Tailored randomized block MCMC methods with application to DSGE models

In this paper we develop new Markov chain Monte Carlo schemes for the estimation of Bayesian models. One key feature of our method, which we call the tailored randomized block Metropolis–Hastings (TaRB-MH) method, is the random clustering of the parameters at every iteration into an arbitrary number of blocks. Then each block is sequentially updated through an M–H step. Another feature is that the proposal density for each block is tailored to the location and curvature of the target density based on the output of simulated annealing, following  and  and Chib and Ergashev (in press). We also provide an extended version of our method for sampling multi-modal distributions in which at a pre-specified mode jumping iteration, a single-block proposal is generated from one of the modal regions using a mixture proposal density, and this proposal is then accepted according to an M–H probability of move. At the non-mode jumping iterations, the draws are obtained by applying the TaRB-MH algorithm. We also discuss how the approaches of Chib (1995) and Chib and Jeliazkov (2001) can be adapted to these sampling schemes for estimating the model marginal likelihood. The methods are illustrated in several problems. In the DSGE model of Smets and Wouters (2007), for example, which involves a 36-dimensional posterior distribution, we show that the autocorrelations of the sampled draws from the TaRB-MH algorithm decay to zero within 30–40 lags for most parameters. In contrast, the sampled draws from the random-walk M–H method, the algorithm that has been used to date in the context of DSGE models, exhibit significant autocorrelations even at lags 2500 and beyond. Additionally, the RW-MH does not explore the same high density regions of the posterior distribution as the TaRB-MH algorithm. Another example concerns the model of An and Schorfheide (2007) where the posterior distribution is multi-modal. While the RW-MH algorithm is unable to jump from the low modal region to the high modal region, and vice-versa, we show that the extended TaRB-MH method explores the posterior distribution globally in an efficient manner.     

Modeling and calculating the effect of treatment at baseline from panel outcomes

We propose and examine a panel data model for isolating the effect of a treatment, taken once at baseline, from outcomes observed over subsequent time periods. In the model, the treatment intake and outcomes are assumed to be correlated, due to unobserved or unmeasured confounders. Intake is partly determined by a set of instrumental variables and the confounding on unobservables is modeled in a flexible way, varying both by time and treatment state. Covariate effects are assumed to be subject-specific and potentially correlated with other covariates. Estimation and inference is by Bayesian methods that are implemented by tuned Markov chain Monte Carlo methods. Because our analysis is based on the framework developed by Chib [2004. Analysis of treatment response data without the joint distribution of counterfactuals. Journal of Econometrics, in press], the modeling and estimation does not involve either the unknowable joint distribution of the potential outcomes or the missing counterfactuals. The problem of model choice through marginal likelihoods and Bayes factors is also considered. The methods are illustrated in simulation experiments and in an application dealing with the effect of participation in high school athletics on future labor market earnings.     

Accept–reject Metropolis–Hastings sampling and marginal likelihood estimation

We describe a method for estimating the marginal likelihood, based on Chib (1995) and C hib and Jeliazkov (2001) , when simulation from the posterior distribution of the model parameters is by the accept–reject Metropolis–Hastings (ARMH) algorithm. The method is developed for one-block and multiple-block ARMH algorithms and does not require the (typically) unknown normalizing constant of the proposal density. The problem of calculating the numerical standard error of the estimates is also considered and a procedure based on batch means is developed. Two examples, dealing with a multinomial logit model and a Gaussian regression model with non-conjugate priors, are provided to illustrate the efficiency and applicability of the method.