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

Similarly, we fail to reject the null hypothesis that βf2 = 0, ∀f and therefore Q is set equal
to 1 in (4.11). Further, we set β₁₁ = 0 to achieve identification.

Since input and output quantities in all distance function specifications may be en-
dogenous, we use an instrumental variables approach. To examine identification issues we
use the Hansen (1982) J test. We found support for the use of the set of instruments con-
taining firm dummies, time period dummies, the interaction of continuous time and firm
dummies, the interaction of continuous time squared and firm dummies, and the interac-
tion of continuous time cubed and firm dummies. This set of moment conditions generated
the largest p-value for the J test statistic. We also confirm that the instruments are highly
correlated with the regressors.

4.3.1 Specification of the Prior

In the specification of the prior, we differ from Zellner (1998) and Zellner and Tobias
(2001), by going beyond a maxent prior to a more informative one. The full prior dis-
tribution is a product of independent priors on the structural parameters of the distance
function, the prior on the covariance matrix of the vector of errors, and a set of indicator
functions that restrict prior support to the region where the theoretical restrictions are
satisfied.

The structural parameters of the distance function are each given a normal prior
distribution with zero mean and variance of 100. This is a very diffuse prior, having
virtually no effect on the posterior means, but does ensure that the prior is proper in any
dimensions that are not restricted to a finite subspace by the indicator function part of
the prior. It also makes the posterior sample density more straightforward to work with
when we begin Gibbs sampling. The prior for Σ (the matrix of variances and covariances
of the four errors appended to the equations to be estimated) is a standard Jeffreys prior.

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