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Unformatted text preview: Imbens/Wooldridge, Lecture Notes 7, NBER, Summer 07 1 Whats New in Econometrics NBER, Summer 2007 Lecture 7, Tuesday, July 31th, 11.00-12.30pm Bayesian Inference 1. Introduction In this lecture we look at Bayesian inference. Although in the statistics literature ex- plicitly Bayesian papers take up a large proportion of journal pages these days, Bayesian methods have had very little impact in economics. This seems to be largely for historial rea- sons. In many empirical settings in economics Bayesian methods appear statistically more appropriate, and computationally more attractive, than the classical or frequentist methods typically used. Recent textbooks discussing modern Bayesian methods with an applied focus include Lancaster (2004) and Gelman, Carlin, Stern and Rubin (2004). One important consideration is that in practice frequentist and Baeysian inferences are often very similar. In a regular parametric model, conventional confidence intervals around maximum likelihood (that is, the maximum likelihood estimate plus or minus 1.96 times the estimated standard error), which formally have the property that whatever the true value of the parameter is, with probability 0.95 the confidence interval covers the true value, can in fact also be interpreted as approximate Bayesian probability intervals (that is, conditional on the data and given a wide range of prior distributions, the posterior probability that the parameter lies in the confidence interval is approximately 0.95). The formal statement of this remarkable result is known as the Bernstein-Von Mises theorem. This result does not always apply in irregular cases, such as time series settings with unit roots. In those cases there are more fundamental differences between Bayesian and frequentist methods. Typically a number of reasons are given for the lack of Bayesian methods in econometrics. One is the difficulty in choosing prior distributions. A second reason is the need for a fully specified parametric model. A third is the computational complexity of deriving posterior distributions. None of these three are compelling. Imbens/Wooldridge, Lecture Notes 7, NBER, Summer 07 2 Consider first the specification of the prior distribution. In regular cases the influence of the prior distribution disappears as the sample gets large, as formalized in the Bernstein-Von Mises theorem. This is comparable to the way in which in large samples normal approx- imations can be used for the finite sample distributions of classical estimators. If, on the other hand, the posterior distribution is quite sensitive to the choice of prior distribution, then it is likely that the sampling distribution of the maximum likelihood estimator is not well approximated by a normal distribution centered at the true value of the parameter in a frequentist analysis....
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