Bayesian analysis of non-linear multivariate econometric models

This thesis aims to investigate Bayesian sampling techniques for estimating parameters of three nonlinear models with different levels of endogeneity and sample selection. These models include a bivariate probit model with an endogenous dummy regressor, an ordered probit model with sample selection, and an ordered probit model with double hurdles of sample selection. We developed Bayesian sampling algorithms to sample parameters in each of these models, and the resulting posterior estimates of parameters were compared with those obtained through a few classical estimation methods such as maximum likelihood estimate (MLE) and a two-step method. Monte Carlo simulations were conducted to check the performance of different estimators for each model. In the bivariate probit model with an endogenous dummy regressor, we discussed the identification conditions especially the effect of exclusion restrictions. The Monte Carlo study reveals that exclusion restrictions are not essential for model identification. However, the existence of exclusion restrictions will make the estimation much easier for all estimators. Moreover, model identification can be improved by increasing the variation of explanatory variables and the number of exogenous regressors. In terms of the performance of the three estimators, MLE is often accurate and efficient except for occasional convergence failures. The Bayesian method can always produce an estimate for each simulated sample and is most efficient. However, it shows same small bias when the correlation coefficient between errors is large. The inconsistent two-step method has less convergence problems than MLE, but has quite large biases when the correlation coefficient between errors is large. In terms of the ordered probit model with binary selection, we used a reparameterization to derive a Gibbs sampler, such that conditional posteriors can be obtained. We also propose a likelihood-based two-step method in a way similar to the derivation of the concentrated likelihood function. The two proposed methods were compared with the full information maximum likelihood (FIML) method and another established two-step method. Monte Carlo results show that the Bayesian method and the likelihood-based two-step method can be alternative methods to FIML, while the other two-step method is not acceptable in models with large error correlation. The absence of exclusion restrictions does not cause big problems for the model estimation. With the FIML and the Bayesian methods, we used the ordered probit model with binary selectivity to model the effect of mental illness on employment and job categories, where exclusion restrictions do not exist. The ordered probit model with double-hurdle selection is an extension of the above model with one additional level of sample selection. We found that FIML has encountered severe convergence-failure problems as the model becomes more complicated. As such, the proposed Bayesian sampling method is of great value because it always produces an estimate of the parameter vector. We propose two Bayesian samplers, one obtained through a standard process currently available in the literature, while the other involved reparameterization. In the Monte Carlo study, we found that both samplers and the FIML provide unbiased and efficient estimates. However, FIML fails to converge for more than half of the simulated samples, while Bayesian samplers can always produce estimates for each simulated sample. The reparameterization-based sampler shows better convergence than the other sampler. We applied the three estimators to the estimation of the double-hurdle ordered probit model investigating the effect of mental illness on labor market outcomes. We found that reparameterization-based sampler is the only estimator that did not encounter convergence problems. The resulting estimates of parameters were used for analyzing marginal effects of mental health variables.