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

The indicator function part of the prior restricts positive prior support to the region,
R, that satisfies a set of conditions derived from economic theory. Monotonicity is required
for all inputs, the good, and the bad. These conditions have to be evaluated at a particular
point in the data set. Due to potential measurement errors, we do not require monotonicity
at 100% of our data points. Instead, we define monotonicity as satisfied when 85% of the
data points meet their required monotonicity conditions.

We can write this prior distribution as the product of its three parts: a multivariate
normal for the γ parameters, the Jeffreys prior on Σ, and an indicator function to represent
the restrictions from economic theory. Write this as

p(γ, ∑) α MVN(go ,Ho)∣∑^-11^-5/2 I(γ, R),

where g_o is the vector of prior means on the parameters in γ , H_o is the prior variance-
covariance matrix on the same parameters, and I (γ, R) represents the indicator function
that equals one when the restrictions are satisfied and zero otherwise.

4.3.2 The Posterior Density

Following Zellner (1998) and Zellner and Tobias (2000), a maxent framework is used to
yield the least informative posterior density that is still proper and consistent with the prior
in (4.16) and the first and second moment conditions specified by our instrumental vari-
ables, generalized method of moment approach. This joint density is a truncated version
of the standard multivariate normal-inverted Wishart distribution common in Bayesian
econometrics,

p(γ, Σ∣data) - MVN-IW(gp,Hp)I(γ, R),

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