William Greene

Stern School of Business

New York University

Stochastic Frontier and
Efficiency Estimation


Professor William. Greene
 e-mail: wgreene@stern.nyu.edu

Home Page: http://people.stern.nyu.edu/wgreene


This course will present the methodology of econometric estimation of economic efficiency. We will examine the stochastic frontier model as an econometric extension of the classical microeconomic theory of production and cost at the individual producer level. Basic models for production, cost and distance will be examined. We will examine major extensions of the models to provide scope for cross firm heterogeneity (such as heteroscedasticity) as well as unobserved heterogeneity captured by the stochastic specification of the model. The second day of the course will turn to more advanced applications, such as Bayesian and classical methods of estimation and, especially, panel data models. In addition to the examination of theoretical and econometric methods, we will study several applications from the recent literature.


The course will include lectures that develop the relevant theory and extensive practical, laboratory applications. Emphasis in the laboratory sessions will be on estimation of stochastic frontier models and using them to compute measures of economic efficiency. Course participants will apply the techniques on their own computers using the LIMDEP/NLOGIT computer program and several real data sets that have been used in applications already in the literature.


Prior knowledge is assumed to include a course in microeconomics, calculus at the level assumed in the first year of a Ph.D. program in economics and a course in econometrics at the beginning Ph.D. level out of a textbook such as Greene, W., Econometric Analysis, 7th edition. Familiarity with LIMDEP will be helpful, but is not necessary.


Students in this course will obtain background in both the theory and methods of estimation for stochastic frontier modeling. This course will provide a gateway to the professional literature as well as practical application of the methods at the level of the contemporary research in the field.


Students are assumed to have had a previous course in Econometrics at the graduate level, using a text at the level of Greene, W., Econometric Analysis.

Course Outline

This is a course in efficiency and productivity analysis. There are a huge variety of models that are used in this context. We will focus on, the fundamental normal-half normal model that began the field then examine a long list of extensions that have accommodated different situations and data. The course will consist of discussions and laboratory sessions which will apply the techniques to live data sets and some time devoted to topics, discussions and laboratory work on student projects. Discussions will cover the topics listed below. Lab sessions will apply the techniques discussed in the preceding sessions. Practicals will consist of directed exercises and student assignments to be completed singly or in groups.

No specific textbook is assigned for the course. A useful reference is


Kumbhakar, S. and K. Lovell, Stochastic Frontier Modeling, Cambridge University Press, 2000, monograph.


A foundation work that contains broad surveys of the field is


Fried, H., K. Lovell and S. Schmidt, The Measurement of Productive Efficiency and Productivity, Oxford University Press, 2007, the Red Book.


Some of the presentation will be based on Econometric Analysis, 7th ed., by Greene, W. (Prentice Hall, 2011). Some chapters are included with the course materials: Left click to activate. Right click to download


Greene, W., Econometric Analysis, 7th Ed.

Greene 11-Panel data methods

Greene 12-Estimation methods

Greene 14-Maximum likelihood estimation

Greene 15-Simulation based estimation and inference

Greene 19-Censoring and truncation (see Section 19.2.4 on stochastic frontiers)


The received literature on frontier models is vast - one could easily compose a list of thousands of articles. Your course materials include a small handful of survey articles:


Recent Developments and Methodology of Stochastic Frontier Analysis,Parmeter and Kumbhakar Survey

The Econometric Approach to Efficiency Estimation Greene Survey (one of the chapters in the Red Book)

Efficiency Analysis Fried, Lovell and Schmidt Survey (Chapter 1 in the Red Book)

Efficiency Analysis, Sena survey (article)

Economic Efficiency and Frontier Techniques Luis R. Murillo-Zamorano


Course Materials and Resources


I. Class Notes: These are Powerpoint slide presentations and background readings (pdf). Left click to activate. Right click to download.


Part 0: Introduction

Part 1: Modeling Efficiency

Microeconomic essentials: Production functions; Functional form, Cobb-Douglas and translog; Isoquants and efficiency measures; implications for least squares estimation of production relationships

Greene (2007 survey, Red Book)

Fried, Lovell, Schmidt (2007 Red Book),

Kumbhakar and Lovell (2000 monograph)


Part 2: Production Frontiers

Frontier modeling; programming estimators; The gamma frontier; modifying least squares; Data envelopment analysis

Greene (1980) gamma frontier, one sided errors

Greene (1990) gamma frontier, maximum likelihood


Part 3: The Stochastic Frontier Model

The stochastic frontier model; Implications for OLS; The half normal model; Maximum likelihood estimation; Exponential and truncated distributions; estimating inefficiency; Confidence intervals; Model specification

Aigner, Lovell, Schmidt (1977), original proposal for stochastic frontier model

Jondrow, Lovell, Materov, Schmidt (1982), how to compute inefficiency

Waldman (1982), wrongward leaning residuals

Greene (1990), MLE for gamma model

Greene (2003b), the last gamma frontier paper

Bera and Sharma (1999), confidence intervals for inefficiency

Tsionas (2012), continues developing the gamma frontier model

Tsionas (2013), Weibull model


Part 4: Production and Cost and Allocative Inefficiency

Production, cost and other models; Cost frontiers; Allocative inefficiency; The Greene problem; Multiple outputs; Distance functions; Profit and revenue functions

Christensen and Greene (1976), translog form for cost function

Tsionas and Kumbhakar (2004), Bayesian solution to the Greene problem

Atkinson and Dorfman (2005), multiple outputs, one undesirable

Tsionas and Kumbhakar (2005), banking industry


Part 5: Modeling Heterogeneity

Review of stochastic frontier modeling; Estimating inefficiency; functional form Heterogeneity in the stochastic frontier model; model extensions; heteroscedasticity; One and two step estimation; A latent class model; Random parameters; Sample Selection

