5. Conclusions
While many researchers have developed and applied methods for estimating the tech-
nical efficiency of firms (or other units of observations), less effort has been expended on
examining the precision of the estimated efficiency scores and the resulting rankings of the
firms studied. Horrace and Schmidt (2000) introduced two multiple comparison techniques
(MCC and MCB) based on sampling theory statistics to this literature. In this paper, we
add a Bayesian approach to the toolkit for measuring the precision of efficiency estimates
and the ability to such estimates to accurately differentiate between the units being ranked.
After presenting the details of how to implement the Bayesian Multiple Compari-
son (BMC) approach, we presented an application to a panel of 43 U.S. electric utilities.
Bayesian estimation of a distance function yields a set of technical efficiency estimates con-
sistent with economic theory that provide an empirical ranking of the 43 firms. Application
of the BMC approach then allows us to analyze which firms can truly be differentiated
from which others at any desired level of probability. That is, we can make statements such
as “there is an 99.6% probability that firm 31 is more efficient than all the firms in group
4” and “there is a 54.7% probability that all the firms in group 1 are more efficient than
all the firms in group 3.” The MCB approach of Horrace and Schmidt was also applied to
the same technical efficiency estimates and provided some contrasting results in terms of
the ability to differentiate firms on the basis of their TE scores.
We believe that the Bayesian results provide more flexibility in terms of multiple
comparisons that are possible, particularly for those researchers who are not statistical
experts. Using the procedure outlined in this paper, it is straightforward to compute the
probability of any firm or group of firms being more efficient than any other firm or group
of firms. This probability provides an exact measure of the ability to rank the groups/firms
26
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