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ISSN 2229 – 5046
DERIVATION OF NORMAL – EXPONENTIAL MODEL
FOR STOCHASTIC COST FRONTIER ANALYSIS
SHAMEENA. H. KHAN*, MARY LOUIS. L
*Ph.D scholar, Department of Mathematics,
Avinashilingam Institute for Home Science & Higher education for women,
Coimbatore-641043, Tamil Nadu, India.
Associate Professor of Mathematics,
Faculty of Engineering, Avinashilingam Institute for Home Science & Higher education for
women, Coimbatore-641043, Tamil Nadu, India.
(Received On: 08-06-17; Revised & Accepted On: 05-07-17)
ABSTRACT
S
tochastic Frontier Analysis in term of cost means to produce a specified level of output in a minimal possible manner. This paper provides detailed steps on the estimation of Normal Exponential model for the estimation of stochastic cost frontier. Stochastic Cost Frontier plays a major role in the estimation of Cost Efficiency Scores in the field ofproduction. Here
v
i~
N
(
0
,
σ
v2)
andu
i~
exp
onential
.The parameters are evaluated using MaximumLikelihood Estimates. The cost efficiency of each producer can be obtained from
( )
ui.
i
CE
=
e
−Key words: Stochastic Frontier Analysis, Stochastic cost frontier, Cost efficiency Scores, Normal Exponential distribution, Cost efficiency scores, Maximum likelihood.
1. INTRODUCTION
Stochastic Cost Frontier Analysis (SCFA) which began with Farell (1957) has wide range of applications in different fields such as hospitals, banks, education sector etc. SCFA is a mathematical model to estimate cost function parameters using regression techniques. Hence comes the importance of distributional assumption. The SCFA model
can be expressed as vi ui
i i
f
x
e
y
=
(
,
β
)
+ wherey
i is the output,x
i,
β
are the input and technology parameterrespectively.
v
i+
u
i is the composed error term withv
i as the random shock andu
icaptures the cost efficiency. Meeusen and Van den Broeck (1977) assigned exponential distribution tou
i and Aigner, Lovell and Schmidt (1977) considered both exponential and half normal. In this paperv
i follows normal distribution andu
ifollows exponential distribution. Alijar (2014) considered truncated and exponential distribution foru
i and proposed that exponential distribution is just like half normal distribution and a special case of truncated normal distribution.In the study relatedto the estimation of technical efficiency of maize farmers, John Ng’ombe and Thomson Kalinda (2015) assumed
exponential distribution for
u
i . Maximum likelihood estimation which was developed by Aigner and chu (1968) canbe used to maximise the log likelihood function of
f
(
ε
)
. Normal-Exponential Stochastic Cost Frontier Model(NESCFM) can be estimated once the conditional probability of
u
i/
ε
i is derived.Corresponding Author: Shameena. H. Khan*,
*Ph.D scholar, Department of Mathematics,
© 2017, IJMA. All Rights Reserved 235
2. THE NORMAL-EXPONENTIAL STOCHASTIC COST FRONTIER MODEL (NESCFM)
In the estimation, of vi ui
i
i
f
x
e
y
=
(
,
β
)
+i)
v
i~
iid
N
(
0
,
σ
v2)
ii)
u
i ~ iid exponentialiii)
v
iandu
i are independently and identically distributed of each other and of the regressors. The probability density function ofv
is2 2
2
2
1
)
(
vv v
e
v
f
σσ
