Ceris-Cnr, W.P. N° 2/2002

from the conditional distribution of uft given ψft, which incorporates whatever
information ψft contains concerning uft.
The conditional density function of uft given Ψft = ψft is given by83
fU | Ψ = ψ (u ) =

f Ψ ,U (ψ , u )
f Ψ (ψ )

,

[A.17]

thus, using equations [A.6B] and [A.9b],
1
exp− {(u − µ * ) 2 / σ *2 }
2
.
f U | Ψ = ψ (u ) =
2π σ * Φ[ µ * / σ * ]

[A.18]

The overall cost efficiency of the f th firm at the tth observation, CEft, may be
expressed as the ratio of stochastic frontier minimum cost (with uft = 0) to observed
cost, which is equal to84
CE ft =

1
= exp{−u ft } .
exp{u ft }

[A.19]

This measure is bounded between zero (uft → ∞) and one (uft = 0), and can be
predicted in a similar way to that described for technical efficiency in the stochastic
production frontier case analyzed by Battese and Coelli (1993). Using the conditional
distribution of uft given ψft defined by equation [A.17], the authors derive an expression
for the conditional expectation of the technical efficiency for the f th firm at the tth
observation, conditional upon the observed value of ψft = (vft - uft). This expression,
E(exp{-uf}|Ψft = ψft), is a generalization of the results presented in Jondrow et al. (1982)
and Battese and Coelli (1988).
The prediction of the individual cost efficiencies relative to a stochastic cost
frontier, i.e. expression [A.19], can be obtained by minor sign alterations of the
technical efficiency point estimator in Battese and Coelli (1993). It is derived using the
conditional density function of uft given Ψft = ψft specified in equation [A.18] and is
given by
 Φ[( µ *ft / σ * ) − σ * ] 
 * 1 2
ˆ
CE ft = (exp{−u ft } | Ψ ft = ψ ft ) = 
 exp− µ ft + σ * 
*
2 
 Φ[ µ ft / σ * ] 


83

84

[A.20]

Again the subscripts, f and t, are omitted in the following expressions for convenience in the
presentation.
Expression [A.19] is appropriate for CEft only if the general specification of the stochastic frontier
cost model is given by equations [1]-[2] in the text.

45