Advanced Econometrics Takeshi Amemiya
Bayesian Solution
The Bayesian solution to the selection-of-regressors problem provides a peda - gogically useful starting point although it does not necessarily lead to a useful solution in practice. We can obtain the Bayesian solution as a special case of the Bayes estimator (defined in Section 2.1.2) for which both в and D consist of two elements. Let the losses be represented as shown in Table 2.1, where Ln is the loss incurred by choosing model 1 when model 2 is the true model and L2l is the loss incurred by choosing model 2 when model 1 is the true model.3 Then, by the result of Section 2.1.2, the Bayesian strategy is to choose model 1 if
(2.1.3)
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where />(%), і = 1 and 2, is the posterior probability that the model і is true given the sample y. The posterior probabilities are obtained by Bayes’s rule as
and similarly for P(2|y), where в, = (fi'(, <rj )',f(y6,) is the joint density of у given в, i) is the prior density of 0, given the model i, and P(i) is the prior probability that the model і is true, for і = 1 and 2.
There is an alternative way to characterize the Bayesian strategy. Let S be a subset of the space of у such that the Bayesian chooses the model 1 if у Є S.
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Then the Bayesian minimizes the posterior risk
Ll2P{2)P{y Є S|2) + L2iP()P(y Є S 1) (2.1.5)
with respect to S, where S is the complement of S. It is easy to show that the posterior risk (2.1.5) is minimized when S is chosen to be the set of у that satisfies the inequality (2.1.3).
The actual Bayesian solution is obtained by specifying/(y|0,), /(0,|/)> and P(i) in (2.1.4) and the corresponding expressions for, P(2|y). This is not done here because our main purpose is to understand the basic Bayesian thought, in light of which we can perhaps more clearly understand some of the classical strategies to be discussed in subsequent subsections. The interested reader should consult Gaver and Geisel (1974) or Zellner (1971, p. 306). Gaver and Geisel pointed out that if we use the standard specifications, that is,/(y|fy) normal, “diffuse” natural conjugate, P(l) = P(2), the Bayesian solution leads to a meaningless result unless Kx = K2.4
