Ceris-Cnr, W.P. N° 2/2002

or, alternatively,
f Ψ (ψ ) =

1
exp− {(ψ − δ ' z ) 2 /(σ v2 + σ u2 )}
2
.
2
2π (σ u + σ v2 )1 2 {Φ[δ ' z / σ u ] / Φ[ µ * / σ * ]}

[A.9b]

The density function for the cost value, vcft, in equation [A.1], is most
conveniently given using the expression in equation [A.9b],
2
1 [vc ft − vc( x ft ; β ) − δ ' z ] 
exp− 

2 
σ v2 + σ u2

,
f VC ft (vc ft ) =
2π (σ u2 + σ v2 )1 2 {Φ[d ft ] / Φ[d *ft ]}

[A.10]

where d ft = δ ' z ft / σ u , d *ft = µ *ft / σ * and µ *ft = [σ v2δ ' z ft + σ u2 (vc ft − vc( x ft ; β )] /(σ u2 + σ v2 ) .
Given that there are Tf observations obtained for the f th firm, where 1 ≤ Tf ≤ T, and
vcf ≡ (vc f 1 , vc f 2 ,..., vc fT f )' denotes the vector of the Tf cost values in equation [A.1], then
the logarithm of the likelihood function for the sample observations, vc ≡ (vc1’, vc2’,…,
vcF’)’, is
1 F
L(Θ * ; vc) = −  ∑ T f
2  f =1
T


 ln 2π + ln(σ u2 + σ v2 )



{

}

{

}

−

1 F f
[vc ft − vc( x ft ; β ) − δ ' z ft ] 2 /(σ u2 + σ v2 )
∑∑
2 f =1 t =1

−

1 F TF
∑∑ ln Φ[d ft ] − ln Φ[d *ft ] ,
2 f =1 t =1

{

}

[A.11]

where Θ* ≡ (β ’, δ ’, σu2, σv2)’.
Using the re-parameterization of the model suggested by Battese and Corra
(1977), involving the parameters σ 2 ≡ (σv2 + σu2) and 0 ≤ γ ≡ σu2/(σv2 + σu2) ≤ 1, the
logarithm of the likelihood function can be expressed by
1 F
L(Θ; vc) = −  ∑ T f
2  f =1
−

T

{

T

{


 ln 2π + ln σ 2



{

}

1 F f
[vc ft − vc( x ft ; β ) − δ ' z ft ] 2 / σ 2
∑∑
2 f =1 t =1

}

1 F f
− ∑∑ ln Φ[d ft ] − ln Φ[d *ft ] ,
2 f =1 t =1

43

}
[A.12]