π
−
=
(1)The probability density function of
u
isu
u u
e
u
f
σσ
−
=
1
)
(
, foru
≥
0
(2)Due to the independence assumption, the joint probability density function of
v
andu
is the product of their individual density functions therefore,)
(
)
(
)
(
u
v
f
u
f
v
f
=
2 2
2
2
1
)
,
(
u vv u v u
e
v
u
f
σ σσ
σ
π
− −
=
(3)Taking
ε
=
v
+
u
, the joint probability density function ofu
andε
is given by2 2
2 ) (
2
1
)
,
(
u vu u v u
e
u
f
σε σ
σ
σ
π
ε
− − −
=
(4)2 2 2
2 ) 2 (
2
1
)
,
(
u vu u u
v u
e
u
f
σε ε σ
σ
σ
π
ε
+ − − −
=
(5)2 2 2 2 2
2 2 2 2
2
1
)
,
(
u v v vu u u
v u
e
u
f
σ σε σ ε σ
σ
σ
π
ε
=
− − + − (6)
+ − − −
=
22 2
2 2
) 1 ( 2 2 1
2
1
)
,
(
v v u vu u
v u
e
u
f
σε σ
σ ε σ
σ
σ
π
ε
(7)
+ − − −
=
22 2
2
) (
2 2 1
2
1
)
,
(
u vv v v v
u u v u
e
u
f
σε σ
σ σ ε σ σ
σ
σ
π
ε
(8)
+ − − −
=
2 2 2
2
) 1 ( 2 2 1
2
1
)
,
(
v v u v v v
u u
v u
e
u
f
σ ε
σ σ σ
ε σ σ
σ
σ
π
ε
(9)Since,
v u
σ
σ
λ
=
+ − − −
=
22 2
2
) 1 ( 2 2 1
2
1
)
,
(
v v v vu u v u
e
u
f
σε λ σ
ε σ σ
σ
σ
π
+
− −
− − −
=
22 2 2
1 )
1 ( 2 1
2
1
)
,
(
v v v vu v
u
e
u
f
σε λ σ
ε λ
σ ε σ
σ
σ
π
ε
(11)
+ − + − −
− − −
=
22 2 2 2 2
1 2 2
1 ) 1 ( 2 1
2
1
)
,
(
v v v v ve
e
u
f
u v u
σ ε λ λ σ
ε σ
ε λ
σ ε σ
σ
σ
π
ε
(12)
−
− − −
=
σλε λ λ
σ ε σ
σ
σ
π
ε
e
v ve
vu
f
u v u
2 1 2 1 ) 1 ( 2 1
2 2
2
1
)
,
(
(13)The marginal density function of
ε
is given by.
)
,
(
)
(
0
∫
∞
=
f
u
du
f
ε
ε
du
e
e
f
v v vu v
u
∫
∞
− − −
−
=
0
) 1 ( 2 1 2
1 2
1 2
2
2
1
)
(
σ λε σ λ
σ ε λ
σ
σ
π
ε
(14)Define u s
v v
= − −( σε λ1 )
σ ;
du
=
σ
vds
(15)When
v
s
u
σ
ε
λ
−
→
→
0
,
1
;u
→
∞
,
s
→
∞
∫
∞− −
−
=
v
v
e
ds
e
f
vs v
u
σ ε λ λ σ
ε
λ
σ
σ
σ
π
ε
1 2 2
1 2
1 2
2
2
1
)
(
(16)∫
∞− −
−
=
v v
ds
e
e
f
vs u
σ ε λ λ
σ ε
λ
σ
π
σ
ε
1 2 2
1 2
1 2
2
2
1
1
)
(
(17)
−
Φ
−
=
−
v u
v
e
f
σ
ε
λ
σ
ε
σ λε
λ
1
1
1
)
(
2 1 2 1
2
(18)
3. MEAN AND VARIANCE
The marginal density function, which is asymmetrically distributed with mean and variance is given by
∫
∫
∞ −
∞
=
=
=
=
+
=
=
0
0
1
)
(
)
(
)
(
)
(
)
(
)
(
du
e
u
u
E
E
du
u
f
u
u
E
u
v
E
E
Mean
u
u u
σ
σ
ε
ε
(19)
Define,
u
s
du
uds
u
σ
σ
=
;
=
s
varies from0
to∞
Therefore,
∫
∞
−
=
0
.
1
)
(
u
s
e
ds
E
us u u
σ
σ
σ
(20)[
− −]
∞−
−
=
(
1
)
0)
(
u
use
se
sE
σ
(21)u u
u
© 2017, IJMA. All Rights Reserved 237
To find Variance,
v
(
ε
)
=
v
(
v
)
+
v
(
u
)
(23)
2 2
0
(
)
( )
E u
u f u du
∞
=
∫
2
0
1 u
u
u
u e σ du
σ
∞ −
=
∫
(24)2 2 2
,
;
u;
uu
u
Again let
s du
σ
ds u
σ
s
σ
=
=
=
s
varies from0
to∞
Hence,
E
u
us
e
s uds
u
σ
σ
σ
− ∞
∫
=
0 2 2
2
1
)
(
(25)ds
e
s
u
E
su
u −
∞
∫
=
0 2 3 2
)
(
σ
σ
(26)
[
−
−−
−+
−
−]
∞=
02 2 2
)
(
2
)
(
2
)
(
)
(
u
us
e
ss
e
ss
e
sE
σ
(27)2 2
2
2
)]
1
0
(
2
0
[
)
(
u
u uE
=
σ
−
−
=
σ
(28) 22 2 2
)
(
2
)
(
u
u u uand
v
v
vv
=
σ
−
σ
=
σ
=
σ
(29) 22
)
(
v uv
ε
=
σ
+
σ
(30)The product of the density function of the individual observations is the likelihood function of the sample, which is given as,
The log likelihood equation for a sample of K producers is
(
)
∑
∑
= =
−
+
−
Φ
−
+
−
=
ki v i k
i v
i u
k
In
In
k
L
In
1 2 1
2
2
1
1
2
σ
λ
ε
λ
σ
ε
λ
σ
(31)Let
ε
i=
y
i−
β
'x
i(
)
∑
∑
= =
−
−
+
−
−
Φ
−
+
−
=
Ki v
i i
K
i v
i i
u
x
y
K
x
y
In
In
K
L
In
1
' 2
1
'
2
(
)
2
)
(
1
1
2
σ
λ
β
λ
σ
β
λ
σ
(32)Using loglikelihood function, cost efficiency can be measured after the parameter estimation.
