A direct Monte Carlo approach for Bayesian analysis of the seemingly unrelated regression model
ABSTRACT 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.
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ABSTRACT: We discuss Bayesian inferential procedures within the family of instrumental variables regression models and focus on two issues: existence conditions for posterior moments of the parameters of interest under a flat prior and the potential of Direct Monte Carlo (DMC) approaches for efficient evaluation of such possibly highly onelliptical posteriors. We show that, for the general case of m endogenous variables under a flat prior, posterior moments of order r exist for the coefficients reflecting the endogenous regressors’ effect on the dependent variable, if the number of instruments is greater than m+r, even though there is an issue of local non-identification that causes non-elliptical shapes of the posterior. This stresses the need for efficient Monte Carlo integration methods. We introduce an extension of DMC that incorporates an acceptance-rejection sampling step within DMC. This Acceptance-Rejection within Direct Monte Carlo (ARDMC) method has the attractive property that the generated random drawings are independent, which greatly helps the fast convergence of simulation results, and which facilitates the evaluation of the numerical accuracy. The speed of ARDMC can be easily further improved by making use of parallelized computation using multiple core machines or computer clusters. We note that ARDMC is an analogue to the well-known 'Metropolis-Hastings within Gibbs' sampling in the sense that one 'more difficult' step is used within an 'easier' simulation method. We compare the ARDMC approach with the Gibbs sampler using simulated data and two empirical data sets, involving the settler mortality instrument of Acemoglu et al. (2001) and father's education's instrument used by Hoogerheide et al. (2012a). Even without making use of parallelized computation, an efficiency gain is observed both under strong and weak instruments, where the gain can be enormous in the latter case.Econometric Reviews 09/2012; · 0.68 Impact Factor
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ABSTRACT: Computationally efficient methods for Bayesian analysis of Seemingly Unrelated Regression (SUR) models with a large number of predictors are developed. One of the most crucial problems in Bayesian modeling of SUR models is how to determine the optimal combination of predictors. In this paper, under a Bayesian hierarchical framework where each regression function is represented as a linear combination of a large number of basis functions, the regression coefficients, the variance matrix of the errors, and a set of predictors to be included in the model are estimated simul-taneously. Usually the Bayesian model estimation problem is solved using Markov Chain Monte Carlo (MCMC) techniques. Herein we show how a direct Monte Carlo (DMC) technique can be employed to solve the variable selection and model param-eter estimation problems more efficiently.JOURNAL OF THE JAPAN STATISTICAL SOCIETY. 01/2012; 41(2).
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ABSTRACT: A process or system under study often requires the measurement of multiple responses. The optimization of multiple response variables has received considerable attention in the literature with the majority focusing on locating optimal operating conditions within the current experimental region and thus often occurs in the later stages of experimentation. This article focuses instead on the initial experiment and the location of additional experimental runs if the region of interest shifts. Considerable trade-off is often required in the multiple response context. In order to account for uncertainty in the model parameters and correlations among the responses, we propose the computation of Bayesian reliabilities to determine optimal factor settings for future experimental runs. The approach will be described in detail for two common design follow-up strategies: steepest ascent (descent) and shifting factor levels. Illustrative examples are provided for each application. Copyright © 2011 John Wiley & Sons, Ltd.Quality and Reliability Engineering 12/2011; 27:1107-1118. · 0.99 Impact Factor