Stochastic Frontier Models

1. William Greene Stern School of Business New York University 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

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/) Fqis 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

11. Inefficiency Estimates

12. Tsionas Fourier Approach to Gamma

13. A 3 Parameter Gamma Model

14. Functional Form • Truncated normal • 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

19. Discrete Outcome Stochastic Frontier

22. 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

26. 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

27. Bayesian Formulation of SF Model Normal – Exponential Model

28. 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).

29. 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.

30. Priors over Parameters

31. Priors for Inefficiencies

32. Posterior

34. Gibbs Sampling: Conditional Posteriors

35. 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

36. Bayesian and Classical Results

37. 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)