4. ESTIMATION OF THE PARAMETERS OF THE NORMAL-EXPONENTIAL STOCHASTIC COST FRONTIER MODEL
Parameters
σ
u2,
σ
v2,
λ
2,
β
can be estimated using the first order conditions of the maximization of log-likelihoodfunction of (32) as follows.
0
1
2
22 2
*
=
−
=
∂
∂
=
u u
u
K
InL
E
σ
σ
σ
(33)(
)
0
)
(
1
1
)
(
1
2
1
2
1
'1 '
'
3 1
3 ' 2
2
*
−
=
−
−
Φ
−
−
−
−
−
=
∂
∂
=
∑
∑
=
= i i
K
i
v i i
v i i
v K
i v
i i
v
v
y
x
x
y
x
y
x
y
InL
E
β
σ
β
λ
σ
β
λ
φ
λσ
σ
β
λ
σ
σ
(34)(
)
22
)
(
)
(
)
(
u
E
u
E
u
0
)
(
1
1
)
(
1
1 1 ' ' *+
=
−
−
Φ
−
−
−
=
∂
∂
=
∑
∑
= = K i v i v i K i v i i v i ix
x
x
y
x
y
InL
E
λ
σ
σ
σ
β
λ
σ
β
λ
φ
β
β
(35)(
)
0
)
(
1
1
)
(
1
2
1
2
' 1 ' ' 3 4 2 2 *−
=
−
−
Φ
−
−
−
+
−
=
∂
∂
=
∑
= i i
K i v i i v i i
x
y
x
y
x
y
k
InL
E
β
σ
β
λ
σ
β
λ
φ
λ
λ
λ
λ
(36)Once the parameters are estimated the inefficiency,
u
i can be obtained. In order to extract the information thatε
i contains onu
ican be done using the conditional distribution ofu
igivenε
i.(Kumbhakar et al. 2003). The conditional distribution ofu
i givenε
i whenu
i follows half normal distribution is given by Jondrow et al.(1982), same condition can be used whenu
iis assumed with exponential distribution.)
(
)
,
(
)
/
(
ε
ε
ε
f
u
f
u
f
=
−
Φ
−
=
− − − − − v u u v u v v v ve
e
e
u
f
σ
ε
λ
σ
σ
σ
π
ε
λ σε λ λ σ ε λ λ σ ε σ1
1
1
2
1
)
/
(
2 1 2 1 2 1 2 1 ) 1 ( 2 1 2 2 2 (37) 2 ) 1 ( 2 1 11
1
2
1
)
/
(
− − − −
−
Φ
−
=
σ λε σ
σ
ε
λ
σ
π
ε
v vu v
v
e
u
f
(38)2 ) 1 ( 2 1 1
1
1
2
1
)
/
(
− − − −
−
Φ
−
=
v u v v u v v u ve
u
f
σ σ σ ε σσ
ε
σ
σ
σ
π
ε
(39)2 2 ) ( 2 1 1 2
1
2
1
)
/
(
− − − −
−
Φ
−
=
u vv v v u v v u v v
e
u
f
σ σσ σ ε σ
σ
ε
σ
σ
σ
σ
π
ε
(40)2 2 ) ( 2 1 1 2
1
2
1
)
/
(
− − − −
−
−
Φ
−
=
u v v u v u v v ve
u
f
σ σ ε σσ
σ
σ
σ
ε
σ
π
ε
(41)Let u v
σ
σ
ε
µ
^=
−
22 ^ ) 2 1 1 ^
1
2
1
)
/
(
− − −
−
Φ
−
=
σ µσ
µ
σ
π
ε
u v v ve
u
© 2017, IJMA. All Rights Reserved 239
5. MEASURE OF COST EFFICIENCY OF NORMAL-EXPONENTIAL STOCHASTIC COST FRONTIER
MODEL
As
f
(
u
/
ε
)
is distributed as(
,
2)
^
v
N
+µ
σ
, the mean of this distribution can serve as a point estimator of which isgiven by
∫
∞=
0)
/
(
)
/
(
u
u
f
u
du
E
ε
ε
(43)∫
∞ − − −
−
Φ
−
=
0 ) 2 1 1^ ^ 2
1
2
1
)
/
(
u
u
e
du
E
u v v v µ σσ
µ
σ
π
ε
(44)du
e
u
u
E
u v v v∫
∞ − − −
−
Φ
−
=
0 ) 2 1 1^ ^ 2
1
2
1
)
/
(
µ σσ
µ
σ
π
ε
(45)Let
;
;
.
