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

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

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

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

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

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

Normal-Gamma Frontier Model

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.

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

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

Inefficiency Estimates

Tsionas Fourier Approach to Gamma

A 3 Parameter Gamma Model

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.

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

Exponential Gamma Rayleigh Half Normal

Rayleigh vs. Half Normal

Spatial Autoregression in a Linear Model

Discrete Outcome Stochastic Frontier

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

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

Bayesian Formulation of SF Model Normal – Exponential Model

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).

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.

Priors over Parameters

Priors for Inefficiencies

Posterior

Gibbs Sampling: Conditional Posteriors

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

Bayesian and Classical Results

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)