ON THE PREDICTION OF TECHNICAL EFFICIENCIES,
GIVEN THE SPECIFICATIONS OF A GENERALIZED
FRONTIER PRODUCTION FUNCTION AND PANEL
DATA ON SAMPLE FIRMS
George E. Battese
No.28 - June 1986
Typist: Val Boland
ISSN
I SBN
On the Prediction of Technical Efficiencies, Given the
Specifications of a Generalized Frontier Production
Function and Panel Data on Sample Firms
by
George E. Battese
University of New England
ABSTRACT
A stochastic frontier production function is defined
for panel data on sample firms, such that the disturbances
associated with observations for a given smaple firm involve
the differences between traditiona! symmetric random errors
and a non-negative random variable, which is associated with
the technical efficiency of the firm. Given that the nonnegative firm effects have a general truncated normal
distribution, we obtain the best predictor for the firm-effect
random variables and, hence, the technical efficiency of
individual firms, given the values of the disturbances in the
model. The results obtained are a generalization of those
presented by Jondrow et al. (1982) for a cross-sectional model
in which the firm effects have half-normal distribution.
1.
Introduction
The stochastic frontier production function, proposed independently by
Aigner, Lovell and $chmidt (1977) and Meeusen and van den Broeck (1977),
has been considered and applied or modified in a number of studies,
including Battese and Corra (1977), Lee and Tyler (1978), Stevenson (1980),
Pitt and Lee (1981), Jondrow et al. (1982), Kalirajan (1982), Kalirajan and
Flinn (1983), Schmidt and Sickles (1984), Waldman (1984) and Coelli and
Battese (1986). The earlier studies involved the estimation of the
parameters of the stochastic frontier production function and the mean
technical efficiency for firms in the industry. It was initially claimed
that technical efficiencies for individual sample firms could not be
predicted. Jondrow et al. (1982) presented two estimators for the technical
efficiency for an individual firm on the assumption that the parameters of
the frontier production function were known and cross-sectional data were
available for sample firms. Schmidt and Sickles (1984) considered a number
of methods of predicting individual firm effects (and hence technical
efficiencies) given that panel data were available on sample firms. Waldman
(1984) investigated the properties of the predictor for firm technical
efficiencies proposed by Jondrow et al. (1982) and two other possible
predictors.
In this paper we present a generalization of the predictor proposed by
Jondrow et al. (1982), under the assumption that panel data on sample firms
are available and that the more general distribution for firm effects,
suggested by Stevenson (1980), applies for the stochastic frontier production
function.
2 o
The Frontier Production Function
Consider the stochastic frontier production function
Yit : ~it~ + Eit
(i)
Eit = Vit - Ui
(2)
and
where Yit denotes the logarithm of the production variable for the i-th
sample firm (i=l,2,...,N) in the t-th time period (t=l,2,...,T);
~it is a (i × k) vector of the logarithms of the inputs associated with
the i-th sample firm in the t-th time period (the first element would
generally be one);
8 is a (k × i) vector of elasticity parameters for the associated
Cobb-Douglas production function;
3o
the Vit-random variables are assumed to be independent and identically
2
distributed as N(O,~V) independent of the U.-random.~_ variables, which are
assumed to be independent and identically distributed non-negative random
variables, defined by the truncation (at zero) of the N(~,o2) distribution.
In addition, it is assumed that the Vit and U1
. random variables are
independently distributed of the input variables in the model.
The density function for U. is defined by
1
exp [’ ~(u-p) 2/o2]
f (u) =
u.1
(2~)½~
[i-%(-~/O) ]
,
u > 0
(3)
where ~(.) denotes the distribution function of the standard normal random
variable. It is readily verified that the mean of U. is
1
E(Ui) = n + d{¢(-U/~)/[l-¢(-~/o)]}
(4)
where ~(.) denotes the density function of the standard normal random variable.
Further, the variance of U. is
1
2
Var(Ui) = 0 -[E(Ui)-P]E(Ui).
(5)
These results are required in subsequent derivations.
