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Benchmark priors for Bayesian models averaging

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  • Carmen Fernandez
  • E Ley
  • Mark F J Steel

Abstract

In contrast to a posterior analysis given a particular sampling model, posterior model probabilities in the context of model uncertainty are typically rather sensitive to the specification of the prior. In particular, 'diffuse' priors on model-specific parameters can lead to quite unexpected consequences. Here we focus on the practically relevant situation where we need to entertain a (large) number of sampling models and we have (or wish to use) little or no subjective prior information. We aim at providing an 'automatic' or 'benchmark' prior structure that can be used in such cases. We focus on the Normal linear regression model with uncertainty in the choice of regressors. We propose a partly noninformative prior structure related to a Natural Conjugate $g$-prior specification, where the amount of subjective information requested from the user is limited to the choice of a single scalar hyperparameter $g_{0j}$. The consequences of different choices for $g_{0j}$ are examined. We investigate theoretical properties, such as consistency of the implied Bayesian procedure. Links with classical information criteria are provided. In addition, we examine the finite sample implications of several choices of $g_{0j}$ in a simulation study. The use of the MC$^3$ algorithm of Madigan and York (1995), combined with efficient coding in Fortran, makes it feasible to conduct large simulations. In addition to posterior criteria, we shall also compare the predictive performance of different priors. A classic example concerning the economics of crime will also be provided and contrasted with results in the literature. The main findings of the paper will lead us to propose a 'benchmark' prior specification in a linear regression context with model uncertainty.

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Bibliographic Info

Paper provided by Edinburgh School of Economics, University of Edinburgh in its series ESE Discussion Papers with number 66.

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Length: 26
Date of creation: Apr 2004
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Handle: RePEc:edn:esedps:66

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Keywords: Bayes factors; Markov chain; Monte Carlo; Posterior odds; Prior elicitation;

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References

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  1. Isaac Ehrlich, 1973. "The Deterrent Effect of Capital Punishment: A Question of Life and Death," NBER Working Papers 0018, National Bureau of Economic Research, Inc.
  2. Cornwell, Christopher & Trumbull, William N, 1994. "Estimating the Economic Model of Crime with Panel Data," The Review of Economics and Statistics, MIT Press, vol. 76(2), pages 360-66, May.
  3. Smith, Michael & Kohn, Robert, 1996. "Nonparametric regression using Bayesian variable selection," Journal of Econometrics, Elsevier, Elsevier, vol. 75(2), pages 317-343, December.
  4. Akaike, Hirotugu, 1981. "Likelihood of a model and information criteria," Journal of Econometrics, Elsevier, Elsevier, vol. 16(1), pages 3-14, May.
  5. Gary S. Becker, 1974. "Crime and Punishment: An Economic Approach," NBER Chapters, in: Essays in the Economics of Crime and Punishment, pages 1-54 National Bureau of Economic Research, Inc.
  6. Atkinson, A. C., 1981. "Likelihood ratios, posterior odds and information criteria," Journal of Econometrics, Elsevier, Elsevier, vol. 16(1), pages 15-20, May.
  7. Chow, Gregory C., 1981. "A comparison of the information and posterior probability criteria for model selection," Journal of Econometrics, Elsevier, Elsevier, vol. 16(1), pages 21-33, May.
  8. Hoeting, Jennifer & Raftery, Adrian E. & Madigan, David, 1996. "A method for simultaneous variable selection and outlier identification in linear regression," Computational Statistics & Data Analysis, Elsevier, Elsevier, vol. 22(3), pages 251-270, July.
  9. Jacek OSIEWALSKI & Mark F.J. STEEL, 1993. "Regression Models under Competing Covariance Structures: A Bayesian Perspective," Annales d'Economie et de Statistique, ENSAE, issue 32, pages 65-79.
  10. Richard, J. F. & Steel, M. F. J., 1988. "Bayesian analysis of systems of seemingly unrelated regression equations under a recursive extended natural conjugate prior density," Journal of Econometrics, Elsevier, Elsevier, vol. 38(1-2), pages 7-37.
  11. Peter C.B. Phillips, 1992. "Bayesian Model Selection and Prediction with Empirical Applications," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 1023, Cowles Foundation for Research in Economics, Yale University.
  12. Min, Chung-ki & Zellner, Arnold, 1993. "Bayesian and non-Bayesian methods for combining models and forecasts with applications to forecasting international growth rates," Journal of Econometrics, Elsevier, Elsevier, vol. 56(1-2), pages 89-118, March.
  13. Poirier, Dale J, 1988. "Frequentist and Subjectivist Perspectives on the Problems of Model Building in Economics," Journal of Economic Perspectives, American Economic Association, vol. 2(1), pages 121-44, Winter.
  14. BAUWENS, Luc, . "The "pathology" of the natural conjugate prior density in the regression model," CORE Discussion Papers RP -962, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  15. Chib, B. & Osiewalski, J. & Steel, M., 1990. "Regression Models Under Competing Covariance Matrices: A Baysian Perspective," Papers, Tilburg - Center for Economic Research 9063, Tilburg - Center for Economic Research.
  16. Ehrlich, Isaac, 1973. "Participation in Illegitimate Activities: A Theoretical and Empirical Investigation," Journal of Political Economy, University of Chicago Press, University of Chicago Press, vol. 81(3), pages 521-65, May-June.
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