^ ^
ds
du
s
u
u
s
v vv
σ
µ
σ
σ
µ
=
+
=
−
=
s
varies fromv
σ
µ
^−
to∞
s
1
2
1
)
/
(
^ 2 -2 ^ v 1 ^∫
∞ − −
+
−
Φ
−
=
vds
e
u
E
v s v v σ µσ
µ
σ
σ
µ
σ
π
ε
(46)1
2
1
)
/
(
^ 2 2 ^ -2 ^ 2 v 1 ^
+
−
Φ
−
=
∫
∞ −∫
∞ − − − v vds
e
ds
e
s
u
E
s s v σ µ σ µµ
σ
σ
µ
π
ε
(47)2
1
1
)
1
(
1
2
1
)
/
(
^ 2 ^ 2 -2 1 ^ ^ 2 1 ^∫
∞ − − ∞ − −
−
Φ
−
+
−
−
Φ
−
=
v vds
e
e
u
E
s v s v v σ µ σµ
σ
π
µ
µ
σ
σ
µ
π
ε
(48)2
1
1
2
1
1
)
/
(
^ 2 2 2 ^ -2 1 ^ ^ 2 1 ^∫
∞ − − − −
−
Φ
−
+
−
Φ
−
=
vv
e
ds
e
u
E
s v v v σ µ σ µπ
σ
µ
µ
π
σ
σ
µ
ε
(49)1
1
1
)
/
(
^ 1 ^ ^ ^ 1 ^
−
Φ
−
−
Φ
−
+
−
Φ
−
=
− − v v v v vu
E
σ
µ
σ
µ
µ
σ
µ
φ
σ
σ
µ
ε
(50)
−
Φ
−
+
=
v v vu
E
σ
µ
σ
µ
φ
σ
µ
ε
^ ^ ^1
)
/
6. CONCLUSION
Once the marginal density function of
ε
iis calculated, using log likelihood functions,σ
u2,
σ
v2,
λ
2,
β
are estimated.The cost efficiency of each producer can be obtained from ,where . If an organisation uses
its resources allocatively and technically efficient then it can be said that it has achieved cost efficiency. We may apply the models to real data sets to get the cost efficiencies for each producers. Similarly, we can estimate the parameters for other distributions and derive cost efficiency. According to the value of the loglikelihood function of each model, we can conclude which model is better. Analysis of the collected data can be done using LIMDEP software.
REFERNCES
1. Alijar Meesters. A note on the assumed distributions in stochastic frontier models. Journal of Productivity Analysis 42 (2014), 171-173.
2. Aigner, D.J. and SF. Chu,On estimating the industry production function, American Economic Review
58(1968), 826-839.
3. Aigner, Lovell and Schmidt.Formulation and estimation of stochastic frontier production function models.
Journal of Econometrics Copyright: North-Holland Publishing Company 6 (1977), 21-37.
4. Farrell Measurement of productive efiiciency.Journal of the Royal Statistical Society: series A (general) 120(1957), 253-290.
5. John Drow, Lovell, Materove and Schimdt. On the estimation of technical inefficiency in the stochastic
frontier production function model. Journal of Econometrics. North Holland Publishing Company 19 (1982), 233-238.
6. John Ng’ombe and Thomson Kalinda, A stochastic frontier analysis of technical efficiency of Maize
production under minimum tillage in Zambia. Sustainable Agriculture Research 4 (2015), 31- 46.
7. Kumbhakar. S and C.A Knox Lovell. A text book of Stochastic Frontier Analysis. Cambridge university press. (2003).
8. Meeusen W, Van den Broeck J. Efficiency estimation from Cobb-Douglas production functions with
composed error. International Economic Review 18 (1977), 435-444.
Source of support: Nil, Conflict of interest: None Declared.
