**Bayesian Econometrics, Bayes Theory Analysis**

Econometrics Lecture Notes

The Bayesian approach is a fundamentally different interpretation of the meaning of statistics. It proposes a subjective interpretation of statistics as opposed to an objective interpretation. Bayesians propose dropping the classical interpretation and, therefore, dropping traditional classical econometrics. Needless to say, there is significant controversy among statisticians about the Bayesian versus the classical method. An understanding of this difference is fundamental to an understanding of many of the confusions that surround econometric testing.

To see the distinction, suppose that we want to estimate the value of a parameter. In classical statistics, one arrives at a point estimate of the parameter that satisfies certain characteristics, such as BLUE criteria (best, linear, unbiased, estimator). In addition, it must have desirable asymptotic properties, so that when large amounts of data are available, the estimate will converge to the true value of the parameter. The total focus in classical analysis is on the estimator and the statistics that characterize it.

In Bayesian analysis the interpretation of an estimator is quite different. Instead of producing a point estimate of data, Bayesian analysis produces a density function for data, which is called a posterior density function. The density function is not a sampling distribution. It can be interpreted only in reference to a prior conviction about what one believed. It is normally discussed as the odds a researcher would give when taking bets on the true value of data. It is a subjective notion of probability rather than an objective or frequentist notion of probability, as is the classical approach.

Thus, in the Bayesian approach one must specify one's initial degree of belief and use empirical evidence as a means of changing that degree of belief. One has both a prior density function and a posterior density function. In the Bayesian analysis one is simply using empirical data to modify one's prior beliefs, whereas in the classical approach one is continually attempting to establish the true nature of the model.

For the most part, economists have not used Bayesian methods. The reasons for this are not so much that they object to the underlying philosophical nature of subjectivist probability; instead, they are practical reasons: (1) formalizing prior beliefs into a formal distribution is difficult; (2) the mechanics of finding the posterior distribution are difficult; and (3) convincing others of the validity of Bayesian results is difficult because they are definitely contaminated or can only be interpreted by personal beliefs. These practical problems notwithstanding, a number of econometricians are seriously committed to Bayesian econometrics.

Baysesian methods have not significantly caught on, but there has been a groundswell of complaints about how little what is taught in econometric courses actually reflects what econometricians do. For example, Intriligator, Bodkin, and Hsiao write:

At least 80 percent of the material in most of the existing textbooks in econometrics focuses purely on economic techniques. By contrast, practicing econometricians typically spend 20 percent or less of their time and effort on econometric techniques per se; the remainder is spent on other aspects of the study, particularly on the construction of a relevant econometric model and the development of appropriate data before estimations and the interpretation of results after estimations.

The reason for this difference is that the professors teaching econometrics often are not the people who are actually doing econometrics. As Magnus and Morgan (1999) emphasize, actual econometric work is learned by doing, not by what is taught. Whether these complaints will lead to better empirical work in the future remains to be seen.