technically efficient than any single firm outside of G4. Our firms are identified by these
groupings in Table 2, with the groups identified by the numbers 1 (most efficient) through
4 (least efficient). The posterior mean estimates of each firm’s TE are also displayed
graphically in Figure 1, sorted from least to most efficient along with their firm IDs and
group numbers. One can clearly see differentiation between the most and least efficient
groups and the firms in the middle two groups. Visually differentiating between G2 and G3
is more problematic. This visual information motivates us to go beyond the firm by firm
analysis briefly mentioned above used in initially categorizing the firms. A use of Bayesian
multiple comparisons will allow us to precisely define which firms can be differentiated
from each other.
4.4.1 Bayesian Multiple Comparisons
To present TE results obtained using the BMC approach we will use the two firms
identified as the most efficient, firm 31 (F31), and the least efficient, firm 1 (F1), and also
the groups of firms (designated G1, G2, G3, and G4 for this section). Because we hold the
comparison units (firm or group) constant, this is analogous to what Horrace and Schmidt
call MCC.8
Results of comparing each of the four efficiency groups to the others are presented in
Table 3. As can be seen in the table, G4 (the least efficient group) can be differentiated
from G2 and G1 with high probability levels, implying that all firms in G4 are almost
surely less efficient than all firms in both G1 and G2. However, G3 is more efficient as
a group than G4 at a probability support level that would not satisfy many researchers
8 The numerical Bayesian approach easily adapts to the MCB algorithm of an unknown “best” firm. To
compute a probability that firm A is less efficient than the “best” or most efficient firm is likely to result
in a probability near 1 given that firm A’s efficiency level cannot be greater than that of the most efficient
firm in any draw, only equal to it.
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