The distribution of the non-negative firm-effect random variables is that
suggested by Stevenson (1980), which is a generalization of the half-normal
distribution (in which p=0). Pitt and Lee (1981) and Schmidt and Sickles
(1984) considered the special case of this model in which the firm effects
had half-normal distribution. As noted by Pitt and Lee (1981, p.46), firms
may discover the extent of their inefficiency, after a period of time, and
adjust their input values accordingly. This is not assumed to be the case in
this paper.
The above model is considered by Coelli and Battese (1986). The
maximum-likelihood estimators are suggested to be approximated by use of the
Davidon-Fletcher-Powell method which does not require the second derivatives
of the logarithm of the likelihood function. The model is applied in the
analysis of three years of data on sample dairy farms obtained in the
Australian Dairy Industry Survey, conducted by the Bureau of Agricultural
Economics. In the empirical results obtained, the estimates for the mean
parameter, ~, associated with the distribution of the firm effects, Ui, were
not significantly different from zero, indicating that the half-normal
distribution is likely to be an appropriate specification.
3.
Prediction of Technical Efficiencies
The stochastic frontier production function (i) - (2) is defined as the
logarithmic form of a Cobb-Douglas production function, so that the ratio of
the given firm’s production, exp(Yit), to its corresponding production
frontier, exp(~it~+ Vit), is given by exp(-Ui), which has value between zero
and one. This measure defines the technical efficiency of the i-th sample
firm.
It is not dependent on the level of the factor inputs for the given
firm.
If a stochastic frontier production function is expressed in the form
(i) - (2), such that the output and input variables are in original (rather
than logarithmic) units, then a measure of technical efficiency of the i-th
sample firm is defined by (~i~ -Ui)/qi~, where ~ is the mean of the input
levels for the i-th firm. This measure of technical efficiency is dependent
on the level of inputs for the given sample firm. Further, a measure of the
(mean) technical efficiency for the firms in the industry is given by
1 - [E(Ui)/~8]’4~ where ~~ is the mean of the input levels for the firms in the
industry.
We consider inference about the firm effect, Ui, on the basis of the
random vector ~i £ (EiI,Ei2,...,EiT),
Theorem I Given the specifications of the stochastic frontier production
function (I) - (2), the conditional distribution of U., given
1
E..l = e.~l ~ (@ii’$i2 ......
eiT)’
is defined by the truncation of the normal
distribution with mean,
2-
2
(6)
v
~i* Z (-~ ei + ~IOv/T)/(°2+ ~ /T),
and variance,
(7)
where
e. =
1
Proof:
T
~ eit/Tt=l
Consider the general truncated distribution of U.
fu (U) =
i
1
2/o2
]
exp[- ~(u-p)
½c~{l-~
(2~)
where a is a constant.
,
1
U > a,
(8)
[ (a-~)/o] }
(The special case involved here is a=0. If a=-~,
then the firm effect has normal distribution.)
The joint density function for U.1 and V.~l z (ViI’Vi2’’’’’ViT)’ is
1
1 , , 2~
exp{- ~[ (u-~)2/o2]- ~v .V/Ov~
f
U.,V.
(u,v) =
1 ~i
(2~) (T+I)/2o Vo~ {l-¢[(a-p)/o]}
where v~ ~ (Vil,Vi2 ..... ViT)’ for u > a and -~ < vit < ~, i=l,2,...,T.
,
It is readily verified that the joint density function for U0 and E. is
1
-i
1
1
v2
exp{- ~[(u-p)2/o2]- -~[ (e~ + u{)’ (e+ u~)/o ] }
fui,~i(u,~) =
(2z) (T+I)/2O o~{i-,9[ (a-p)/o] }
where ~ is the (Tx i) vector with all elements equal to one.
The marginal density function for E, is expressed by
~i
fE. (-ei) =
~l
2 2 ½{
(2~)T/2 ovT-I(ov+T~
) l-~[(a-p)/o]}[l-~(zi)]-I
T
2
=
~
where e1.
~ eit/T and Z.l
(a- ~~)/0,, and ~i and o, are as defined by
t=l
(6) and (7), respectively.
