1. [Part 3] 1/49 Stochastic FrontierModels Stochastic Frontier Model Stochastic Frontier Models William Greene Stern School of Business New York University 0Introduction 1Efficiency Measurement 2Frontier Functions 3Stochastic Frontiers 4Production and Cost 5Heterogeneity 6Model Extensions 7Panel Data 8Applications

2. [Part 3] 2/49 Stochastic FrontierModels Stochastic Frontier Model 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. [Part 3] 3/49 Stochastic FrontierModels Stochastic Frontier Model The Stochastic Frontier Model u i > 0, but v i may take any value. A symmetric distribution, such as the normal distribution, is usually assumed for v i. Thus, the stochastic frontier is  +  ’x i +v i and, as before, u i represents the inefficiency.

4. [Part 3] 4/49 Stochastic FrontierModels Stochastic Frontier Model Least Squares Estimation Average inefficiency is embodied in the third moment of the disturbance ε i = v i - u i. So long as E[v i - u i ] is constant, the OLS estimates of the slope parameters of the frontier function are unbiased and consistent. (The constant term estimates α-E[u i ]. The average inefficiency present in the distribution is reflected in the asymmetry of the distribution, which can be estimated using the OLS residuals:

5. [Part 3] 5/49 Stochastic FrontierModels Stochastic Frontier Model Application to Spanish Dairy Farms InputUnitsMeanStd. Dev. MinimumMaximum MilkMilk production (liters) 131,108 92,539 14,110727,281 Cows# of milking cows 2.12 11.27 4.5 82.3 Labor# man-equivalent units 1.67 0.55 1.0 4.0 LandHectares of land devoted to pasture and crops. 12.99 6.17 2.0 45.1 FeedTotal amount of feedstuffs fed to dairy cows (tons) 57,94147,9813,924.1 4 376,732 N = 247 farms, T = 6 years (1993-1998)

6. [Part 3] 6/49 Stochastic FrontierModels Stochastic Frontier Model Example: Dairy Farms

7. [Part 3] 7/49 Stochastic FrontierModels Stochastic Frontier Model The Normal-Half Normal Model

8. [Part 3] 8/49 Stochastic FrontierModels Stochastic Frontier Model Normal-Half Normal Variable

9. [Part 3] 9/49 Stochastic FrontierModels Stochastic Frontier Model The Skew Normal Variable

10. [Part 3] 10/49 Stochastic FrontierModels Stochastic Frontier Model Standard Form: The Skew Normal Distribution

11. [Part 3] 11/49 Stochastic FrontierModels Stochastic Frontier Model Battese Coelli Parameterization

12. [Part 3] 12/49 Stochastic FrontierModels Stochastic Frontier Model Estimation: Least Squares/MoM  OLS estimator of β is consistent  E[u i ] = (2/π) 1/2 σ u, so OLS constant estimates α+ (2/π) 1/2 σ u  Second and third moments of OLS residuals estimate  Use [a,b,m 2,m 3 ] to estimate [ , ,  u,  v ]

13. [Part 3] 13/49 Stochastic FrontierModels Stochastic Frontier Model 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. [Part 3] 14/49 Stochastic FrontierModels Stochastic Frontier Model Airlines Data – 256 Observations

15. [Part 3] 15/49 Stochastic FrontierModels Stochastic Frontier Model Least Squares Regression

16. [Part 3] 16/49 Stochastic FrontierModels Stochastic Frontier Model

17. [Part 3] 17/49 Stochastic FrontierModels Stochastic Frontier Model Alternative Models: Half Normal and Exponential

18. [Part 3] 18/49 Stochastic FrontierModels Stochastic Frontier Model Normal-Exponential Likelihood

19. [Part 3] 19/49 Stochastic FrontierModels Stochastic Frontier Model Normal-Truncated Normal

20. [Part 3] 20/49 Stochastic FrontierModels Stochastic Frontier Model Truncated Normal Model: mu=.5

21. [Part 3] 21/49 Stochastic FrontierModels Stochastic Frontier Model Effect of Differing Truncation Points From Coelli, Frontier4.1 (page 16)

22. [Part 3] 22/49 Stochastic FrontierModels Stochastic Frontier Model 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. [Part 3] 23/49 Stochastic FrontierModels Stochastic Frontier Model A Possible Problem with the Method of Moments  Estimator of σ u is [m 3 /-.21801] 1/3  Theoretical m 3 is < 0  Sample m 3 may be > 0. If so, no solution for σ u. (Negative to 1/3 power.)

