one-way model. Journal of the American Statistical Association 58 (1965): 918–932.CrossRefGoogle Scholar
25.Hill, B.M.Correlated errors in the random model. Journal of the American Statistical Association 62 (1967): 1387–1400.CrossRefGoogle Scholar
26.Hill, B.M.Posterior distribution of percentiles: Bayes theorem for sampling from a finite population. Journal of the American Statistical Association 63 (1968): 677–691.Google Scholar
27.Hill, B.M.Foundations for the theory of least squares. Journal of the Royal Statistical Society B 31 (1969): 89–97.Google Scholar
28.Hill, B.M. On coherence, inadmissibility and inference about many parameters in the theory of least squares. In Fienberg, S. & Zellner, A. (eds.), Studies in Bayesian Econometrics and Statistics in Honor of L.J. Savage, pp. 555–584. Amsterdam: North Holland, 1974.Google Scholar
29.Hill, B.M.The rank frequency form of Zipf's law. Journal of the American Statistical Association 69 (1974): 1017–1026.CrossRefGoogle Scholar
30.Hill, B.M.A simple general approach to inference about the tail of a distribution. Annals of Statistics 3 (1975): 1163–1174.CrossRefGoogle Scholar
31.Hill, B.M. Decision theory. In Hogg, R.V. (ed.), Studies in Statistics, Volume 19, pp. 168–209. Mathematical Association of America, 1978.Google Scholar
32.Hill, B.M. Robust analysis of the random model and weighted least squares regression. In Kmenta, J. & Ramsey, J. (eds.), Evaluation of Econometric Models, pp. 197–217. New York: Academic Press, 1980.CrossRefGoogle Scholar
33.Hill, B.M.Comment on “Lindley's Paradox,” by G. Shafer. Journal of the American Statistical Association 77 (1982): 344–347.Google Scholar
34.Hill, B.M.Some subjective Bayesian considerations in the selection of models. Econometric Reviews 4 (1985–1986): 191–288.CrossRefGoogle Scholar
35.Hill, B.M. De Finetti's theorem, induction, and An, or Bayesian nonparametric predictive inference. In Bernardo, J.M., Degroot, M.H., Lindley, D.V. & Smith, A.F.M. (eds.), Bayesian Statistics 3, pp. 211–241. London: Oxford University Press, 1988.Google Scholar
36.Hill, B.M. A theory of Bayesian data analysis. In Geisser, S., Hodges, J.S., Press, S.J. & Zellner, A. (eds.), Bayesian and Likelihood Methods in Statistics and Econometrics: Essays in Honor of George A. Barnard, pp. 49–73. Amsterdam: North-Holland, 1990.Google Scholar
37.Hill, B.M.Comment on “An ancillarity paradox that appears in multiple linear regression,” by L.D. Brown. Annals of Statistics 18 (1990): 513–523.CrossRefGoogle Scholar
38.Hill, B.M. Dutch books, the Jeffreys–Savage theory of hypothesis testing, and Bayesian reliability. In Barlow, R., Clarotti, C. & Spizzichino, F. (eds.), Reliability and Decision Making, pp. 31–85. Chapman and Hall, 1993.CrossRefGoogle Scholar
39.Hill, B.M. Bayesian nonparametric prediction and statistical inference. In Goel, P.K. & Iyengar, N.S. (eds.), Bayesian Analysis in Statistics and Econometrics, pp. 43–94. New York: Springer-Verlag, 1992.CrossRefGoogle Scholar
40.Hill, B.M. On Steinian shrinkage estimators: The finite/infinite problem and formalism in probability and statistics. In Freeman, P. & Smith, A.F.M. (eds.), Aspects of Uncertainty. London: Wiley, 1994.Google Scholar
41.Hill, B.M.Parametric models of An: Splitting processes and mixtures. Journal of the Royal Statistical Society B 55 (1993): 423–433.Google Scholar
42.Hill, B.M., Lane, D. & Sudderth, W.. A strong law for some generalized urn processes. Annals of Probability 8 (1980): 214–226.CrossRefGoogle Scholar
