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Possibly Ill-behaved Posteriors in Econometric Models

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  • Lennart Hoogerheide

    ()
    (Erasmus University Rotterdam)

  • Herman K. van Dijk

    ()
    (Erasmus University Rotterdam)

Abstract

Highly non-elliptical posterior distributions may occur in several econometric models, in particular, when the likelihood information is allowed to dominate and data information is weak. We explain the issue of highly non-elliptical posteriors in a model for the effect of education on income using data from the well-known Angrist and Krueger (1991) study and discuss how a so-called Information Matrix or Jeffreys' prior may be used as a `regularization prior' that in combination with the likelihood yields posteriors with desirable properties. We further consider an 8-dimensional bimodal posterior distribution in a 2-regime mixture model for the real US GNP growth. In order to perform a Bayesian posterior analysis using indirect sampling methods in these models, one has to find a good candidate density. In a recent paper - Hoogerheide, Kaashoek and Van Dijk (2007) - a class of neural network functions was introduced as candidate densities in case of non-elliptical posteriors. In the present paper, the connection between canonical model structures, non-elliptical credible sets, and more sophisticated neural network simulation techniques is explored. In all examples considered in this paper – a bimodal distribution of Gelman and Meng (1991) and posteriors in IV and mixture models - the mixture of Student's t distributions is clearly a much better candidate than a Student's t candidate, yielding far more precise estimates of posterior means after the same amount of computing time, whereas the Student's t candidate almost completely misses substantial parts of the parameter space.

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Bibliographic Info

Paper provided by Tinbergen Institute in its series Tinbergen Institute Discussion Papers with number 08-036/4.

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Date of creation: 08 Apr 2008
Date of revision: 18 Apr 2008
Handle: RePEc:dgr:uvatin:20080036

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Web page: http://www.tinbergen.nl

Related research

Keywords: instrumental variables; vector error correction model; mixture model; importance sampling; Markov chain Monte Carlo; neural network;

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References

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  1. Joshua D. Angrist & Guido W. Imbens, 1995. "Identification and Estimation of Local Average Treatment Effects," NBER Technical Working Papers 0118, National Bureau of Economic Research, Inc.
  2. Geweke, John, 2007. "Interpretation and inference in mixture models: Simple MCMC works," Computational Statistics & Data Analysis, Elsevier, Elsevier, vol. 51(7), pages 3529-3550, April.
  3. Hoogerheide, L.F. & van Dijk, H.K., 2006. "A reconsideration of the Angrist-Krueger analysis on returns to education," Econometric Institute Research Papers EI 2006-15, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
  4. HOOGERHEIDE, Lennart F. & KAASHOEK, Johan F. & VAN DIJK, Herman K., 2005. "On the shape of posterior densities and credible sets in instrumental variable regression models with reduced rank: An application of flexible sampling methods using neural networks," CORE Discussion Papers, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) 2005029, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  5. Fruhwirth-Schnatter S., 2001. "Markov Chain Monte Carlo Estimation of Classical and Dynamic Switching and Mixture Models," Journal of the American Statistical Association, American Statistical Association, American Statistical Association, vol. 96, pages 194-209, March.
  6. Geweke, John, 1989. "Bayesian Inference in Econometric Models Using Monte Carlo Integration," Econometrica, Econometric Society, Econometric Society, vol. 57(6), pages 1317-39, November.
  7. Charles S. Bos & Ronald J. Mahieu & Herman K. van Dijk, 2000. "Daily Exchange Rate Behaviour and Hedging of Currency Risk," Econometric Society World Congress 2000 Contributed Papers 0504, Econometric Society.
  8. HOOGERHEIDE, Lennart F. & VAN DIJK, Herman K. & VAN OEST, Rutger D., 2007. "Simulation based Bayesian econometric inference: principles and some recent computational advances," CORE Discussion Papers, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) 2007015, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  9. van Dijk, H. K. & Kloek, T., 1980. "Further experience in Bayesian analysis using Monte Carlo integration," Journal of Econometrics, Elsevier, Elsevier, vol. 14(3), pages 307-328, December.
  10. Kleibergen, F.R. & van Dijk, H.K., 1997. "Bayesian Simultaneous Equations Analysis using Reduced Rank Structures," Econometric Institute Research Papers EI 9714/A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
  11. Donald W.K. Andrews, 1988. "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 877R, Cowles Foundation for Research in Economics, Yale University, revised Jul 1989.
  12. Joshua D. Angrist & Alan B. Krueger, 1990. "Does Compulsory School Attendance Affect Schooling and Earnings?," NBER Working Papers 3572, National Bureau of Economic Research, Inc.
  13. DREZE, Jacques H., . "Bayesian regression analysis using poly-t densities," CORE Discussion Papers RP -316, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  14. Dreze, Jacques H, 1976. "Bayesian Limited Information Analysis of the Simultaneous Equations Model," Econometrica, Econometric Society, Econometric Society, vol. 44(5), pages 1045-75, September.
  15. Donald W.K. Andrews & Christopher J. Monahan, 1990. "An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 942, Cowles Foundation for Research in Economics, Yale University.
  16. Kloek, Tuen & van Dijk, Herman K, 1978. "Bayesian Estimates of Equation System Parameters: An Application of Integration by Monte Carlo," Econometrica, Econometric Society, Econometric Society, vol. 46(1), pages 1-19, January.
  17. Hoogerheide, L.F. & Kleibergen, F.R. & van Dijk, H.K., 2006. "Natural conjugate priors for the instrumental variables regression model applied to the Angrist-Krueger data," Econometric Institute Research Papers EI 2006-02, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
  18. Kleibergen, Frank & van Dijk, Herman K., 1994. "On the Shape of the Likelihood/Posterior in Cointegration Models," Econometric Theory, Cambridge University Press, vol. 10(3-4), pages 514-551, August.
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Citations

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Cited by:
  1. David Ardia & Lennart F. Hoogerheide, 2010. "Efficient Bayesian Estimation and Combination of GARCH-Type Models," Tinbergen Institute Discussion Papers 10-046/4, Tinbergen Institute.
  2. de Pooter, M.D. & Ravazzolo, F. & Segers, R. & van Dijk, H.K., 2008. "Bayesian near-boundary analysis in basic macroeconomic time series models," Econometric Institute Research Papers EI 2008-13, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
  3. Ardia, David & Hoogerheide, Lennart F. & van Dijk, Herman K., 2008. "AdMit: Adaptive Mixtures of Student-t Distributions," DQE Working Papers, Department of Quantitative Economics, University of Freiburg/Fribourg Switzerland 10, Department of Quantitative Economics, University of Freiburg/Fribourg Switzerland, revised 07 Jan 2009.
  4. David Ardia & Lennart F. Hoogerheide & Herman K. van Dijk, . "Adaptive Mixture of Student-t Distributions as a Flexible Candidate Distribution for Efficient Simulation: The R Package AdMit," Journal of Statistical Software, American Statistical Association, American Statistical Association, vol. 29(i03).
  5. David Ardia & Lennart F. Hoogerheide & Herman K. van Dijk, 2008. "Adaptive Mixture of Student-t distributions as a Flexible Candidate Distribution for Efficient Simulation," Tinbergen Institute Discussion Papers 08-062/4, Tinbergen Institute, revised 15 Dec 2008.

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