Bayesian Analysis of Instrumental Variable Models: Acceptance-Rejection within Direct Monte Carlo

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 non-elliptical 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.

Keywords:

• Acceptance-Rejection
• Bayesian inference
• Direct Monte Carlo
• Instrumental variables
• Numerical standard errors

JEL Classification:

• C11
• C15
• C26
• C36

ACKNOWLEDGMENT

This paper started through intense, lively discussions between Arnold Zellner and Herman K. van Dijk in April 2010, when the latter was visiting Chicago. We note that, given the untimely death of Arnold Zellner in August 2010, he has not been involved in the empirical illustrations and the revision of this paper. However, all co-authors feel that Arnold's influence on the topic of this paper has been so important that as a credit to his enormous positive activity on the topic we wish to maintain him as co-author although he is not responsible for any errors in the empirical analysis. The authors are indebted to the editors Essie Maasoumi and Ehsan Soofi and two anonymous referees for very helpful comments that greatly helped in the revision of an earlier version of this paper.

Notes

The model (1)–(2) may include exogenous explanatory variables w _t (1 × n) in both equations. In that case, we assume a flat prior for the coefficients at w _t , and these coefficients are marginalized out of the posterior distribution using analytical integration. This amounts to replacing y _t , x _t and z _t by their residuals after regression on w _t , and replacing T by T − n.

The intimate link between Importance Sampling and the independence chain MH algorithm is pointed out by Liu (1996).

NSE = Numerical Standard Error of estimated posterior mean

s.c. = first order serial correlation in Gibbs sequence

ESS = Effective Sample Size (for estimating the posterior mean)

NSE = Numerical Standard Error of estimated posterior mean

s.c. = first order serial correlation in Gibbs sequence of conditional posterior mean

ESS = Effective Sample Size (for estimating the posterior mean)

NSE, s.c., ESS: see Table 1.

NSE, s.c., ESS: see Table 2.