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 βfq 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 βfq 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 vft , 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
Ut = 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(eft)), adding and sub-
tracting Ut yields
ʌ Z X . . . ^ . Z X . ... Z . . X
0 = lnDi(y, x,t) + Vft - Uft + ^t - Ut = lnDi (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 Uft = ^ft — Ut ≥ 0.
Using (4.11), we estimate firm f’s level of technical efficiency in period t, TEft, as
(4.13)
TEft = exp(-uf t),
where our normalization of Uft guarantees that 0 < TEft ≤ 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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