43.Hill, B.M., Lane, D. & Sudderth, W.. Exchangeable urn processes. Annals of Probability 15 (1987): 1586–1592.CrossRefGoogle Scholar
44.Jeffreys, H.Theory of Probability, 3rd ed.London: Oxford University Press, 1961.Google Scholar
45.Jeffreys, H. Some general points in probability theory. In Zellner, A. (ed.), Bayesian Analysis in Probability and Statistics, pp. 451–453. Amsterdam: North-Holland, 1980.Google Scholar
46.Lindley, D.Probability and Statistics 2. Inference. Cambridge: Cambridge University Press, 1965.Google Scholar
47.Malkiel, B.G.A Random Walk Down Wall Street. New York: W.W. Norton, 1990.Google Scholar
48.Mandelbrot, B.B.The Fractal Geometry of Nature. New York: W.H. Freeman, 1977.Google Scholar
49.Maret, X. Estimating the Structure of Stochastic Dynamic Linear Systems: A Bayesian Approach and Economic Applications. Ph.D. Dissertation, University of Michigan, 1988.Google Scholar
50.Markov, A.A.O nekotorth predelnth formulah iscistenta veroiatnostei. Izvestia Akad. Nauk. Seria VI 11 (1917): 177–186.Google Scholar
51.Meeden, G. & Arnold, B.C.. The admissibility of a preliminary test estimator when the loss incorporates a complexity cost. Journal of the American Statistical Association 74 (1979): 872–874.CrossRefGoogle Scholar
52.Meinhold, R.J. & Singpurwalla, N.D.. Robustification of Kalman filter models. Journal of the American Statistical Association 84 (1989): 479–486.CrossRefGoogle Scholar
53.Miller, R.G.Simultaneous Statistical Inference. New York: McGraw-Hill, 1966.Google Scholar
54.Pemantle, R.Nonconvergence to unstable points in urn models and stochastic approximations. Annals of Probability 18 (1990): 698–712.CrossRefGoogle Scholar
55.Phillips, P.C.B.Time series regression with a unit root. Econometrica 55 (1987): 277–301.CrossRefGoogle Scholar
56.Phillips, P.C.B.To criticize the critics: An objective Bayesian analysis of stochastic trends. Journal of Applied Econometrics 6 (1992): 333–364.CrossRefGoogle Scholar
57.Phillips, P.C.B. & Ploberger, W.. Posterior-odds testing for a unit root with data-based model selection. Econometric Theory 10 (1994): 774–808.CrossRefGoogle Scholar
58.Poirier, D.A Bayesian view of nominal money and real output through a new classical macroeconomic window. Journal of Business and Economic Statistics 7 (1991): 125–161.CrossRefGoogle Scholar
59.Ramsey, F.The Foundations of Mathematics and Other Logical Essays, Braithwaite, R.B. (ed.). New York: Humanities Press, 1950.Google Scholar
60.Savage, L.J. et al. The Foundations of Statistical Inference. London: Methuen, 1962.Google Scholar
61.Savage, L.J.The Foundations of Statistics, 2nd rev. ed.New York: Dover, 1972.Google Scholar
62.Sims, C.A.Comment on “Some subjective Bayesian considerations in the selection of models,” by B.M. Hill. Econometric Reviews 4 (1985–1986): 269–275.Google Scholar
63.Stancu, D.D.On the Markov probability distribution. Bulletin Mathématique de la Soc. Sci. Math, de la R.S. de Roumanie 12 (1968): 203–208.Google Scholar
64.West, M. & Harrison, J.. Bayesian Forecasting and Dynamic Models. New York: Springer-Verlag, 1989.CrossRefGoogle Scholar
65.Wilks, S.S.Mathematical Statistics. New York: Wiley, 1962.Google Scholar
66.Wold, H.A Study in the Analysis of Stationary Time Series, 2nd ed.Uppsala: Almqvist and Wiksells, 1938.Google Scholar
67.Wrinch, D. & Jeffreys, H.. On certain fundamental principles of scientific inquiry. Philosophical Magazine 42 (1921): 369–390.Google Scholar