Efficiency Measurement

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Efficiency Measurement

• William Greene
• Stern School of Business
• New York University

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Session 6

• Model Extensions

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Model Extensions

• Simulation Based Estimators
• Normal-Gamma Frontier Model
• Bayesian Estimation of Stochastic Frontiers
• A Discrete Outcomes Frontier
• Similar Model Structures
• Similar Estimation Methodologies
• Similar Results

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Functional Forms

• Normal-half normal and normal-exponential
  Restrictive functional forms for the inefficiency
  distribution

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Normal-Truncated Normal
More flexible. Inconvenient, sometimes ill
behaved log-likelihood function.
MU-.5
MU0
MU.5
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Normal-Gamma
Very flexible model. VERY difficult log
likelihood function. Bayesians love it.
Conjugate functional forms for other model parts
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Normal-Gamma Model
z N-?i ?v2/?u, ?v2.
q(r,ei) is extremely difficult to compute
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Normal-Gamma Frontier Model
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Simulating the Log Likelihood

• ?i yi - ?xi,
• ?i -?i - ?v2/?u,
• ?v, and
• PL ?(-?i/?)
• Fq is a draw from the continuous uniform(0,1)
  distribution.

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Application to CG Data
This is the standard data set for developing and
testing Exponential, Gamma, and Bayesian
estimators.
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Application to CG Data
Descriptive Statistics for JLMS Estimates of
Eue Based on Maximum Likelihood Estimates of
Stochastic Frontier Models
Model Mean Std.Dev. Minimum Maximum
Normal .1188 .0609 .0298 .3786
Exponential .0974 .0764 .0228 .5139
Gamma .0820 .0799 .0149 .5294
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Inefficiency Estimates
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Tsionas Fourier Approach to Gamma
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Discrete Outcome Stochastic Frontier
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Chanchala Ganjay Gadge

• CONTRIBUTIONS TO THE INFERENCE ON STOCHASTIC
  FRONTIER MODELS
• DEPARTMENT OF STATISTICS AND CENTER FOR ADVANCED
  STUDIES,
• UNIVERSITY OF PUNE
• PUNE-411007, INDIA

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Bayesian Estimation

• Short history first developed post 1995
• Range of applications
• Largely replicated existing classical methods
• Recent applications have extended received
  approaches
• Common features of the applications

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Bayesian Formulation of SF Model
Normal Exponential Model
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Bayesian Approach
vi ui yi - ? - ?xi. Estimation proceeds (in
principle) by specifying priors over ?
(?,?,?v,?u), then deriving inferences from the
joint posterior p(?data). In general, the joint
posterior for this model cannot be derived in
closed form, so direct analysis is not feasible.
Using Gibbs sampling, and known conditional
posteriors, it is possible use Markov Chain Monte
Carlo (MCMC) methods to sample from the marginal
posteriors and use that device to learn about the
parameters and inefficiencies. In particular,
for the model parameters, we are interested in
estimating E?data, Var?data and, perhaps
even more fully characterizing the density
f(?data).
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On Estimating Inefficiency

• One might, ex post, estimate Euidata
  however, it is more natural in this setting to
  include (u1,...,uN) with ?, and estimate the
  conditional means with those of the other
  parameters. The method is known as data
  augmentation.

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Priors over Parameters
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Priors for Inefficiencies
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Posterior
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Gibbs Sampling Conditional Posteriors
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Bayesian Normal-Gamma Model

• Tsionas (2002)
• Erlang form Integer P
• Random parameters
• Applied to CG (Cross Section)
• Average efficiency 0.999
• River Huang (2004)
• Fully general
• Applied (as usual) to CG

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Bayesian and Classical Results
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A 3 Parameter Gamma Model
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Methodological Comparison

• Bayesian vs. Classical
• Interpretation
• Practical results Bernstein von Mises Theorem
  in the presence of diffuse priors
• Kim and Schmidt comparison (JPA, 2000)
• Important difference tight priors over ui in
  this context.
• Conclusions
• Not much change in existing results
• Extensions to new models (e.g., 3 parameter gamma)