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Noninformative Priors and Bayesian Testing for the AR(1) Model

Published online by Cambridge University Press:  11 February 2009

James O. Berger  and
Ruo-Yong Yang
James O. Berger
Affiliation:
Purdue University
Ruo-Yong Yang
Affiliation:
Purdue University
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Abstract

Various approaches to the development of a noninformative prior for the AR(1) model are considered and compared. Particular attention is given to the reference prior approach, which seems to work well for the stationary case but encounters difficulties in the explosive case. A symmetrized (proper) version of the stationary reference prior is ultimately recommended for the problem. Bayesian testing of the unit root, stationary, and explosive hypotheses is considered also. Bounds on the Bayes factors are developed and shown to yield answers that appear to conflict with classical tests.

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Articles
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Econometric Theory , Volume 10 , Issue 3-4 , August 1994 , pp. 461 - 482
Copyright
Copyright © Cambridge University Press 1994

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References

REFERENCES

1.Anderson, T.W.On asymptotic distributions of estimates of parameters of stochastic difference equations. Annals of Mathematical Statistics 30 (1959): 676–687.CrossRefGoogle Scholar
2.Bayes, T.R. Essay towards solving a problem in the doctrine of chances (1763). Reprinted in Biometrika 45 (1958): 243–315.CrossRefGoogle Scholar
3.Berger, J.O.Statistical Decision Theory and Bayesian Analysis. New York: Springer-Verlag, 1985.CrossRefGoogle Scholar
4.Berger, J. & Bernardo, J.M.. Estimating a product of means: Bayesian analysis with reference priors. Journal of American Statistical Association 84 (1989): 200–207.CrossRefGoogle Scholar
5.Berger, J. & Bernardo, J.M.. Reference priors in a variance components problem. In Goel, P. & Iyengar, N.S. (eds.), Bayesian Analysis in Statistics and Econometrics. New York: Springer-Verlag, 1992.Google Scholar
6.Berger, J. & Bernardo, J.M.. Ordered group reference priors with application to a multi-nomial problem. Biometrika 79 (1992): 25–37.CrossRefGoogle Scholar
7.Berger, J. & Bernardo, J.M.. On the development of the reference prior method. In Bernardo, J.M., Berger, J.O., Lindley, D.V. & Smith, A.F.M. (eds.), Bayesian Statistics 4. London: Oxford University Press, 1992.Google Scholar
8.Berger, J., Bernardo, J.M. & Mendoza, M.. On priors that maximize expected information. In Klein, J.P. & Lee, J.C. (eds.), Recent Developments in Statistics and Their Applications. Seoul: Freedom Academy Publishing, 1989.Google Scholar
9.Berger, J. & Sellke, T.. Testing of a point null hypothesis: The irreconcilability of significance levels and evidence (with discussion). Journal of American Statistical Association 82 (1987): 112–139.Google Scholar
10.Berger, J. & Delampady, M.. Testing precise hypotheses. Statistical Science 2 (1987): 317–352.Google Scholar
11.Berger, J. & Berry, D.. Analyzing data: Its objectivity possible. American Scientist 76 (1988): 159–165.Google Scholar
12.Bernardo, J.M.Reference posterior distributions for Bayes inference. Journal of Royal Statistical Society B 41 (1979): 113–147.Google Scholar
13.Box, G.E.P. & Jenkins, G.M.. Time Series Analysis: Forecasting and Control. San Francisco: Holden Day, 1976.Google Scholar
14.Casella, G. & Berger, R.. Reconciling Bayesian and frequentist evidence in the one-sided testing problem. Journal of American Statistical Association 82 (1987): 106–111.CrossRefGoogle Scholar
15.Jeffreys, H.Theory of Probability. London: Oxford University Press, [1937] 1961.Google Scholar
16.Laplace, P.Mémoire sur la probabilityé des causes par les évenemens Mem. Acad. R. Sci. Presentés par Divers Savans 6 (1774): 621–656 (translated in Statistical Science 1 [1986]: 359–378).Google Scholar
17.Laplace, P.Theorie Analytique des Probabilites. Paris: Courcier, 1812.Google Scholar
18.Mitchell, A. Discussion of paper by Good, I.J.. Journal of Royal Statistical Society B 29 (1967): 423–424.Google Scholar
19.Lindley, D.V.On a measure of the information provided by an experiment. Annals of Mathematical Statistics 27 (1956): 986–1005.CrossRefGoogle Scholar
20.Monette, G., Fraser, D.A.S. & Ng, K.W.. Marginalization, likelihood, and structural models. In Krishnaiah, P.R. (ed.), Multivariate Analysis VI. Amsterdam: North-Holland, 1984.Google Scholar
21.Phillips, P.C.B.To criticize the critics: An objective Bayesian analysis of stochastic trends. Journal of Applied Econometrics 6 (1991): 333–364.CrossRefGoogle Scholar
22.Rubin, H. Consistency of maximum likelihood estimates in the explosive case. In Koopmans, T.C. (ed.), Statistical Inference in Dynamic Economic Models, pp. 356–364. New York: Wiley, 1950.Google Scholar
23.Schotman, P.C. & Van Dijk, H.K.. On Bayesian routes to unit roots. Journal of Applied Econometrics 6 (1991): 387–401.CrossRefGoogle Scholar
24.Stein, C. Approximation of improper prior measures by prior probability measures. In Bernoulli–Bayes–Laplace Festschrift, pp. 217–240. New York: Springer-Verlag, 1965.Google Scholar
25.Uhlig, H.What macroeconomists should know about unit roots: A Bayesian perspective. Econometric Theory 10 (1994): 645–671.CrossRefGoogle Scholar
26.Welch, B.L. & Peers, H.W.. On formulae for confidence points based on intervals of weighted likelihoods. Journal of Royal Statistical Society B 25 (1963): 318–329.Google Scholar
27.White, J.S.The limiting distribution of the serial correlation coefficient in the explosive case. Annals of Mathematical Statistics 29 (1958): 1188–1197.CrossRefGoogle Scholar
28.Ye, K.Y. & Berger, J.. Noninformative priors for inference in exponential regression models. Biometrika 78 (1991): 645–656.CrossRefGoogle Scholar
29.Zellner, A.An Introduction to Bayesian Inference in Econometrics. New York: Wiley, 1971.Google Scholar
30.Zellner, A. Maximal data information prior distributions. In Aykac, A. (ed.), New Methods in the Application of Bayesian Methods. Amsterdam: North-Holland, 1977.Google Scholar
31.Zellner, A. & Siow, A.. Posterior odds for selected regression hypotheses. In Bernardo, J.M., DeGroot, M.H., Lindley, D.V. & Smith, A.F.M. (eds.), Bayesian Statistics. Valencia: University Press, 1980.Google Scholar