This volume in the Econometric Exercises series contains questions and answers to provide students with useful practice, as they attempt to master Bayesian econometrics. In addition to many theoretical exercises, this book contains exercises designed to develop the computational tools used in modern Bayesian econometrics. The latter half of the book contains exercises that show how these theoretical and computational skills are combined in practice, to carry out Bayesian inference in a wide variety of models commonly used by econometricians. Aimed primarily at advanced undergraduate and graduate students studying econometrics, this book may also be useful for students studying finance, marketing, agricultural economics, business economics or, more generally, any field which uses statistics. The book also comes equipped with a supporting website containing all the relevant data sets and MATLAB computer programs for solving the computational exercises.

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We offer an empirical test of a theoretical result in the contingent valuation literature. Specifically, it has been argued from a theoretical point of view that survey participants who perceive a survey to be "consequential" will respond to questions truthfully regardless of the degree of perceived consequentiality. Using survey data from the Iowa Lakes Project, we test this supposition. Specifically, we employ a Bayesian treatment effect model in which the degree of perceived consequentiality, measured as an ordinal response, is permitted to have a structural impact on willingness to pay (WTP) for a hypothetical environmental improvement. We test our theory by determining if the WTP distributions are the same for each value of the ordinal response.In our survey data, a subsample of individuals were randomly assigned supporting information suggesting that their responses to the questionnaires were important and will have an impact on policy decisions. In conjunction with a Bayesian posterior simulator, we use this source of exogenous variation to identify the structural impacts of consequentiality perceptions on willingness to pay, while controlling for the potential of confounding on unobservables. We find evidence consistent with the "knife-edge" theoretical results, namely that the willingness to pay distributions are equal among those believing the survey to be at least minimally consequential, and different for those believing that the survey is irrelevant for policy purposes.

Even a cursory look at the empirical literature in most fields of economics reveals that a majority of applications use simple parametric approaches such as ordinary least squares regression or two-stage least squares accompanied by simple descriptive statistics. The use of such methods has persisted despite the development of more general nonparametric techniques in the recent (and perhaps not-so-recent) statistics and econometrics literatures.At least two possible explanations for this come to mind. First, given the challenges-or lack of-provided by economic theories with empirical content, the parametric toolkit is more than sufficient. Where serious first-order problems in nonexperimental inference exist, they are in the inadequacy of the research design and data, not in the limitations of the parametric approach. Second, the predominant use of parametric approaches may reflect the lack of sufficient computational power or the difficulty of computation with off-the-shelf statistical software. Given the recent advances in computing power and software (as well as the development of the necessary theoretical foundation), only the first point remains an open question. The purpose of this article is to make the case that nonparametric techniques need not be limited to use by econometricians.Our discussion is divided into two parts. In the first part, we focus on "density estimation"-estimation of the entire distribution of a variable or set of variables. In the second part, we discuss nonparametric regression, which concerns estimation of regression functions without the straightjacket of a specific functional form.

SUMMARYWe generalize the specifications used in previous studies of the effect of body mass index (BMI) on earnings by allowing the potentially endogenous BMI variable to enter the log wage equation nonparametrically. We introduce a Bayesian posterior simulator for fitting our model that permits a nonparametric treatment of the endogenous BMI variable, flexibly accommodates skew in the BMI distribution, and whose implementation requires only Gibbs steps. Using data from the 1970 British Cohort Study, our results indicate the presence of nonlinearities in the relationships between BMI and log wages that differ across men and women, and also suggest the importance of unobserved confounding for our sample of males.

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This note derives simply computed closed-form expressions for the average treatment effect, the effect of treatment on the treated, the local average treatment effect, and the marginal treatment effect in a latent-variable framework for both normal and nonnormal models. Asymptotic standard errors for versions of these parameters that average over observed characteristics are also obtained. The performances of the derived estimators are also evaluated in Monte Carlo experiments under correct specification and misspecification. © 2003 President and Fellows of Harvard College and the Massachusetts Institute of Technology.

Using data from the National Longitudinal Survey of Youth (NLSY) we introduce and estimate various Bayesian hierarchical models that investigate the nature of unobserved heterogeneity in returns to schooling. We consider a variety of possible forms for the heterogeneity, some motivated by previous theoretical and empirical work and some new ones, and let the data decide among the competing specifications. Empirical results indicate that heterogeneity is present in returns to education. Furthermore, we find strong evidence that the heterogeneity follows a continuous rather than a discrete distribution, and that bivariate normality provides a very reasonable description of individual-level heterogeneity in intercepts and returns to schooling