The conditional density function for U.,1 given E.~l = e.,
is
~i
1
, 2 2
exp{- ~(u-pi) /o,}
,
½
fui E~i=e_i(u) =
(2~) o, [l-i (zi) ]
:
u > a.
(9)
This is the density function of the normal distribution with mean, N*.,
1
2
and variance, ~,, truncated at the point, a. Clearly if a= 0, as for the
stochastic frontier production function (1) - (2), then the conditional
density function, given by (9), is the truncation of the normal with mean
and variance, given by (6) and (7), respectively. §
The result of (9) gives the conditional distribution of Ui
. , given
E. = V. - U.~,
~ for a general truncated norma! distribution for U.,1 such that
~i
~l
i
the well-known result involving normal random variables, is a special case,
i % N namely, if a =-~, then IEi]
’
2
1
211
+o~/ T
implies
i
*
It is evident that the mean, ~i’
defined by (6), is of order one and
T
converges, as T ÷ ~, to the limit, Lim - ~ eit/T, which is the sample value
T÷~ t=l
2
of the random effect for the i-th firm. Further, the variance, ~,, defined
by (7), is of order T
-1
and converges to zero as T ÷ ~.
It is readily seen that the results of Jondrow et al. (1982) for the
half-normal case (and for cross-sectional data only) are obtained by
substituting ~ = 0 and T= 1 in the expressions for the mean and the variance
of the normal distribution specified in Theorem i.
The mean of the conditional distribution of U., given E. = e., can be
1
~l
~l
shown to be
(i0)
Based on this result, we consider the predictor for U., defined by
1
U. =- M*. + O,{%(-M*.Io,)I[I-I(-M*./O,)]}_
~_
_
~_ _
(ii)
where
M~ =-_ (-o2~i+ ~Ov2/T)/(O2 + Ov21T).
(12)
Note that MS is the random variable which is the counterpart of the parameter
1
in (6).
Theorem 2 The predictor, Ui, defined by (ii), for the random variable, Ui, in
the stochastic frontier production function (i) - (2) is
(a) the minumum squared error predictor of U., given E.;
1
~i
^
(b) an unbiased predictor for U.,
1 in the sense that E(U.)
1 = E(Ui); and
(c) a consistent predictor for U., given that T ÷ ~.
1
(a) Consider any predictor for U1
Proof:
Now
, based on E~i’ say g(Ei).
E[g(~Ei) - Ui]2 : E{[g(.Ei) - E(UiI~Ei)] + [E(UiI_Ei) - Ui]}2
: E[g(Ei)~ - E(UilEi)]2. + E[E(UilEi)~ - U.]~
because E[g(~Ei) - E(UiIE.i)] [E(UilE.i) - Ui] = 0.
Thus
E[g(Ei)~ - U’]2
unless
g(E_i) = E(UiI~Ei) almost surely.
(b) Now
E(M~) : [-o2E(~.)i + UOv/T]/(~2+O /T)
2 2
: [o2E(Ui) + U~v21T]I(~ +~vlT)
because
~.
1 : ~.
1 - U.
1 and E(~i) : 0.
Further, it can be shown that
°v
i (-ulo)
(Ov2+To2) ½ [l-! (-D/~) ]
This result requires the use of the density function of ~., which is
1
expressed by
i exp[- ~(ei+u)2/(
02+O~/T)]
i
(27) ½(o2+o~IT)½[1-i(-ulo)]/[l-!(-U[/o,)]
Using these results we obtain that
E(Ui)
=
^
o2E (Ui)
2
2 + 2~°v/T
+°*
2 2°V
½ 1 -i(-~/o)
V
I
o
+ o /T
2
(Ov+TO)
2
o
+ o /T
2
~ + ~ /T
because [E(Ui) - ~]/~ = i(-~/o)/[l-i(-~/o)] from (4).
E(U.)
= E(U.).
1
1
(c)
The predictor, Ui, is consistent for Ui because as T + ~ the random
variable, M~ defined by (12), converges in probability to Plim - ~ = U
i’
i
i
T÷~
and the random variable, ~(-ME/o.)/[1-O(-ME/O,)] , converges in probability
to zero.
