Efficiency and Productivity Measurement:
Stochastic Frontier Analysis
D.S. Prasada Rao
School of Economics
The University of Queensland, Australia
1
Stochastic Frontier Analysis
• It is a parametric technique that uses standard
production function methodology.
• The approach explicitly recognises that production
function represents technically maximum feasible
output level for a given level of output.
• The Stochastic Frontier Analysis (SFA) technique
may be used in modelling functional relationships
where you have theoretical bounds:
– Estimation of cost functions and the study of cost efficiency
– Estimation of revenue functions and revenue efficiency
– This technique is also used in the estimation of multi-output and
multi-input distance functions
– Potential for applications in other disciplines
2
Some history….
• Much of the work on stochastic frontiers began in
70’s. Major contributions from Aigner, Schmidt,
Lovell, Battese and Coelli and Kumbhakar.
• Ordinary least squares (OLS) regression
production functions:
– fit a function through the centre of the data
– assumes all firms are efficient
• Deterministic production frontiers:
– fit a frontier function over the data
– assumes there is no data noise
• SFA production frontiers are a “mix” of these two.
3
Production Function
• It is a relationship between output and a set of
input quantities.
– We use this when we have a single output
– In case of multiple outputs:
• people often use revenue (adjusted for price differences) as
an output measure
• It is possible to use multi-output distance functions to study
production technology.
• The functional relationship is usually written in
the form:
q  f ( x1 , x2 ,..., xN )  v
4
Production Function Specification
• A number of different functional forms are used in the
literature to model production functions:
–
–
–
–
Cobb-Douglas (linear logs of outputs and inputs)
Quadratic (in inputs)
Normalised quadratic
Translog function
N
1 N N
ln q  0   n ln xn   nm ln xn ln xm  u
2 n1 m1
n 1
– Translog function is very commonly used – it is a generalisation of the
Cobb-Douglas function
– It is a flexible functional form providing a second order approximation
– Cobb-Douglas and Translog functions are linear in parameters and can
be estimated using least squares methods.
– It is possible to impose restrictions on the parameters (homogeneity
5
conditions)
Cobb-Douglas Functional form
• Linear in logs
• Advantages:
– easy to estimate and interpret
– requires estimation of few parameters: K+3
• Disadvantages:
– simplistic - assumes all firms have same
production elasticities and that subsitution
elasticities equal 1
6
Translog Functional form
• Quadratic in logs
• Advantages:
– flexible functional form - less restrictions on
production elasticities and subsitution
elasticities
• Disadvantages:
– more difficult to interpret
– requires estimation of many parameters:
K+3+K(K+1)/2
– can suffer from curvature violations
7
Functional forms
Cobb-Douglas:
lnqi = 0 + 1lnx1i + 2lnx2i + vi - ui
Translog:
lnqi = 0 + 1lnx1i + 2lnx2i + 0.511(lnx1i)2
+ 0.522(lnx2i)2 + 12lnx1ilnx2i + vi - ui
8
Interpretation of estimated parameters
Cobb-Douglas:
Production elasticity for j-th input is: Ej = j
Scale elasticity is:  = E1+E2
Translog:
Production elasticity for i-th firm and j-th input is:
Eji = j+ j1lnx1i+ j2lnx2i
Scale elasticity for i-th firm is: i = E1i+E2i
Note: If we use transformed data where inputs are
measured relative to their means, then Translog
elasticities at means would simply be i.
9
Test for Cobb-Douglas versus Translog
• Using sample data file which comes with the
FRONTIER program
• H0: 11=22=12=0, H1: H0 false
• Compute -2[LLFo-LLF1] which is distributed as
Chi-square (r) under Ho.
• For example, if:
LLF1=-14.43, LLF0=-17.03
LR=-2[-17.03-(-14.43)]=5.20
Since 32 5% table value = 7.81 => do not reject H0
10
Deterministic Frontier models
• In this model all the errors are assumed to be
due to technical inefficiency – no account is
taken of noise.
• Consider the following simple specification:
ln yi  xi   ui for firmsi  1,2,...N
xi’s are in logs and include a constant
• ui is a non-negative random variable.
Therefore ln yi  xi .
• Given the frontier nature of the model we can
measure technical efficiency using:
11
Deterministic Frontier models
We have
yi
exp(xi   ui )
TEi 

