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Efficiency and Productivity Measurement: Stochastic Frontier Analysis. D.S. Prasada Rao School of Economics The University of Queensland, Australia. Stochastic Frontier Analysis. It is a parametric technique that uses standard production function methodology.
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D.S. Prasada Rao
School of Economics
The University of Queensland, Australia
lnqi = 0 + 1lnx1i + 2lnx2i + vi - ui
lnqi = 0 + 1lnx1i + 2lnx2i + 0.511(lnx1i)2 + 0.522(lnx2i)2 + 12lnx1ilnx2i + vi - ui
Production elasticity for j-th input is: Ej = j
Scale elasticity is: = E1+E2
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.
Since 32 5% table value = 7.81 => do not reject H0
xi’s are in logs and include a constant
Estimation of Parameters:
where is the OLS residual for i-th firm.
OLS: qi = 0 + 1xi + vi
Deterministic : qi = 0 + 1xi - ui
SFA: qi = 0 + 1xi + vi - ui
vi = “noise” error term - symmetric (eg. normal distribution)
ui = “inefficiency error term” - non-negative (eg. half-normal distribution)
In general, we write the stochastic frontier model with several inputs and a general functional form (which is linear in parameters) as
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.
A more general specification:
This forms the basis for the inefficiency effects model where
Aigner, Lovell and Schmidt
Battese and Corra
andEstimation of SF Models
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 zero
log likelihood function = -0.74409920E+02
Predicting Firm Level Efficiencies:
Once the SF model is estimated using MLE method, we compute the following:
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.
technical efficiency estimates :
mean efficiency = 0.72941885
e.g., Is there significant technical inefficiency?
H0: =0 versus H1: >0
t-ratio = (parameter estimate) / (standard error)
[note that the above hypothesis is one-sided - therefore must use Kodde and Palm critical values (not chi-square) for LR test
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)
The LR statistic has mixed Chi-square distribution
SEi = exp[(1-i)2/2]
where i is the scale elasticity of the i-th firm and
We note the following points with respect to SFA models
Some Special cases:
Sign of is important. As t goes to T, uitgoes to ui.
In FRONTIER Program, this is under Error Components Model.
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.
In this case firm-level technical efficiency levels predicted will vary with traditional inputs and environmental variables.
where is a vector of parameters to be estimated. In the FRONTIER program, this is the TEEFFECTS model
DEA and SFA Methods