24. [Part 3] 24/49 Stochastic FrontierModels Stochastic Frontier Model Now Include LM in the Production Model

25. [Part 3] 25/49 Stochastic FrontierModels Stochastic Frontier Model

26. [Part 3] 26/49 Stochastic FrontierModels Stochastic Frontier Model 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.

27. [Part 3] 27/49 Stochastic FrontierModels Stochastic Frontier Model

28. [Part 3] 28/49 Stochastic FrontierModels Stochastic Frontier Model Estimating u i  No direct estimate of u i  Data permit estimation of y i – β’x i. Can this be used? ε i = y i – β’x i = v i – u i Indirect estimate of u i, using E[u i |v i – u i ] This is E[u i |y i, x i ]  v i – u i is estimable with e i = y i – b’x i.

29. [Part 3] 29/49 Stochastic FrontierModels Stochastic Frontier Model Fundamental Tool - JLMS We can insert our maximum likelihood estimates of all parameters. Note: This estimates E[u|v i – u i ], not u i.

30. [Part 3] 30/49 Stochastic FrontierModels Stochastic Frontier Model Other Distributions

31. [Part 3] 31/49 Stochastic FrontierModels Stochastic Frontier Model Technical Efficiency

32. [Part 3] 32/49 Stochastic FrontierModels Stochastic Frontier Model Application: Electricity Generation

33. [Part 3] 33/49 Stochastic FrontierModels Stochastic Frontier Model Estimated Translog Production Frontiers

34. [Part 3] 34/49 Stochastic FrontierModels Stochastic Frontier Model Inefficiency Estimates

35. [Part 3] 35/49 Stochastic FrontierModels Stochastic Frontier Model Inefficiency Estimates

36. [Part 3] 36/49 Stochastic FrontierModels Stochastic Frontier Model Estimated Inefficiency Distribution

37. [Part 3] 37/49 Stochastic FrontierModels Stochastic Frontier Model Estimated Efficiency

38. [Part 3] 38/49 Stochastic FrontierModels Stochastic Frontier Model Confidence Region Horrace, W. and Schmidt, P., Confidence Intervals for Efficiency Estimates, JPA, 1996.

39. [Part 3] 39/49 Stochastic FrontierModels Stochastic Frontier Model Application (Based on Electricity Costs)

40. [Part 3] 40/49 Stochastic FrontierModels Stochastic Frontier Model 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. [Part 3] 41/49 Stochastic FrontierModels Stochastic Frontier Model Airlines Application

42. [Part 3] 42/49 Stochastic FrontierModels Stochastic Frontier Model Efficiency Distributions

43. [Part 3] 43/49 Stochastic FrontierModels Stochastic Frontier Model Nonparametric Methods - DEA

44. [Part 3] 44/49 Stochastic FrontierModels Stochastic Frontier Model DEA is done using linear programming

45. [Part 3] 45/49 Stochastic FrontierModels Stochastic Frontier Model

46. [Part 3] 46/49 Stochastic FrontierModels Stochastic Frontier Model Methodological Problems with DEA  Measurement error  Outliers  Specification errors  The overall problem with the deterministic frontier approach

47. [Part 3] 47/49 Stochastic FrontierModels Stochastic Frontier Model 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

48. [Part 3] 48/49 Stochastic FrontierModels Stochastic Frontier Model

49. [Part 3] 49/49 Stochastic FrontierModels Stochastic Frontier Model Comparing the Two Methods.