1. Efficiency Measurement William Greene Stern School of Business New York University

2. Session 6 Model Extensions

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

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

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

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

7. Normal-Gamma Model z ~ N[- i +  v 2 / u,  v 2 ]. q(r,ε i ) is extremely difficult to compute

8. Normal-Gamma Frontier Model

9. Simulating the Log Likelihood  i = y i - ’x i,  i = - i -  v 2 / u, =  v, and P L = (- i /) F q is a draw from the continuous uniform(0,1) distribution.

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

11. Application to C&G Data ModelMeanStd.Dev.MinimumMaximum Normal.1188.0609.0298.3786 Exponential.0974.0764.0228.5139 Gamma.0820.0799.0149.5294 Descriptive Statistics for JLMS Estimates of E[u|e] Based on Maximum Likelihood Estimates of Stochastic Frontier Models

12. Inefficiency Estimates

13. Tsionas Fourier Approach to Gamma

14. Discrete Outcome Stochastic Frontier

17. 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

21. 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

22. Bayesian Formulation of SF Model Normal – Exponential Model

23. Bayesian Approach v i – u i = y i -  - ’x i. 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).

24. On Estimating Inefficiency One might, ex post, estimate E[u i |data] however, it is more natural in this setting to include (u 1,...,u N ) with , and estimate the conditional means with those of the other parameters. The method is known as data augmentation.

25. Priors over Parameters

26. Priors for Inefficiencies

27. Posterior

29. Gibbs Sampling: Conditional Posteriors

30. 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

31. Bayesian and Classical Results

32. A 3 Parameter Gamma Model

33. 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 u i in this context.  Conclusions Not much change in existing results Extensions to new models (e.g., 3 parameter gamma)