MULTIPLE COMPARISONS WITH THE BEST: BAYESIAN PRECISION MEASURES OF EFFICIENCY RANKINGS

4.3 Bayesian Estimation Procedure

We estimate the model in a Bayesian generalized method of moments (BGMM) frame-
work. To do so, we must specify priors for the unknown parameters and a set of moment
conditions for the data. We can then follow earlier work to find the maximum entropy
(maxent) density that is compatible with our prior and moment information. In total, we
have a system of four equations: the distance function and three first-order conditions for
the three inputs. Our BGMM approach follows and extends Zellner (1998) and Zellner
and Tobias (2001) which both present estimates using a Bayesian method of moments
(BMOM) approach. Our extensions allow the use of instruments to address the endogene-
ity inherent in estimation of a distance function, the nonlinear nature of our system of
equations, and the incorporation of informative priors on the random parameters while
still yielding exact finite sample posterior moments for the parameters of interest (Zellner,
1998). To implement the BGMM algorithm, we combine two sets of moment conditions
and a proper prior density, yielding a proper posterior density for the unknown random
parameters.⁶

Our estimated distance system consists of (4.9), subject to (4.10) and (4.11), and
a set of first order conditions in (4.4), for a total of four equations. As indicated, we
impose symmetry and linear homogeneity. An additive iid error term, w_f^k_t , k = 1, . . . , 3,
is appended to each equation in (4.4). After setting Q = 2 in (4.11), we test a number
of null hypotheses by computing a quasi-likelihood ratio test statistic that equals the
sample size times the difference between the restricted and unrestricted criterion functions,
which is asymptotically distributed as chi-square. At the .01 level we fail to reject the
null hypothesis that βf0 = 0, ∀f and subsequently drop the corresponding firm dummies.

⁶ For full details of the estimation algorithm, see Atkinson and Dorfman, 2001.

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