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A direct Monte Carlo approach for Bayesian analysis of the seemingly unrelated regression model. (English) Zbl 1431.62295
Summary: Computationally efficient methods for Bayesian analysis of seemingly unrelated regression (SUR) models are described and applied that involve the use of a direct Monte Carlo (DMC) approach to calculate Bayesian estimation and prediction results using diffuse or informative priors. This DMC approach is employed to compute Bayesian marginal posterior densities, moments, intervals and other quantities, using data simulated from known models and also using data from an empirical example involving firms’ sales. The results obtained by the DMC approach are compared to those yielded by the use of a Markov Chain Monte Carlo (MCMC) approach. It is concluded from these comparisons that the DMC approach is worthwhile and applicable to many SUR and other problems.

62J05 Linear regression; mixed models
62-08 Computational methods for problems pertaining to statistics
62F15 Bayesian inference
62P20 Applications of statistics to economics
65C40 Numerical analysis or methods applied to Markov chains
Full Text: DOI
[1] Ando, T., Bayesian predictive information criterion for the evaluation of hierarchical Bayesian and empirical Bayes models, Biometrika, 94, 443-458, (2007) · Zbl 1132.62005
[2] Ando, T.; Zellner, A., Hierarchical Bayesian analysis of the seemingly unrelated regression and simultaneous equation models, Bayesian analysis, 5, 65-96, (2010) · Zbl 1330.62108
[3] Box, G.E.P.; Tiao, G.C., Bayesian inference in statistical analysis, (1973), Addison-Wesley Reading, MA · Zbl 0178.22003
[4] Brooks, S.P.; Gelman, A., General methods for monitoring convergence of iterative simulations, Journal of computational and graphical statistics, 7, 434-455, (1997)
[5] Carroll, R.J.; Doug, M.; Larry, F.; Victor, K., Seemingly unrelated measurement error models, with application to nutritional epidemiology, Biometrics, 62, 75-84, (2006) · Zbl 1091.62110
[6] Chib, S.; Greenberg, E., Hierarchical analysis of SUR models with extensions to correlated serial errors and time-varying parameter models, Journal of econometrics, 68, 339-360, (1995) · Zbl 0833.62103
[7] Frasera, D.A.S.; Rekkasb, M.; Wong, A., Highly accurate likelihood analysis for the seemingly unrelated regression problem, Journal of econometrics, 127, 17-33, (2005) · Zbl 1336.62140
[8] Gallant, R., Seemingly unrelated nonlinear regressions, Journal of econometrics, 3, 35-50, (1975) · Zbl 0296.62053
[9] Gelman, A.; Rubin, D.B., Inference from iterative simulation using multiple sequences, Statistical science, 7, 457-511, (1992) · Zbl 1386.65060
[10] Geweke, J., Evaluating the accuracy of sampling-based approaches to calculating posterior moments, (), 169-193
[11] Geweke, J., Contemporary Bayesian econometrics and statistics, (2005), Wiley New York · Zbl 1093.62107
[12] Greene, W.H., Econometric analysis, (2002), Prentice-Hall New Jersey
[13] Heidelberger, P.; Welch, P.D., Simulation run length control in the presence of an initial transient, Operations research, 31, 1109-1144, (1983) · Zbl 0532.65097
[14] Jacquier, E.; Polson, N.G.; Rossi, P.E., Bayesian analysis of stochastic volatility models with fat-tails and correlated errors, Journal of econometrics, 122, 185-212, (2004) · Zbl 1328.91254
[15] Jeffreys, H., An invariant form for the prior probability in estimation problems, Proceedings of the royal society of London, series A, 196, 453-461, (1946) · Zbl 0063.03050
[16] Jeffreys, H., Theory of probability, (1961), Oxford University Press Oxford · Zbl 0116.34904
[17] Judge, G.; Hill, R.; Griffiths, W.; Lutkepohl, H.; Lee, T., Introduction to the theory and practice of econometrics, (1988), Wiley New York · Zbl 0731.62155
[18] Kim, J.K.S.; Shephard, N.; Chib, S., Stochastic volatility likelihood inference and comparison with ARCH models, Review of economic studies, 65, 361-393, (1998) · Zbl 0910.90067
[19] Kowalski, J.R.; Mendoza-Blanco, X.; Tu, M.; Gleser, L.J., On the difference in inference and prediction between the joint and independent \(t\)-error models for seemingly unrelated regressions, Communications in statistics, part A—theory and methods, 28, 2119-2140, (1999) · Zbl 1055.62519