 exp(ui )
exp(xi  )
exp(xi  )
• Frontier is deterministic since it is given by
exp(xi) which is non-random.
Estimation of Parameters:
• Linear programming approach
Min  ui   ln yi  xi  
i
i
subject toui  0.
12
Deterministic Frontier models
• Aigner and Chu suggested a Quadratic
prrogramming approach
Min  ui    ln yi  xi  
2
i
2
i
subject to ui 2  0.
• Afriat (1972) suggested the use of a Gamma
distribution for ui and the use of maximum
likelihood estimation.
• Corrected ordinary least squares [COLS]
– If we apply OLS, intercept estimate is biased
downwards, all other parameters are unbiased.
13
Deterministic Frontier models
• So COLS suggests that the OLS estimator from
OLS be corrected.
• If we do not wish to make use of any probability
distribution for yi then
ˆo (COLS)  ˆo (OLS)  maximumi uˆi : i  1,2,...,N 
where uˆ i is the OLS residual for i-th firm.
• If we assume that ui is distributed as Gamma
then
ˆo (COLS )  ˆo (OLS )  ˆ 2[OLS ]
• It is a bit more complicated if ui follows halfnormal distribution.
14
Production functions/frontiers
Deterministic
q
×
×
×
×
×
×
×
×
×
SFA
OLS
×
x
15
Production functions/frontiers
OLS:
qi = 0 + 1xi + vi
Deterministic :
qi = 0 + 1xi - ui
SFA:
qi = 0 + 1xi + vi - ui
where
vi = “noise” error term - symmetric
(eg. normal distribution)
ui = “inefficiency error term” - non-negative
(eg. half-normal distribution)
16
Stochastic Frontier: Model Specification
• We start with the general production function as before and add a
new term that represents technical inefficiency.
− This means that actual output is less than what is postulated
by the production function specified before.
− We achieve this my subtracting u from the production
function
− Then we have
q  f ( x1 , x2 ,..., xN )  v  u
− In the Cobb-Douglas production function with one input we
can write the stochastic frontier function for the i-th firm as:
ln qi  0  1 ln xi  vi  ui
qi  exp(0  1 ln xi  vi  ui )
17
Stochastic frontiers
deterministic frontier
qi = exp(β0 + β1 ln xi)
yi
q*A ? exp(β0 + β1ln xA + vA)
q*B ? exp(β0 + β1ln xB + vB)
noise effect
noise effect
inefficiency effect
qB ? exp(β0 + β1ln xB + vB – uB)
inefficiency
effect
qA ? exp(β0 + β1ln xA + vA – uA)
xA
xB
18
Stochastic Frontier: Model Specification
qi  exp( 0  1 ln xi )  exp(vi )  exp(ui )
noise
inefficiency
deterministic
component
In general, we write the stochastic frontier model with several inputs and
a general functional form (which is linear in parameters) as
ln qi  xi β  vi  ui
• We stipulate that ui is a non-negative random variable
• By construction the inefficiency term is always between 0 and 1.
• This means that if a firm is inefficient, then it produces less than what is
expected from the inputs used by the firm at the given technology.
• We can define technical efficiency as the ratio of “observed” or “realised
output” to the stochastic frontier output
qi
exp( xi β  vi  ui )
TEi 

 exp(ui )
exp( xi β  vi )
exp(xi β  vi )
19
Stochastic Specifications
• The SF model is specified as:
ln qi  xi β  vi  ui

The following are the assumptions made on the distributions of v
and u.

Standard assumptions of zero mean, homoskedasticity and
independence is assumed for elements of vi.

We assume that ui’s are identically and independently
distributed non-negative random variables.

Further we assume that vi and ui are independently distributed.

