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. Range of Applications
Regulated industries – railroads, electricity, public services
Health care delivery – nursing homes, hospitals, health care systems (WHO)
Banking and Finance
Many, many (many) other industries. See Lovell and Schmidt survey…

3. Discrete Variables Count data frontier
Outcomes inside the frontier: Preserve discrete outcome
Patents (Hofler, R. “A Count Data Stochastic Frontier Model,”
Infant Mortality (Fe, E., “On the Production of Economic Bads…”)

4. Count Frontier P(y*|x)=Poisson Model for optimal outcome
Effects the distribution: P(y|y*,x)=P(y*-u|x)= a different count model for the mixture of two count variables
Effects the mean:E[y*|x]=λ(x) while E[y|x]=u λ(x) with 0 < u < 1. (A mixture model)
Other formulations.

5. Alvarez, Arias, Greene Fixed Management
Yit = f(xit,mi*) where mi* = “management”
Actual mi = mi* - ui. Actual falls short of “ideal”
Translates to a random coefficients stochastic frontier model
Estimated by simulation
Application to Spanish dairy farms

6. Fixed Management as an Input Implies Time Variation in Inefficiency

7. Random Coefficients Frontier Model
[Chamberlain/Mundlak: Correlation mi* (not mi-mi*) with xit]

8. Estimated Model
First order production coefficients (standard errors). Quadratic terms not shown.

9. Inefficiency Distributions
Without Fixed Management
With Fixed Management

10. Holloway, Tomberlin, Irz: Coastal Trawl Fisheries
Application of frontier to coastal fisheries
Hierarchical Bayes estimation
Truncated normal model and exponential
Panel data application
Time varying inefficiency
The “good captain” effect vs. inefficiency

11. Sports Kahane: Hiring practices in hockey
Output=payroll, Inputs=coaching, franchise measures
Efficiency in payroll related to team performance
Battese/Coelli panel data translog model
Koop: Performance of baseball players
Aggregate output: singles, doubles, etc.
Inputs = year, league, team
Policy relevance? (Just for fun)

12. Macro Performance Koop et al.
Productivity Growth in a stochastic frontier model
Country, year, Yit = ft(Kit,Lit)Eitwit
Bayesian estimation
OECD Countries,

13. Mutual Fund Performance
Standard CAPM
Stochastic frontier added
Excess return=a+b*Beta +v – u
Sub-model for determinants of inefficiency
Bayesian framework
Pooled various different distribution estimates

14. Energy Consumption Derived input to household and community production
Cost analogy
Panel data, statewide electricity consumption: Filippini, Farsi, et al.

15. Hospitals Usually cost studies Multiple outputs – case mix
“Quality” is a recurrent theme
Complexity – unobserved variable
Endogeneity
Rosko: US Hospitals, multiple outputs, panel data, determinants of inefficiency = HMO penetration, payment policies, also includes indicators of heterogeneity
Australian hospitals: Fit both production and cost frontiers. Finds large cost savings from removing inefficiency.

16. Law Firms Stochastic frontier applied to service industry
Output=Revenue
Inputs=Lawyers, associates/partners ratio, paralegals, average legal experience, national firm
Analogy drawn to hospitals literature – quality aspect of output is a difficult problem

17. Farming Hundreds of applications
Major proving ground for new techniques
Many high quality, very low level micro data sets
O’Donnell/Griffiths – Philippine rice farms
Latent class – favorable or unfavorable climate
Panel data production model
Bayesian – has a difficult time with latent class models. Classical is a better approach

18. Railroads and other Regulated Industries
Filippini – Maggi: Swiss railroads, scale effects etc. Also studied effect of different panel data estimators
Coelli – Perelman, European railroads. Distance function. Developed methodology for distance functions
Many authors: Electricity (C&G). Used as the standard test data for Bayesian estimators

19. Banking Dozens of studies Typically multiple output cost functions
Wheelock and Wilson, U.S. commercial banks
Turkish Banking system
Banks in transition countries
U.S. Banks – Fed studies (hundreds of studies)
Typically multiple output cost functions
Development area for new techniques
Many countries have very high quality data available

20. Sewers New York State sewage treatment plants
200+ statewide, several thousand employees
Used fixed coefficients technology
lnE = a + b*lnCapacity + v – u; b < 1 implies economies of scale (almost certain)
Fit as frontier functions, but the effect of market concentration was the main interest

21. Summary

22. Inefficiency

23. Methodologies Data Envelopment Analysis Stochastic Frontier Modeling
HUGE User base
Largely atheoretical
Applications in management, consulting, etc.
Stochastic Frontier Modeling
More theoretically based – “model” based
More active technique development literature
Equally large applications pool

24. SFA Models Normal – Half Normal Normal-Gamma, Exponential, Rayleigh
Truncation
Heteroscedasticity
Heterogeneity in the distribution of ui
Normal-Gamma, Exponential, Rayleigh
Classical vs. Bayesian applications
Flexible functional forms for inefficiency
There are yet others in the literature

25. Modeling Settings Production and Cost Models Multiple output models
Cost functions
Distance functions, profits and revenue functions

26. Modeling Issues Appropriate model framework Cost, production, etc.
Functional form
How to handle observable heterogeneity – “where do we put the zs?”
Panel data
Is inefficiency time invariant?
Separating heterogeneity from inefficiency
Dealing with endogeneity
Allocative inefficiency and the Greene problem

27. Range of Applications
Regulated industries – railroads, electricity, public services
Health care delivery – nursing homes, hospitals, health care systems (WHO, AHRQ)
Banking and Finance
Many other industries. See Lovell and Schmidt “Efficiency and Productivity”
27 page bibliography.
Table of over 200 applications since 2000