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. Stochastic Frontier Models
Motivation:
Factors not under control of the firm
Measurement error
Differential rates of adoption of technology
Frontier is 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 ( )
Input
Units
Mean
Std. Dev.
Minimum
Maximum
Milk
Milk production (liters)
131,108
92,539
14,110
727,281
Cows
# of milking cows
2.12
11.27
4.5
82.3
Labor
# man-equivalent units
1.67
0.55
1.0
4.0
Land
Hectares of land devoted to pasture and crops.
12.99
6.17
2.0
45.1
Feed
Total amount of feedstuffs fed to dairy cows (tons)
57,941
47,981
3,924.14
376,732

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 the Method of Moments
Estimator of σu is [m3/ ]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.