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Markov Chain Monte Carlo Simulation Methods in Econometrics

Published online by Cambridge University Press:  11 February 2009

Siddhartha Chib  and
Edward Greenberg
Siddhartha Chib
Affiliation:
Washington University
Edward Greenberg
Affiliation:
Washington University
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Abstract

We present several Markov chain Monte Carlo simulation methods that have been widely used in recent years in econometrics and statistics. Among these is the Gibbs sampler, which has been of particular interest to econometricians. Although the paper summarizes some of the relevant theoretical literature, its emphasis is on the presentation and explanation of applications to important models that are studied in econometrics. We include a discussion of some implementation issues, the use of the methods in connection with the EM algorithm, and how the methods can be helpful in model specification questions. Many of the applications of these methods are of particular interest to Bayesians, but we also point out ways in which frequentist statisticians may find the techniques useful.

Type
Research Article
Information
Econometric Theory , Volume 12 , Issue 3 , August 1996 , pp. 409 - 431
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

Albert, J. & Chib, S. (1993a) Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association 88, 669–679.CrossRefGoogle Scholar
Albert, J. & Chib, S. (1993b) Bayes inference via Gibbs sampling of autoregressive time series subject to Markov mean and variance shifts. Journal of Business and Economic Statistics 11, 1–15.CrossRefGoogle Scholar
Carlin, B.P. & Chib, S. (1995) Bayesian model choice via Markov chain Monte Carlo. Journal of the Royal Statistical Society B 57, 473–484.Google Scholar
Carlin, B.P., Gelfand, A.E., & Smith, A.F.M. (1992) Hierarchical Bayesian analysis of change point problems. Journal of the Royal Statistical Society C 41, 389–405.Google Scholar
Carlin, B.P. & Poison, N.G. (1991) Inference for nonconjugate Bayesian models using the Gibbs sampler. Canadian Journal of Statistics 19, 399–405.CrossRefGoogle Scholar
Carlin, B.P., N.G. Poison, & Stoffer, D.S. (1992) A Monte Carlo approach to nonnormal and nonlinear state-space modeling. Journal of the American Statistical Association 87, 493–500.CrossRefGoogle Scholar
Carter, C. & Kohn, R. (1994) On Gibbs sampling for state space models. Biometrika 81, 541–554.CrossRefGoogle Scholar
Casella, G. & George, E. (1992) Explaining the Gibbs sampler. American Statistician 46, 167–174.CrossRefGoogle Scholar
Celeux, G. & Oiebolt, J. (1985) The SEM algorithm: A probabilistic teacher algorithm derived from the EM algorithm for the mixture problem. Computational Statistics Quarterly 2, 73–82.Google Scholar
Chan, K.S. (1993) Asymptotic behavior of the Gibbs sampler. Journal of the American Statistical Association 88, 320–326.Google Scholar
Chib, S. (1992a) An Accelerated Gibbs Sampler for State Space Models and Other Results. Unpublished manuscript, Washington University.Google Scholar
Chib, S. (1992b) Bayes regression for the tobit censored regression model. Journal of Econometrics 51, 79–99.CrossRefGoogle Scholar
Chib, S. (1993a) Bayes regression with autocorrelated errors: A Gibbs sampling approach. Journal of Econometrics 58, 275–294.CrossRefGoogle Scholar
Chib, S. (1993b) Calculating posterior distributions and modal estimates in Markov mixture models. Journal of Econometrics, forthcoming.Google Scholar
Chib, S. (1995) Marginal likelihood from the Gibbs output. Journal of the American Statistical Association 90, 1313–1321.CrossRefGoogle Scholar
Chib, S. & Greenberg, E. (1994) Bayes inference for regression models with ARMA(p,?) errors. Journal of Econometrics 64, 183–206.CrossRefGoogle Scholar
Chib, S. & Greenberg, E. (1995a) Hierarchical analysis of SUR models with extensions to correlated serial errors and time varying parameter models. Journal of Econometrics 68, 339–360.CrossRefGoogle Scholar
Chib, S. & Greenberg, E. (1995b) Understanding the Metropolis-Hastings algorithm. American Statistician 49, 327–335.CrossRefGoogle Scholar
de Jong, P. & Shephard, N. (1995) The simulation smoother for time series models. Biometrika 82, 339–350.Google Scholar
Dempster, A.P., Laird, N., & Rubin, D.B. (1977) Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society B 39, 1–38.Google Scholar
Diebolt, J. & Robert, C.P. (1994) Estimation of finite mixture distributions through Bayesian sampling. Journal of the Royal Statistical Society B 56, 363–375.Google Scholar
