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On Jeffreys Prior when Using the Exact Likelihood Function

Published online by Cambridge University Press:  11 February 2009

Harald Uhlig
Harald Uhlig
Affiliation:
Center for Economic Research, Tilburg University
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Abstract

In this paper, we calculate Jeffreys prior for an AR(1) process with and without a constant and a time trend when using the exact likelihood function. We show how this prior can be calculated for the explosive region, even though the unconditional variance of the process is infinite. The calculations lend additional support to the Schotman-van Dijk [6] procedure for restricting the location and the variance of the time trend coefficient. The results show that flat priors are reasonable for the nonexplosive region in an AR(1) without a constant and a time trend where the variance is known and the initial observation is zero, i.e., for the special case studied by Sims and Uhlig [7]. Differences to a flat prior analysis remain in particular for nonzero initial observations, however. For the explosive region, the unconditional prior diverges as the root diverges, supporting findings by Phillips [4]. This paper thus provides a useful perspective as well as some reconciliation for the different stands taken in the literature about priors and Bayesian inference for potentially nonstationary time series.

Type
Articles
Information
Econometric Theory , Volume 10 , Issue 3-4 , August 1994 , pp. 633 - 644
Copyright
Copyright © Cambridge University Press 1994

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References

REFERENCES

1.DeJong, D.N., Nankervis, J.C., Savin, N.E. & Whiteman, C.H.. Integration versus trend stationarity in time series. Econometrica 60 (1992): 423–433.CrossRefGoogle Scholar
2.Geweke, J.Priors for macroeconomic time series and their application. Econometric Theory 10 (1994): 609–632.CrossRefGoogle Scholar
3.Learner, E.E.Specification Searches. New York: Wiley, 1978.Google Scholar
4.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
5.Phillips, P.C.B.Bayesian routes and unit roots: De rebus prioribus semper est disputandum. Journal of Applied Econometrics 6 (1991): 435–474.CrossRefGoogle Scholar
6.Schotman, P.C. & van Dijk, H.K.. On Bayesian routes to unit roots. Journal of Applied Econometrics 6 (1991): 387–402.CrossRefGoogle Scholar
7.Sims, C.A. & Uhlig, H.. Understanding unit rooters: A helicopter tour. Econometrica 59 (1991): 1591–1600.CrossRefGoogle Scholar
8.Thornber, H.Finite sample Monte Carlo studies: An autoregressive illustration. Journal of the American Statistical Association 62 (1967): 801–818.CrossRefGoogle Scholar
9.Uhlig, H.What macroeconomists should know about unit roots: A Bayesian perspective. Econometric Theory 10 (1994): 645–671.CrossRefGoogle Scholar
10.Zellner, A.An Introduction to Bayesian Inference in Econometrics. New York: Wiley, 1971.Google Scholar
11.Zivot, E.A Bayesian analysis of the unit root hypothesis within an unobserved components model. Econometric Theory 10 (1994): 552–578.CrossRefGoogle Scholar