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. Stochastic Frontier Models • Motivation: • Factors not under control of the firm • Measurement error • Differential rates of adoption of technology • Frontieris randomly placed by the whole collection of stochastic elements which might enter the model outside the control of the firm. • Aigner, Lovell, Schmidt (1977), Meeusen, van den Broeck (1977), Battese, Corra (1977)

3. The Stochastic Frontier Model ui > 0, but vi may take any value. A symmetric distribution, such as the normal distribution, is usually assumed for vi. Thus, the stochastic frontier is +’xi+vi and, as before,ui represents the inefficiency.

4. Least Squares Estimation Average inefficiency is embodied in the third moment of the disturbance εi= vi - ui. So long as E[vi - ui] is constant, the OLS estimates of the slope parameters of the frontier function are unbiased and consistent. (The constant term estimates α-E[ui]. The average inefficiency present in the distribution is reflected in the asymmetry of the distribution, which can be estimated using the OLS residuals:

5. Application to Spanish Dairy Farms N = 247 farms, T = 6 years (1993-1998)

6. Example: Dairy Farms

7. The Normal-Half Normal Model

8. Normal-Half Normal Variable

9. The Skew Normal Variable

10. Standard Form: The Skew Normal Distribution

11. Battese Coelli Parameterization

12. Estimation: Least Squares/MoM • OLS estimator of β is consistent • E[ui] = (2/π)1/2σu, so OLS constant estimates α+ (2/π)1/2σu • Second and third moments of OLS residuals estimate • Use [a,b,m2,m3] to estimate [,,u, v]

13. Log Likelihood Function Waldman (1982) result on skewness of OLS residuals: If the OLS residuals are positively skewed, rather than negative, then OLS maximizes the log likelihood, and there is no evidence of inefficiency in the data.

14. Airlines Data – 256 Observations

15. Least Squares Regression

17. Alternative Models:Half Normal and Exponential

18. Normal-Exponential Likelihood

19. Normal-Truncated Normal

20. Truncated Normal Model: mu=.5

21. Effect of Differing Truncation Points From Coelli, Frontier4.1 (page 16)

22. Other Models • Other Parametric Models (we will examine several later in the course) • Semiparametric and nonparametric – the recent outer reaches of the theoretical literature • Other variations including heterogeneity in the frontier function and in the distribution of inefficiency

23. A Possible Problem with theMethod of Moments • Estimator of σu is [m3/-.21801]1/3 • Theoretical m3 is < 0 • Sample m3 may be > 0. If so, no solution for σu. (Negative to 1/3 power.)

24. Now Include LM in the Production Model

26. Test for Inefficiency? • Base test on u = 0 <=>  = 0 • Standard test procedures • Likelihood ratio • Wald • Lagrange • Nonstandard testing situation: • Variance = 0 on the boundary of the parameter space • Standard chi squared distribution does not apply.

28. Estimating ui • No direct estimate of ui • Data permit estimation of yi – β’xi. Can this be used? • εi = yi – β’xi= vi – ui • Indirect estimate of ui, using E[ui|vi – ui] • This is E[ui|yi,xi] • vi – ui is estimable with ei = yi – b’xi.

29. Fundamental Tool - JLMS We can insert our maximum likelihood estimates of all parameters. Note: This estimates E[u|vi – ui], not ui.

30. Other Distributions

31. Technical Efficiency

32. Application: Electricity Generation

33. Estimated Translog Production Frontiers

34. Inefficiency Estimates

35. Inefficiency Estimates

36. Estimated Inefficiency Distribution

37. Estimated Efficiency

38. Confidence Region Horrace, W. and Schmidt, P., Confidence Intervals for Efficiency Estimates, JPA, 1996.

39. Application (Based on Electricity Costs)

40. A Semiparametric Approach • Y = g(x,z) + v - u [Normal-Half Normal] • (1) Locally linear nonparametric regression estimates g(x,z) • (2) Use residuals from nonparametric regression to estimate variance parameters using MLE • (3) Use estimated variance parameters and residuals to estimate technical efficiency.

41. Airlines Application

42. Efficiency Distributions

43. Nonparametric Methods - DEA

44. DEA is done using linear programming

46. Methodological Problems with DEA • Measurement error • Outliers • Specification errors • The overall problem with the deterministic frontier approach

47. DEA and SFA: Same Answer? • Christensen and Greene data • N=123 minus 6 tiny firms • X = capital, labor, fuel • Y = millions of KWH • Cobb-Douglas Production Function vs. DEA

49. Comparing the Two Methods.