Friihwirth-Schnatter, S. (1994) Data augmentation and dynamic linear models. Journal of Time Series Analysis 15, 183–202.CrossRefGoogle Scholar
Gelfand, A.E. & Smith, A.F.M. (1990) Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association 85, 398–409.Google Scholar
Gelfand, A.E. & Smith, A.F.M. (1992) Bayesian statistics without tears: A sampling-resampling perspective. American Statistician 46, 84–88.Google Scholar
Gelfand, A.E., A.F.M. Smith, & Lee, T.M. (1992) Bayesian analysis of constrained parameter and truncated data problems. Journal of the American Statistical Association 87, 523–532.CrossRefGoogle Scholar
Gelman, A. & Rubin, D.B. (1992) Inference from iterative simulation using multiple sequences. Statistical Science 4, 457–472.Google Scholar
Geman, S. & Geman, D. (1984) Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence 12, 609–628.CrossRefGoogle Scholar
George, E.I. & McCulloch, R.E. (1993) Variable selection via Gibbs sampling. Journal of the American Statistical Association 88, 881–889.CrossRefGoogle Scholar
Geweke, J. (1989) Bayesian inference in econometric models using Monte Carlo integration. Econometrica 57, 1317–1340.CrossRefGoogle Scholar
Geweke, J. (1991) Efficient simulation from the multivariate normal and student; distributions subject to linear constraints. In Keramidas, E. & Kaufman, S. (eds.), Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface, pp. 571–578. Fairfax Station, Virginia: Interface Foundation of North America.Google Scholar
Geweke, J. (1992) Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In Bernardo, J.M.Berger, J.O.Dawid, A.P., & Smith, A.F.M. (eds.), Proceedings of the Fourth Valencia International Conference on Bayesian Statistics, pp. 169–193. New York: Oxford University Press.CrossRefGoogle Scholar
Geweke, J. (1993a) A Bayesian analysis of reduced rank regressions. Journal of Econometrics, forthcoming.Google Scholar
Geweke, J. (1993b) Bayesian treatment of the independent students linear model. Journal of Applied Econometrics 8, S19–S40.CrossRefGoogle Scholar
Geweke, J., Keane, M., & Runkle, D. (1994) Alternative computational approaches to inference in the multinomial probit model. Review of Economics and Statistics 76, 609–632.Google Scholar
Geyer, C. (1992) Practical Markov chain Monte Carlo. Statistical Science 4, 473–482.Google Scholar
Gilks, W.R. & Wild, P. (1992) Adaptive rejection sampling for Gibbs sampling. Applied Statistics 41, 337–348.CrossRefGoogle Scholar
Harvey, A.C. (1981) Time Series Models. London: Philip Allan.Google Scholar
Hastings, W.K. (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109.CrossRefGoogle Scholar
Hsiao, C. (1986) Analysis of Panel Data. New York: Cambridge University Press.Google Scholar
Jacquier, E., N.G. Poison, & Rossi, P.E. (1994) Bayesian analysis of stochastic volatility models (with discussion). Journal of Business and Economic Statistics 12, 371–417.Google Scholar
Kass, R. & Raftery, A.E. (1994) Bayes factors and model uncertainty. Journal of the American Statistical Association 90, 773–795.CrossRefGoogle Scholar
Kloek, T. & van Dijk, H.K. (1978) Bayesian estimates of equation system parameters: An application of integration by Monte Carlo. Econometrica 46, 1–20.CrossRefGoogle Scholar
Koop, G. (1994) Recent progress in applied Bayesian econometrics. Journal of Economic Surveys 8, 1–34.Google Scholar
Koop, G., Osiewalski, J., & Steel, M.F.J. (1994) Bayesian efficiency analysis with a flexible form: The AIM cost function. Journal of Business and Economic Statistics 12, 339–346.CrossRefGoogle Scholar
Learner, E.E. (1978) Specification Searches: Ad-hoc Inference with Non-experimental Data. New York: John Wiley and Sons.Google Scholar
Liu, J.S., Wong, W.W., & Kong, A. (1994) Covariance structure of the Gibbs sampler with applications to the comparison of estimators and augmentation schemes. Biometrika 81, 27–40.CrossRefGoogle Scholar
Marriott, J., Ravishanker, N., & Gelfand, A.E. (1995) Bayesian analysis of ARMA processes: Complete sampling-based inferences under full likelihoods. In Berry, D.Chaloner, K., & Geweke, J. (eds.), Bayesian Statistics and Econometrics: Essays in Honor of Arnold Zellner.Google Scholar
McCulloch, R.E. & Rossi, P.E. (1994) Exact likelihood analysis of the multinomial probit model. Journal of Econometrics 64, 207–240.CrossRefGoogle Scholar
McCulloch, R.E. & Tsay, R.S. (1994) Statistical analysis of economic time series via Markov switching models. Journal of Time Series Analysis 15, 523–539.CrossRefGoogle Scholar
Mengersen, K.L. & Tweedie, R.L. (1993) Rates of Convergence of the Hastings and Metropolis Algorithms. Unpublished manuscript, Colorado State University.Google Scholar
