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. 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, 1979-1988

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 • 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 • 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 • 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