[20] Kurata, H., On the efficiencies of several generalized least squares estimators in a seemingly unrelated regression model and a heteroscedastic model, Journal of multivariate analysis, 70, 86-94, (1999) · Zbl 0951.62053
[21] Lancaster, T., Introduction to modern Bayesian econometrics, (2004), Cambridge University Press New Jersey · Zbl 1096.62140
[22] Liu, A., Efficient estimation of two seemingly unrelated regression equations, Journal of multivariate analysis, 82, 445-456, (2002) · Zbl 1078.62518
[23] Mandy, D.M.; Martins-Filho, C., Seemingly unrelated regressions under additive heteroscedasticity: theory and share equation applications, Journal of econometrics, 58, 315-346, (1993) · Zbl 0806.62099
[24] McCulloch, R.E.; Polson, N.G.; Rossi, P.E., Bayesian analysis of the multinomial probit with fully identified parameters, Journal of econometrics, 99, 173-193, (2000) · Zbl 0958.62029
[25] Neudecker, H.; Windmeijer, F.A.G., \(R^2\) in seemingly unrelated regression equations, Statistica neerlandica, 45, 405-411, (1991) · Zbl 0744.62095
[26] Ng, V.M., Robust Bayesian inference for seemingly unrelated regressions with elliptical errors, Journal of multivariate analysis, 83, 409-414, (2002) · Zbl 1025.62008
[27] Percy, D., Predictions for seemingly unrelated regressions, Journal of the royal statistical society, series B, 54, 243-252, (1992)
[28] Percy, D., Zellner’s influence on multivariate linear models, (), 203-214
[29] Press, S.J., Applied multivariate analysis, (1972), Holt, Rinehart and Winston, Inc. New York · Zbl 0253.60026
[30] Raftery, A.E.; Lewis, S.M., One long run with diagnostics: implementation strategies for Markov chain – monte Carlo, Statistical science, 7, 493-497, (1992)
[31] Rocke, D.M., Bootstrap bartlett adjustment in seemingly unrelated regression, Journal of the American statistical association, 84, 598-601, (1989)
[32] Rossi, P.E.; Allenby, G.; McCulloch, R., Bayesian statistics and marketing, (2005), John Wiley and Sons NJ · Zbl 1094.62037
[33] Schruben, L.W., Detecting initialization bias in simulation experiments, Operations research, 30, 569-590, (1982) · Zbl 0484.65043
[34] Smith, M.; Kohn, R., Nonparametric seemingly unrelated regression, Journal of econometrics, 98, 257-282, (2000) · Zbl 0957.62033
[35] Spiegelhalter, D.J.; Best, N.G.; Carlin, B.P.; van der Linde, A., Bayesian measures of model complexity and fit (with discussion), Journal of the royal statistical society, series B, 64, 583-639, (2002) · Zbl 1067.62010
[36] Srivastava, V.K.; Giles, D.E.A., Seemingly unrelated regression equations models, (1987), Dekker New York · Zbl 0568.62066
[37] van der Merwe, A., Viljoen, C., 1988. Bayesian analysis of the seemingly unrelated regression model. Manuscript. University of the Free State, Department of Mathematical Statistics.
[38] Zellner, A., An efficient method of estimating seemingly unrelated regression equations and tests for aggregation bias, Journal of the American statistical association, 57, 348-368, (1962) · Zbl 0113.34902
[39] Zellner, A., Estimators for seemingly unrelated regression equations: some exact finite sample results, Journal of the American statistical association, 58, 977-992, (1963) · Zbl 0129.11203
[40] Zellner, A., An introduction to Bayesian inference in econometrics, (1971), Wiley New York · Zbl 0246.62098
[41] Zellner, A.; Ando, T., Bayesian and non-Bayesian analysis of the seemingly unrelated regression model with student-\(t\) errors and its application for forecasting, International journal of forecasting, 26, 413-434, (2009)
[42] Zellner, A.; Ando, T., Rejoinder, International journal of forecasting, 26, 439-442, (2009)
[43] Zellner, A., Ando, T., 2010. A direct Monte Carlo approach for Bayesian analysis of the seemingly unrelated regression model with informative prior. Journal of the Japan Statistical Society (in Japanese) (in press). · Zbl 1431.62295
[44] Zellner, A.; Bauwens, L.; Van Dijk, H.K., Bayesian specification analysis and estimation of simultaneous equation models using Monte Carlo methods, Journal of econometrics, 38, 39-72, (1988)
[45] Zellner, A.; Chen, B., Bayesian modeling of economies and data requirements, Macroeconomic dynamics, 5, 673-700, (2001) · Zbl 1003.91512
[46] Zellner, A.; Min, C.K., Gibbs sampler convergence criteria, Journal of the American statistical association, 90, 921-927, (1995) · Zbl 0842.62018
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