1. Stochastic Frontier Models
0 Introduction
1 Efficiency Measurement
2 Frontier Functions
3 Stochastic Frontiers
4 Production and Cost
5 Heterogeneity
6 Model Extensions
7 Panel Data
8 Applications
William Greene
Stern School of Business
New York University

2. 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

3. Functional Forms
Normal-half normal and normal-exponential: Restrictive functional forms for the inefficiency distribution

4. Normal-Truncated Normal
More flexible. Inconvenient, sometimes ill behaved log-likelihood function.
MU=-.5
MU=0
Exponential
MU=+.5

5. Normal-Gamma
Very flexible model. VERY difficult log likelihood function.
Bayesians love it. Conjugate functional forms for other model parts

6. Normal-Gamma Model z ~ N[-i + v2/u, v2].
q(r,εi) is extremely difficult to compute

7. Normal-Gamma Frontier Model

8. 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.

9. Application to C&G Data
This is the standard data set for developing and testing Exponential, Gamma, and Bayesian estimators.

10. Application to C&G Data
Descriptive Statistics for JLMS Estimates of E[u|e] 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

11. Inefficiency Estimates

12. Tsionas Fourier Approach to Gamma

13. A 3 Parameter Gamma Model

14. Functional Form Truncated normal Rayleigh model
Has the advantage of a place to put the z’s
Strong functional disadvantage – discontinuity. Difficult log likelihood to maximize
Rayleigh model
Parameter affects both mean and variance
Convenient model for heterogeneity
Much simpler to manipulate than gamma.

15. Stochastic Frontiers with a Rayleigh Distribution
Gholamreza Hajargasht, Department of Economics, University of Melbourne, 2013

17. Exponential
Gamma
Rayleigh
Half Normal

18. Rayleigh vs. Half Normal

20. Spatial Autoregression in a Linear Model

26. Discrete Outcome Stochastic Frontier

29. Chanchala Ganjay Gadge
CONTRIBUTIONS TO THE INFERENCE ON STOCHASTIC FRONTIER MODELS
DEPARTMENT OF STATISTICS AND CENTER FOR ADVANCED STUDIES,
UNIVERSITY OF PUNE
PUNE , INDIA

33. 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

34. Bayesian Formulation of SF Model
Normal – Exponential Model

35. 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).

36. On Estimating Inefficiency
One might, ex post, estimate E[ui|data] 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.

37. Priors over Parameters

38. Priors for Inefficiencies

39. Posterior

41. Gibbs Sampling: Conditional Posteriors

42. Bayesian Normal-Gamma Model
Tsionas (2002)
Erlang form – Integer P
“Random parameters”
Applied to C&G (Cross Section)
Average efficiency 0.999
River Huang (2004)
Fully general
Applied (as usual) to C&G

43. Bayesian and Classical Results

44. 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)