The latter result follows because -M~/o, is expressible as
22
~ ~ Ei- ~Ov/T ] ½
<~ 0V (
v/T) ]
which would be negative for T large enough. §
The mean squared error for the predictor, Ui, cannot be expressed in a
closed form and would require numerica! integration for given values of the
parameters. However, an upper bound on the mean squared error of U1 is
defined by the mean squared error for the alternative consistent predictor
for Ui, namely
Mi,* defined by (12).
After some tedious algebra, the latter
is found to be
2 2
E(M~ Ui)2
~ (Ov/T)[ [
-(~Io)
i-¢(-~Io (o IT) .
[C2+O2/T]v 20 +
In practice, the parameters of the frontier production function are
unknown and so the predictor, Ui, can only be approximated by substituting
(say) the maximum-likelihood estimators for the parameters in (ii). /
Obtaining estimates for the mean squared error for this feasible predictor
would be quite difficult and remains a challenge for future research.
I0.
References
Aigner, D.J., Lovell, C.A.K. and Schmidt, P. (1977), "Formulation and
Estimation of Stochastic Frontier Production Function Models",
Journal of Econometrics, 6, 21-37.
Battese, G.E. and Corra, G.S. (1977), "Estimation of a Production Frontier
Model: With Application to the Pastoral Zone of Eastern Australia",
Australian Journal of Agricultural Economics, 21, 169-179.
Coelli, T.J. and Battese, G.E. (1986), "A Frontier Production Function
for Panel Data: With Application to the Australian Dairy Industry",
Working Papers in Econometrics and Applied Statistics,
NO.24,~
Department of Econometrics, University of New England, Armidale.
Jondrow, J., Lovell, C.A.K., Materov, I.S. and Schmidt, P. (1982), "On the
Estimation of Technical Inefficiency in the Stochastic Frontier
Production Function Model", Journal
of
Econometrics, 19, 233-238.
Kalirajan, K. (1982), "On Measuring Yield Potential of the High Yielding
Varieties Technology at Farm Level", Journal
of
Agricultural Economics,
XXXlll, 227-235.
Kalirajan, K.P. and Flinn, J.C. (1983), "The Measurement of Farm Specific
Technical Efficiency", Pakistan Journal of Applied Economics, II,
167-180.
Lee, L.F. and Tyler, W.G. (1978), "A Stochastic Frontier Production Function
and Average Efficiency: An Empirical Analysis", Journal of Econometrics,
7, 385-390.
Meeusen, W. and van den Broeck, J. 1977), "Efficiency Estimation from
Cobb-Douglas Production Functions with Composed Error", International
Economic Review, 18, 435-444.
Pitt, M.M. and Lee, L.F. (1981), "Measurement and Sources of Technical
Inefficiency in the Indonesian Weaving Industry", Jour~al of
Development Economics, 9, 43-64.
ii.
Schmidt, P. and Sickles, R.C. (1984), "Production Frontiers and Panel
Data", ~/o~n~ ~]" Bus~nes~ ~ E~:~o~om~. S~os, 2, 367-374.
Stevenson, R.E. (1980), "Likelihood Functions for Generalized Stochastic
Frontier Estimation", Jour~4~Z o~ Eoo~ome~{o8~ 13, 57-66.
Waldman, D.M. (1984), "Properties of Technical Efficiency Estimators in
the Stochastic Frontier Model", Jour, n~Z o~ Eoonomet~{o8~ 25, 353-364.
WORKING PAPERS IN ECONOMETRICS AND APPLIED STATISTICS
The Prior Likelihood and Best Linear Unbiased Prediction in Stochastic
Coefficient Linear Models° Lung-Fei Lee and William E. Griffiths,
No. 1 - March 1979.
Stability Conditions in the Use of Fixed Requirement Approach to Manpower
Planning Models. Howard E. Doran and Rozany R. Deen, No. 2 - March
1979.
A Note on A Bayesian Estimator in an Autocorrelated Error Model.
William Griffiths and Dan Dao, No. 3 - April 1979.
On R2-Statistics for the General Linear Model with Nonscalar Covariance
Matrix. G.E. Battese and W.E. Griffiths, No. 4 - April 1979.