The distributional assumptions are crucial to the estimation of
the parameters. Standard distributions used are:
− Half-normal (truncated at zero)
ui
iidN  (0,  u2 ).
− Exponential
− Gamma distribution
20
Truncated normal distribution for u
var = 1
var= 4
var = 9
-0.5
0.0
0.5
1.0
Distribution of u: ui
1.5
2.0
2.5
3.0
3.5
iidN  (0,  u2 ).
We note that: As u is truncated from a normal distribution
with mean equal to 0, E(u) is “towards” zero and therefore
technical efficiency tends to be high just by model
construction.
21
Truncated normal with non-zero means
2.5
2
mu = -2
1.5
f(x)
mu = -1
mu = 0
1
mu = 1
0.5
mu = 2
0
0
1
2
3
4
5
x
A more general specification: u N (  , u )
This forms the basis for the inefficiency effects model where

2
u N  ( i ,  u2 )
i   0    k Z ki
k
22
Estimation of SF Models
• Parameters to be estimated in a standard SF model
are: β, v2 and  u2
• Likelihood methods are used in estimating the
unknown parameters. Coelli (1995)’s Montecarlo
study shows that in large samples MLE is better than
COLS.
• Usually variance parameters are reparametrized in
the following forms.
 2   v2   u2
and
 2   u2  v2  0.
2
2
 2 and    u /  .
Aigner, Lovell and Schmidt
Battese and Corra
• Testing for the presence of technical inefficiency
depends upon the parametrization used.
23
Estimation of SFA Models
• In the case of translog model, it is a good idea to
transform the data – divide each observation by its
mean
− Then the coefficients of ln Xi can be interpreted
as elasticities.
• Most standard packages such as SHAZAM and
LIMDEP.
• FRONTIER by Coelli is a specialised program for
purposes of estimating SF models.
─ Available for free downloads from CEPA
website: www.uq.edu.au/economics/cepa
24
FRONTIER Instruction File
Table The FRONTIER Instruction File
1
chap9.txt
chap9_2.out
1
y
344
1
344
10
n
n
n
1=ERROR COMPONENTS MODEL, 2=TE EFFECTS MODEL
DATA FILE NAME
OUTPUT FILE NAME
1=PRODUCTION FUNCTION, 2=COST FUNCTION
LOGGED DEPENDENT VARIABLE (Y/N)
NUMBER OF CROSS-SECTIONS
NUMBER OF TIME PERIODS
NUMBER OF OBSERVATIONS IN TOTAL
NUMBER OF REGRESSOR VARIABLES (Xs)
MU (Y/N) [OR DELTA0 (Y/N) IF USING TE EFFECTS MODEL]
ETA (Y/N) [OR NUMBER OF TE EFFECTS REGRESSORS (Zs)]
STARTING VALUES (Y/N)
• Here MU refers to inefficiency effects models and ETA refers to
time-varying inefficiency effects (we will come to this shortly)
• The program uses the ratio of variances as the transformation
• It allows for the use of single cross-sections as well as panel data
sets
25
FRONTIER output
the final mle estimates are :
coefficient
beta
beta
beta
beta
beta
beta
0
1
2
3
4
5
standard-error
t-ratio
0.27436347E+00
0.15110945E-01
0.53138167E+00
0.23089543E+00
0.20327381E+00
-0.47586195E+00
0.39600416E-01 0.69282978E+01
0.67544802E-02 0.22371736E+01
0.79213877E-01 0.67081892E+01
0.74764329E-01 0.30883101E+01
0.44785423E-01 0.45388387E+01
0.20221150E+00 -0.23532883E+01
beta 6
0.60884085E+00
beta 7
0.61740289E-01
beta 8
-0.56447322E+00
beta 9
-0.13705357E+00
beta10
-0.72189747E-02
sigma-squared 0.22170997E+00
0.16599693E+00 0.36677839E+01
0.13839069E+00 0.44613038E+00
0.26523510E+00 -0.21281996E+01
0.14081595E+00 -0.97328160E+00
0.92425705E-01 -0.78105703E-01
0.24943636E-01 0.88884383E+01
gamma
0.88355629E+00 0.36275231E-01
mu is restricted to be zero
eta is restricted to be zero
log likelihood function = -0.74409920E+02
0.24357013E+02
26
SF Models - continued
Predicting Firm Level Efficiencies:
Once the SF model is estimated using MLE method, we
compute the following:
ui*  (ln qi  xi β) u2 /  2
and
 *2   v2 u2 /  2 .
We use estimates of unknown parameters in these equations
and compute the best predictor of technical efficiency for
each firm i :
2
*
  ui*







u
*
i
TEˆi  E exp(ui ) qi       *      exp   ui*  .

