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Efficiency and Productivity Measurement:Stochastic Frontier Analysis

D.S. Prasada RaoSchool of Economics

The University of Queensland, Australia

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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

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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.

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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:

1 2( , ,..., )Nq f x x x v

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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

– 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

conditions)

01 1 1

1ln ln ln ln

2

N N N

n n nm n mn n m

q x x x u

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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

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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

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Functional forms

Cobb-Douglas:

lnqi = 0 + 1lnx1i + 2lnx2i + vi - ui

Translog:

lnqi = 0 + 1lnx1i + 2lnx2i + 0.511(lnx1i)2 +

0.522(lnx2i)2 + 12lnx1ilnx2i + vi - ui

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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.

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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

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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:

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:

Niuxy iii ,...2,1 firmsfor ln

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Deterministic Frontier models

We have

• Frontier is deterministic since it is given by exp(xi) which is non-random.

Estimation of Parameters:• Linear programming approach

)exp()exp(

)exp(

)exp( ii

ii

i

ii u

x

ux

x

yTE

.0 subject to

lnMin

i

i iiii

u

xyu

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Deterministic Frontier models

• Aigner and Chu suggested a Quadratic prrogramming approach

• 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.

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2

Min ln

subject to 0.

i i ii i

i

u y x

u

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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

where is the OLS residual for i-th firm.

• If we assume that ui is distributed as Gamma then

• It is a bit more complicated if ui follows half-normal distribution.

NiuimumOLSCOLS iioo ,...,2,1:ˆmax)(ˆ)(ˆ iu

2ˆ ˆ ˆ( ) ( ) [ ]o oCOLS OLS OLS

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Production functions/frontiers

OLS

x

×

× ×

×

××

×

×

q

×

Deterministic

SFA

×

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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)

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Stochastic Frontier: Model Specification

1 2( , ,..., )Nq f x x x v u

• 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

− In the Cobb-Douglas production function with one input we can write the stochastic frontier function for the i-th firm as:

0 1ln lni i i iq x v u

0 1exp( ln )i i i iq x v u

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Stochastic frontiers

xB xA

yi

qB* ? exp(β0 + β1ln xB + vB)

deterministic frontier qi = exp(β0 + β1 ln xi)

noise effect qA

* ? exp(β0 + β1ln xA + vA)

qB ? exp(β0 + β1ln xB + vB – uB)

qA ? exp(β0 + β1ln xA + vA – uA)

noise effect

inefficiency effect

inefficiency effect

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Stochastic Frontier: Model Specification

0 1exp( ln ) exp( ) exp( )i i i iq x v u

deterministic component

noise inefficiency

• 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

In general, we write the stochastic frontier model with several inputs and a general functional form (which is linear in parameters) as

ln i i i iq v u x β

exp( )exp( )

exp( ) exp( )i i i i

i ii i i i

q v uTE u

v v

x β

x β x β

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Stochastic Specifications• The SF model is specified as:

ln i i i iq v u x β 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)− Exponential− Gamma distribution

2(0, ).i uu iidN

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Truncated normal distribution for u

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

var = 1

var= 4

var = 9

2(0, ).i uu iidN Distribution of u:

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.

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Truncated normal with non-zero means

A more general specification: 2( , )uu N

This forms the basis for the inefficiency effects model where2

0

( , )i u

i k kik

u N

Z

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5

x

f(x)

mu = -2

mu = -1

mu = 0

mu = 1

mu = 2

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Estimation of SF Models

• Parameters to be estimated in a standard SF model are:

• 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.

• Testing for the presence of technical inefficiency depends upon the parametrization used.

2 2, andv u β

2 2 2v u

and

2 2 2 0.u v Aigner, Lovell and Schmidt

2 2 2/ .u

and

Battese and Corra

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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

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FRONTIER Instruction File

Table The FRONTIER Instruction File

1 1=ERROR COMPONENTS MODEL, 2=TE EFFECTS MODEL chap9.txt DATA FILE NAME chap9_2.out OUTPUT FILE NAME 1 1=PRODUCTION FUNCTION, 2=COST FUNCTION y LOGGED DEPENDENT VARIABLE (Y/N) 344 NUMBER OF CROSS-SECTIONS 1 NUMBER OF TIME PERIODS 344 NUMBER OF OBSERVATIONS IN TOTAL 10 NUMBER OF REGRESSOR VARIABLES (Xs) n MU (Y/N) [OR DELTA0 (Y/N) IF USING TE EFFECTS MODEL] n ETA (Y/N) [OR NUMBER OF TE EFFECTS REGRESSORS (Zs)] n 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

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FRONTIER output

the final mle estimates are : coefficient standard-error t-ratio

beta 0 0.27436347E+00 0.39600416E-01 0.69282978E+01 beta 1 0.15110945E-01 0.67544802E-02 0.22371736E+01 beta 2 0.53138167E+00 0.79213877E-01 0.67081892E+01 beta 3 0.23089543E+00 0.74764329E-01 0.30883101E+01 beta 4 0.20327381E+00 0.44785423E-01 0.45388387E+01 beta 5 -0.47586195E+00 0.20221150E+00 -0.23532883E+01

beta 6 0.60884085E+00 0.16599693E+00 0.36677839E+01 beta 7 0.61740289E-01 0.13839069E+00 0.44613038E+00 beta 8 -0.56447322E+00 0.26523510E+00 -0.21281996E+01 beta 9 -0.13705357E+00 0.14081595E+00 -0.97328160E+00 beta10 -0.72189747E-02 0.92425705E-01 -0.78105703E-01 sigma-squared 0.22170997E+00 0.24943636E-01 0.88884383E+01

gamma 0.88355629E+00 0.36275231E-01 0.24357013E+02 mu is restricted to be zero eta is restricted to be zerolog likelihood function = -0.74409920E+02

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SF Models - continued

Predicting Firm Level Efficiencies:

Once the SF model is estimated using MLE method, we compute the following:

* 2 2(ln ) /i i i uu q x β and 2 2 2 2* / .v u

We use estimates of unknown parameters in these equations and compute the best predictor of technical efficiency for each firm i :

We use standard normal density and distribution functions to evaluate technical efficiency.

2* *

***

* *

ˆ exp( ) exp .2

i ii i i i

u uTE E u q u

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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

2

ˆ exp( ) 2 exp .2

ui uTE E u

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.

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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”.

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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”

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Likelihood ratio (LR) testsSteps:

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)

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• 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

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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 half-normal distribution: |N(0,2)|

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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

• If the frontier is concave in inputs then <0. Then SE is in the range 0 to 1.

K

1j

K

1kjk

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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

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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)

),0();,0(;ln 22uitvititititit NuNvuvxy

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Panel data models

Some Special cases:1. Firm specific effects are time invariant: uit = ui .

2. Time varying effects: Kumbhakar (1990)

3. Time-varying effects with convergence – Battese and Coelli (1992)

Sign of is important. As t goes to T, uit goes to ui.

In FRONTIER Program, this is under Error Components Model.

iit uctbtu12 )exp(1

iit uTtu (exp

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Time profiles of efficiencies

0

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

K90 ( = .5, = -.04) K90 ( = = -.02)

BC92 ( = -.01) 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.

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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.

• Inefficiency effects model (Battese, Coelli 1995)

where is a vector of parameters to be estimated. In the FRONTIER program, this is the TEEFFECTS model

iiiii uvzxq ln

2ln ; ( , )it it it it it it uy x v u u N z

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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

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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

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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.

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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 MethodsPeeled DEA

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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

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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.