Metropolis, N., A.W. Rosenbluth, M.N.Rosenbluth, A.H. Teller, & Teller, E. (1953) Equations of state calculations by fast computing machines. Journal of Chemical Physics 21, 1087–1092.Google Scholar
Meyn, S.P. & Tweedie, R.L. (1993) Markov Chains and Stochastic Stability. London: Springer-Verlag.CrossRefGoogle Scholar
Morris, C.N. (1987) Comment: Simulation in hierarchical models. Journal of the American Statistical Association 82, 542–543.CrossRefGoogle Scholar
Miiller, P. (1991) A Generic Approach to Posterior Integration and Gibbs Sampling. Technical report 91-09, Department of Statistics, Purdue University.Google Scholar
Newey, W.K. & West, K.D. (1987) A simple positive-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703–708.CrossRefGoogle Scholar
Newton, M,A. & Raftery, A.E. (1994) Approximate Bayesian inference with the weighted likelihood bootstrap. Journal of the Royal Statistical Society B 56, 3–48.Google Scholar
Nummelin, E. (1984) General Irreducible Markov Chains and Non-negative Operators. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Percy, D.F. (1992) Prediction for seemingly unrelated regressions. Journal of the Royal Statistical Society B 54, 243–252.Google Scholar
Poison, N.G. (1994) Convergence of Markov chain Monte Carlo algorithms. In Bernardo, J.M.Berger, J.O.Dawid, A.P., & Smith, A.F.M. (eds.), Proceedings of the Fifth Valencia International Conference on Bayesian Statistics, forthcoming.Google Scholar
Raftery, A.E. & Lewis, S.M. (1992) How many iterations in the Gibbs sampler? In Bernardo, J.M.Berger, J.O.Dawid, A.P., & Smith, A.F.M. (eds.), Proceedings of the Fourth Valencia International Conference on Bayesian Statistics, pp. 763–774. New York: Oxford University Press.CrossRefGoogle Scholar
Ripley, B. (1987) Stochastic Simulation. New York: John Wiley & Sons.CrossRefGoogle Scholar
Ritter, C. & Tanner, M.A. (1992) The Gibbs stopper and the griddy Gibbs sampler. Journal of the American Statistical Association 87, 861–868.CrossRefGoogle Scholar
Roberts, G.O. & Smith, A.F.M. (1994) Some convergence theory for Markov chain Monte Carlo. Stochastic Processes and Applications 49, 207–216.CrossRefGoogle Scholar
Roberts, G.O. & Tweedie, R.L. (1994) Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika, forthcoming.Google Scholar
Rubin, D.B. (1988) Using the SIR algorithm to simulate posterior distributions. In Bernardo, J.M.Berger, J.O.Dawid, A.P., & Smith, A.F.M. (eds.), Proceedings of the Fourth Valencia International Conference on Bayesian Statistics, pp. 395–402. New York: Oxford University Press.Google Scholar
Rubinstein, R.Y. (1981) Simulation and the Monte Carlo Method. New York: John Wiley & Sons.Google Scholar
Ruud, P.A. (1991) Extensions of estimation methods using the EM algorithm. Journal of Econometrics 49, 305–341.CrossRefGoogle Scholar
Shephard, N. (1994) Partial non-Gaussian state space models. Biometrika 81, 115–131.CrossRefGoogle Scholar
Smith, A.F.M. & Roberts, G.O. (1993) Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. Journal of the Royal Statistical Society B 55, 3–24.Google Scholar
Tanner, M.A. (1993) Tools for Statistical Inference, 2nd ed.New York: Springer-Verlag.CrossRefGoogle Scholar
Tanner, M.A. & Wong, W.H. (1987) The calculation of posterior distributions by data augmentation. Journal of the American Statistical Association 82, 528–549.CrossRefGoogle Scholar
Tierney, L. (1994) Markov chains for exploring posterior distributions (with discussion). Annals of Statistics 22, 1701–1762.Google Scholar
Tobin, J. (1958) Estimation of relationships for limited dependent variables. Econometrica 26, 24–36.CrossRefGoogle Scholar
Wakefield, J.C., Gelfand, A.E.Racine Poon, A., & Smith, A.F.M. (1994) Bayesian analysis of linear and nonlinear population models using the Gibbs sampler. Applied Statistics 43, 201–221.CrossRefGoogle Scholar
Wie, G.C.G. & Tanner, M.A. (1990) A Monte Carlo implementation of the EM algorithm and the poor man's data augmentation algorithm. Journal of the American Statistical Association 85, 699–704.Google Scholar
Zangari, P.J. & Tsurumi, H. (1994) A Bayesian Analysis of Censored Autocorrelated Data on Exports of Japanese Passenger Cars to the U.S. Unpublished manuscript, Rutgers University.Google Scholar
Zeger, S.L. & Karim, M.R. (1991) Generalized linear models with random effects: A Gibbs sampling approach. Journal of the American Statistical Association 86, 79–86.CrossRefGoogle Scholar
Zellner, A. (1984) Basic Issues in Econometrics. Chicago: University of Chicago Press.Google Scholar
Zellner, A. & Min, C. (1995) Gibbs sampler convergence criteria (GSC2). Journal of the American Statistical Association 90, 921–927.CrossRefGoogle Scholar