  * 
 2

  *
We use standard normal density and distribution functions to
evaluate technical efficiency.
27
SF Models - continued
Industry efficiency:
• Industry efficiency can be computed as the average of
technical efficiencies of the firms in the sample
• Industry efficiency can be seen as the expected value of a
randomly selected firm from the industry. Then we have
 u2 
ˆ
TE  E exp(ui )  2   u  exp   .
 2 
Confidence intervals for technical efficiency scores (for the
firms and the industry as a whole) can also be computed.
•
We note that there are no firms with a TE score of 1 as in
the case of DEA.
•
No concept of peers exists in the case of SFA.
28
FRONTIER output
technical efficiency estimates :
firm
eff.-est.
1
0.77532384
2
0.72892751
3
0.77332991
341
0.76900626
342
0.92610064
343
0.81931012
344
0.89042718
mean efficiency = 0.72941885
• Mean efficiency can be interpreted as the “industry
efficiency”.
29
Tests of hypotheses
e.g., Is there significant technical inefficiency?
H0: =0 versus H1: >0
Test options:
• t-test
t-ratio = (parameter estimate) / (standard error)
• Likelihood ratio (LR) test
[note that the above hypothesis is one-sided therefore must use Kodde and Palm critical
values (not chi-square) for LR test
• LR test “safer”
30
Likelihood ratio (LR) tests
Steps:
1) Estimate unrestricted model (LLF1)
2) Estimate restricted model (LLF0)
(eg. set =0)
3) Calculate LR=-2(LLF0-LLF1)
4) Reject H0 if LR>R2 table value,
where R = number of restrictions
(Note: Kodde and Palm tables must be
used if test is one-sided)
31
• Example - estimate translog production
function using sample data file which
comes with the FRONTIER program 344 firms
• t-ratio for  = 24.36, and N(0,1) critical
value at 5% = 1.645 => reject H0
• Or the LR statistic = 28.874, and Kodde
and Palm critical value at 5% = 2.71 =>
reject H0
The LR statistic has mixed Chi-square
distribution
32
Distributional assumptions –
the truncated normal distribution
• N(,2) truncated at zero
• More general patterns
• Can test hypothesis that =0 using t-test
or LR test
• The restriction =0 produces the halfnormal distribution: |N(0,2)|
33
Scale efficiency
• For a Translog Production Function (Ray,
1998)
• An output-orientated scale efficiency measure
is:
SEi = exp[(1-i)2/2]
where i is the scale elasticity of the i-th firm
and
K K
    jk
j1 k 1
• If the frontier is concave in inputs then <0.
Then SE is in the range 0 to 1.
34
Stochastic Frontier Models: Some
Comments
We note the following points with respect to SFA
models
• It is important to check the regularity conditions
associated with the estimated functions – local and
global properties
– This may require the use of Bayesian approach to impose
inequality restrictions required to impose convexity and
concavity conditions.
• We need to estimate distance functions directly in the
case of multi-output and multi-input production
functions.
• It is possible to estimate scale efficiency in the case of
translog and Cobb-Douglas specifications
35
Panel data models
•
•
•
•
•
Data on N firms over T time periods
Investigate technical efficiency change (TEC)
Investigate technical change (TC)
More data = better quality estimates
Less chance of a one-off event (eg. climatic) influencing
results
• Can use standard panel data models
– no need to make distributional assumption
– but must assume TE fixed over time
• The model: i=1,2,…N (cross-section of firms); t=1,2…T
(time points)
ln yit  xit   vit  uit ; vit  N (0, v2 );uit  N  (0, u2 )
36
Panel data models
Some Special cases:
1.
2.
Firm specific effects are time invariant: uit = ui .
Time varying effects: Kumbhakar (1990)