Greene survey (2007, Red Book)

Greene (2004a), Health Economics, heterogeneity

Hadri (1999), heteroscedasticity

Wang and Schmidt (2002), one step vs. two step estimation of inefficiency

Orea and Kumbhakar (2004), a latent class model

Greene (2005), modeling heterogeneity in frontier functions

Besstremyannaya (2011), latent class stochastic frontier model, health economics

Mutter et al., AHRQ (2012), omitted endogenous quality in an SF model

Greene (2010) Sample selection model; maximum simulated likelihood

Kathuria, Raj, Sen (2013) Application of sample selection model

Kumbhakar, Parmeter, Tsionas (2013), latent class stochastic frontier


Part 6: Model Extensions

Bayesian and Maximum simulated likelihood estimation; The normal-gamma, Weibull and Rayleigh models; Discrete outcomes

Greene (2003b), MSL estimation of the gamma model

Kim and Schmidt (2000), Bayesian vs. classical estimation

Kleit and Terrell (2001), Bayesian application to electricity

Huang (2004), Bayesian estimation of gamma frontier model

Koop, Osiewalski and Steel (1997), Bayesian estimation of the frontier model

Nemoto and Furumatsu (2013), distance function approach


Part 7: Panel Data

(Part 9: True Fixed Effects: Discussion of a Panel Data Frontier Model)

Panel data; Fixed and random effects; FE and RE in the stochastic frontier model; True random and fixed effects models, spatial effects

Pitt and Lee (1981), original random effects model

Schmidt and Sickles (1984), fixed effects models

Koop, Osiewalski and Steel (1997), Bayesian treatment of fixed effects

Greene (2005), fixed and random effects models

Battese and Coelli (1988, 1992, 1995), time variation in inefficiency models

Greene 2004b, heterogeneity

Farsi, Fillipini, Kuenzle (2003), true random and fixed effects models

Wang and Ho (2010), more on fixed effects

Kumbhakar Lien Hardaker (2013), Survey of panel data approaches

Filippini and Greene (2015), Stochastic Frontier Model with Time Varying and Time Invariant Inefficiency

Faust and Baranzini (2014) True Random Effects Cost Frontier Analysis

Greene (2014) Discussion of Estimation of True Random Effects Models

Glass-Kenjegalievay-Sickles(2013) Spatial Stochastic Frontier Model for U.S. States

Glass-Kenjegalievay-Sickles(2013) Spatial Stochastic Frontier Model for European Countries


Part 8: Applications; Summary and Conclusions

Applications from the literature

Alvarez, Arias, Greene (2005), latent variable model, management

Farsi, Filippini, Kuenzle (2003), fixed and random effects, railways

Holloway (2005), Bayesian application, hierarchical analysis fishing trawlers

Kumbhakar and Wang (2005) Macro application - cross country growth

Kleit and Terrell (2001), electricity market


II. Lab Exercises


Assignments: Exercises and Practicals for Stochastic Frontier Modeling. Left click to activate. Right click to download. Note there are two parts to each, the .pdf file for the assignment and the .lim file that contains the NLOGIT commands. Each will also use one or more of the data sets.


Assignment 1: (NLOGIT Commands for Assignment 1)

Assignment 2: (NLOGIT Commands for Assignment 2)

Assignment 3: (NLOGIT Commands for Assignment 3)

Assignment 4: (NLOGIT Commands for Assignment 4)


Data Sets: These are NLOGIT project (.lpj) files. Each is accompanied by a corresponding NLOGIT command (.lim) file that contains a description of the variables in the file. Left click to activate. Right click to download.


Airlines data

Banking data

Dairy Farms data

Electricity data

Swiss Railways data

WHO Health Systems data


III. Program Notes: These are Powerpoint slide presentations that explain using NLOGIT and how to do the assignments with NLOGIT. Left click to activate. Right click to download.


Lab 1: Getting Started

Lab 2: Stochastic Frontier Models and Technical Inefficiency; Model Building, Production and Cost Models, Estimating Efficiency

Lab 3: Stochastic Frontier Models with Heterogeneity

Lab 4: Frontier Models and Panel Data


IV. LIMDEP/NLOGIT Software:: The short introduction is a getting started. The LIMDEP manual explains the basics of using LIMDEP and NLOGIT. (LIMDEP is embedded in NLOGIT). The setup file contains an installation kit for installing a copy of NLOGIT made specially for this course on your own computer. You should download the setup file to your own computer and execute it there, rather than launching it from your web browser. Left click to activate. Right click to download.


These chapters from the LIMDEP/NLOGIT manual are specifically for frontier estimation and efficiency analysis.

Chapter E62: Estimating Stochastic Frontier Models

Chapter E63: Heteroscedasticity and Truncation

Chapter E64: Panel Data Stochastic Frontier Models

Chapter E65: Data Envelopment Analysis

These are descriptions of new frontier and efficiency estimation tools that were added to LIMDEP and NLOGIT in 2013
New Stochastic Frontier Specifications
Extensions of the Data Envelopment Analysis Program

These are short manuals that document how to use the program:

Quickstart Introduction to NLOGIT (Command script file to use with Quickstart) (Data file to use with Quickstart)

Short Introduction to NLOGIT

LIMDEP Student User Manual

NLOGIT Student User Manual

This is the installation kit for installing the program. Download this file to your computer before you use it to install NLOGIT on your computer.

NLOGIT Software Setup for Installing NLOGIT on Your Computer