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On Jeffreys Prior when Using the Exact Likelihood Function

  • Harald Uhlig (a1)
  • DOI: http://dx.doi.org/10.1017/S0266466600008707
  • Published online: 01 February 2009
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.

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1.D.N. DeJong , J.C. Nankervis , N.E. Savin & C.H. Whiteman . Integration versus trend stationarity in time series. Econometrica 60 (1992): 423433.

4.P.C.B. Phillips To criticize the critics: An objective Bayesian analysis of stochastic trends. Journal of Applied Econometrics 6 (1991): 333364.

5.P.C.B. Phillips Bayesian routes and unit roots: De rebus prioribus semper est disputandum. Journal of Applied Econometrics 6 (1991): 435474.

6.P.C. Schotman & H.K. van Dijk . On Bayesian routes to unit roots. Journal of Applied Econometrics 6 (1991): 387402.

7.C.A. Sims & H. Uhlig . Understanding unit rooters: A helicopter tour. Econometrica 59 (1991): 15911600.

8.H. Thornber Finite sample Monte Carlo studies: An autoregressive illustration. Journal of the American Statistical Association 62 (1967): 801818.

9.H. Uhlig What macroeconomists should know about unit roots: A Bayesian perspective. Econometric Theory 10 (1994): 645671.

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Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
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