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

where t is a trend and df is a dummy variable equal to one for firm f and zero for the other
firms.² With a fixed effects approach, the βf_q are firm-specific parameters to be estimated.
This avoids the distributional and exogeneity assumptions that would otherwise be required
in a random effects setup. Thus, the estimated equation is obtained by substituting (4.11)
into (4.10), which in turn is substituted into (4.9), so that the βf_q are fit directly with the
other parameters.

In the application that follows, we undertake a Bayesian method of moments estima-
tion based partially on the moment conditions E (vft | zft) = 0, where zft is a vector of
instruments. In distance function applications, it is highly unlikely that (ln yft, lnxft) will
be uncorrelated with vf_t , thus pointing to the need for an instrumental variables approach.

Since we do not impose one-sidedness (non-negativity) on the uft in estimation,
we need to do so after estimation, by adding and subtracting from the fitted model
U_t = minf(Uft), which defines the frontier intercept. With lnD(y,x,t) representing the
estimated translog portion of (4.9) (i.e., those terms other than h(ef_t)), adding and sub-
tracting Ut yields

ʌ     Z                     X                                  .                    .                .                          ^   . Z                     X                                  .  ...                                       Z .         . X

0 = lnDi(y, x,t) + Vft - Uft + ^t - Ut = lnD_i (y, x,t) + vf - ^ft,         (4.12)
where ln D* (y, x, t) = ln Di (y, x, t) — Ut is the estimated frontier distance function in period
t and Uf_t = ^f_t — U_t ≥ 0.

Using (4.11), we estimate firm f’s level of technical efficiency in period t, TEft, as

TEft = exp(^-uf t),

where our normalization of Uf_t guarantees that 0 < TEf_t ≤ 1.

² An alternative approach by Koop (2001) parameterizes the mean of an exponential technical ineffi-
ciency distribution using a vector of variables thought to correlate with firm-specific effects.

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