1
uit  1  exp( bt  ct ) ui
3.
2
Time-varying effects with convergence – Battese and Coelli
(1992)
uit  exp (t  T ui
Sign of  is important. As t goes to T, uit goes to ui.
In FRONTIER Program, this is under Error Components
Model.
37
Time profiles of efficiencies
1
0
1
2
3
4
5
6
7
K90 ( = .5,  = -.04)
BC92 (  = -.01)
8
9
10
11
12 13
K90 ( =
14
15
 = -.02)
BC92 ( = .1)
Note: These are all smooth functions of trends of technical efficiency

over
time. These trends are also independent of any other data on the
firms. There is scope for further work in this area.
38
Accounting for Production Environment
• Technical efficiency is influenced by exogenous factors that
characterise the environment in which production takes place
– Government regulation, ownership, education level of the farmer, etc.
• Non-stochastic Environmental Variables
In this case firm-level technical efficiency levels predicted will vary
with traditional inputs and environmental variables.
ln qi  xi   zi  vi  ui
• Inefficiency effects model (Battese, Coelli 1995)
ln yit  xit   vit  uit ; uit
N ( zit , u2 )
where  is a vector of parameters to be estimated. In the FRONTIER
program, this is the TEEFFECTS model
39
Current research
• There is scope to conduct a lot of research in this area.
Some areas where work is being done and still be done
are:
• Modelling movements in inefficiency over time –
incorporating exogenous factors driving inefficiency
effects
• Possibility of covariance between the random disturbance
and the distribution of the inefficiency term
• Endogeniety due to possible effects of inefficiency and
technical change on the choice of input variables
• Modelling risk into efficiency estimation and
interpretation
40
Current research
• We have seen how technical efficiency can be computed,
but it is difficult to compute standard errors.
• Peter Schmidt and his colleagues have been working on a
number of related topics here.
– Bootstrap estimators and confidence intervals for
efficiency levels in SF models with pantel data
– Testing whether technical inefficiency depends on firm
characteristics
– On the distribution of inefficiency effects under
different assumptions
• Bayesian estimation of stochastic frontier models
– Posterior distribution of technical efficiencies
– Estimation of distance functions
41
Application to Residential Aged Care
Facilities
• Residential aged care is a multi-billion dollar industry in
Australia
• Considered even more important in view of an ageing
population.
• Aged care facilities are funded by the Commonwealth
government and are run by local government,
community/religious and private organisations.
• Efficiency of residential aged care facilities is considered
quite important in view of reducing costs.
• CEPA conducted a study for the Commonwealth
Department of Health and aged Care.
• Data was collected by the Department from the
residential aged care facilities.
42
Application to Residential Aged Care
Facilities
Data:
• 912 Aged Care Facilities
• 30% response rate – response bias
• Data validation
• Actual observations used: 787
Methods:
DEA and SFA Methods
Peeled DEA
43
Application to Residential Aged Care
Facilities
Preferred model:
Output variables:
– High care weighted bed days
– Low care weighted bed days
Input variables:
– Floor area in square meters
– Labour costs
– Other costs
44
General findings
• Average technical efficiency was calculated to be 0.83. 17%
cost savings could be achieved if all ACFs operated on the
frontier.
• Results consistent between SFA and DEA models
• Found variation across ACFs in different states (lowest 0.79
in Victoria and 0.87 in NSW/ACT).
• Privately run ACFs has higher mean TE of 0.89.
• Average scale efficiency of 0.93 was found.
• An average size of 30 to 60 beds was found to be near optimal
scale.
• A second stage Tobit model was run to see the factors driving
inefficiency.
• Potential cost savings was calculated to be $316 m for the
number of firms in the sample –greater savings for the
industry!!
• Economies of Scope were examined using new methodology
developed – but no significant economies were found.
45