Construction of Cost-Of-Living Index Numbers - A Unified Approach.
D.S. Prasada Rao, No. 5 - April 1979.
Omission of the Weighted First Observation in an Autocorrelated Regression
Model: A Discussion of Loss of Efficiency. Howard E. Doran, No. 6 June 1979.
Estimation of Household Expenditure Functions: An Application of a Glass
of Heteroscedastic Regression Models. George E. Battese and
Bruce P. Bonyhady, No. 7 - September 1979.
The Demand for Sawn Timber: An Application of the Diewert Cost Function.
Howard E. Doran and David F. Williams, No. 8 - September 1979.
A New System of Log-Change Index Numbers for Multilateral Comparisons.
D.S. Prasada Rao, No. 9 - October 1980.
A Comparison of Purchasing Power Parity Between the Pound Sterling and
the Australian Dollar - 1979. W.F. Shepherd and D.S. Prasada Rao,
No. I0 - October 1980.
Using Time-Series and Cross-Section Data to Estimate a Production Function
with Positive and Negative Marginal Risks. W.E. Griffiths and
J.R. Anderson, No. ii - December 1980.
A Lack-Of-Fit Test in the Presence of Heteroscedasticity. Howard E. Doran
and Jan Kmenta, No. 12 - April 1981.
~
On the Relative Efficiency of Estimators Which Include the Initial
Observations in the Estimation of Seemingly Unrelated Regressions
with First Order Autoregressive Disturbances. H.E. Doran and
W.E. Griffiths, No. 13 - June 1981.
An Analysis of the Linkages Between the Consumer Price Index and the
Average Minimum Weekly Wage Rate. Pauline Beesley, No. 14 - July 1981.
An Error Components Model for Prediction of County Crop Areas Using Survey
and Satellite Data. George E. Battese and Wayne A. Fuller, No. 15 February 1982.
Networking or Transhipment? Optimisation Alte~tives for Plant Location
Decisions. H.I. Tort and P.A. Cassidy, No. 16 - February 1985.
l)t~a~jmmtic Tests for the Partial Adjustment and Adal,tioe Expeatoat;ion~
Models. H.E. Doran, No. 17 - February 1985.
A ~rther Consideration of Causal Relationships Between Wages and Prices.
J.W.B. Guise and P.A.A. Beesley, No. 18 - February 1985.
~ ~
A Monte Carlo Evaluation of the Power of Some Tests For Heteroscedasticity.
W.E. Griffiths and K. Surekha, No. 19 - August 1985.
A Walrasian Exchange Equilibrium Interpretation of the Geary-Khamis
International Prices. ~.~. ~as~ ~ao, ~o. ~0 - 0~o~e~ 1985.
On.Using O~rbin’s h-Test to Validate the Partial-Adjustment Model.
H.E. Doran~ No. 21 - November 1985.
An Investigation into the Small Sample Properties of Covariance Matrix
and Pre-Test Estimators for the Probit Model. ~i11±a~
R. Carter Hill and Peter J. Pope, No. 22 - November 1985.
A Bayesian Framework for Optimal Input Allocation with an Uncertain
Stochastic Production Function. ~±11±a~ ~. ~±~±~so ~o. ~ February 1986.
A Frontier Production Function for Panel Data: With Application to the~
Australian Dairy Industry. T.J. Coelli and G.E. Battese, No. 24 February 1986.
Identification and Estimation of Elasticities of Substitution for FirmLevel Production Function~ Using Aggregative Data. George ~. ~a~e~e
and Sohail a. Malik, No. 25 - April 1986.
Estimation of Elasticities of Substitution for CES Production Functions
Using Aggregative Data on Selected Manufacturing Industries in Pakistan.
George E. Battese and Sohail J. Malik, No.26 - April 1986.
Estimation of Elasticities of Substitution for CES and VES Production
Functions Using Firm-Level Data for Food-Processing Industries in
Pakistan. George E. Battese and Sohail a. Malik, No.27 - May 1986.
On the Prediction of Technical Efficiencies, Given the Specifications of a
Generalized Frontier Production Function and Panel Data on Sample Firms.
George E. Battese, No.28 - June 1986.