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

where MVN is a multivariate normal density, IW is an inverted Wishart density,

Hp^-1 = Ho^-1 + Hd^-1                            (4.18)

and

gp = Hp[Ho^-1go +H_d^-1gd],                           (4.19)

where gd is the conventional GMM estimator of γ and Hd is the conventional GMM
estimated covariance matrix of γ .

Because the joint posterior density is complicated to deal with due to the prior restric-
tions, we use Gibbs sampling to generate draws sequentially from conditional distributions
of parameter subsets.⁷ In this model, we only need two subsets. First, we can draw the
covariances from an inverted Wishart distribution conditional on the previous draw for the
γ vector. Then the γ vector can be drawn from a truncated multivariate normal distribu-
tion conditional on the drawn value of the Σ matrix. In terms of a “recipe,” the Gibbs
sampler in our application is comprised of the following steps:

0. Obtain initial value for covariance matrix of errors, S⁽⁰⁾ , either through conventional
GMM estimation as ê^zê/v, where the (4FT × 1) column vector e = (v^,,w^,₁ ,w2₂,w3₃)^z,
or after arbitrary choice of all parameters.

1. Draw Σ⁽ⁱ⁾ from IW(S⁽ⁱ⁾, ν), where ν = FT - K, and K is the number of estimated
parameters (Draw system covariance matrix conditional on covariance estimate in 1.)
See the Appendix for futher details on this step.

2. Compute g⁽ⁱ⁾ = GMM(y,X, Z∣Σ(ⁱ)), ( Compute GMM estimate conditional on Σ(ⁱ))
This requires iterating until convergence using GMM with the covariance of the errors
held constant at Σ⁽ⁱ⁾.

⁷ For a good and simple explanation of Gibbs sampling for the non-Bayesian, see Casella and George
(1992).

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