Estimation Frameworks in Econometrics

Compare the fully parametric and semiparametric approaches to estimation of a discrete choice model such as the multinomial logit model discussed in Chapter 21. What are the benefits and costs of the semiparametric approach  

A fully parametric model/estimator provides consistent, efficient, and comparatively precise results. The semiparametric model/estimator, by comparison, is relatively less precise in general terms. But, the payoff to this imprecision is that the semiparametric formulation is more likely to be robust to failures of the assumptions of the parametric model. Consider, for example, the binary probit model of Chapter 21, which makes a strong assumption of normality and homoscedasticity. If the assumptions are coirect, the probit estimator is the most efficient use of the data. However, if the normality assumption or the homoscedasticity assumption are incorrect, then the probit estimator becomes inconsistent in an unknown fashion. Lewbel s semiparametric estimator for the binary choice model, in contrast, is not very precise in comparison to the probit model. But, it will remain consistent if the normality assumption is violated, and it is even robust to certain kinds of heteroscedasticity. 

Asymptotics take on a different meaning in the Bayesian estimation context, since parameter estimators do not converge to a population quantity. Nonetheless, in a Bayesian estimation setting, as the sample size increases, the likelihood function will dominate the posterior density. What does this imply about the Bayesian estimator when this occurs. 

The Bayesian estimator must converge to the maximum likelihood estimator as the sample size grows. The posterior mean will generally be a mixture of the prior and the maximizer of the likelihood function. We do note, however, that the likelihood will only dominate an informative prior asymptotically - the Bayesian estimator in this case will ultimately be a mixture of a prior with a finite precision and a likelihood based estimator whose variance converges to zero (thus, whose precision grows infinitely). Thus, the domination will not be complete in a finite sample. 

Referring to the situation in question 2, one might think that an informative prior would outweigh the effect of the increasing sample size. With respect to the Bayesian analysis of the linear regression, analyze the way in which the likelihood and an informative prior will compete for dominance in the posterior mean. 

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