Topics in Microeconometrics Professor William Greene Stern School of Business, New York University

1. Topics in Microeconometrics Professor William Greene Stern School of Business, New York University at Curtin Business School Curtin University Perth July 22-24, 2013

2. 3. Frontier Model Building

3. Functional Form and Efficiency Measurement

4. Single Output Stochastic Frontier 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.

5. The Normal-Half Normal Model

6. 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] = E[ui|yi,xi] • vi – ui is estimable with ei = yi – b’xi.

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

8. Multiple Output Frontier • The formal theory of production departs from the transformation function that links the vector of outputs, y to the vector of inputs, x; T(y,x) = 0. • As it stands, some further assumptions are obviously needed to produce the framework for an empirical model. By assuming homothetic separability, the function may be written in the form A(y) = f(x).

9. Multiple Output Production Function Inefficiency in this setting reflects the failure of the firm to achieve the maximum aggregate output attainable. Note that the model does not address the economic question of whether the chosen output mix is optimal with respect to the output prices and input costs. That would require a profit function approach. Berger (1993) and Adams et al. (1999) apply the method to a panel of U.S. banks – 798 banks, ten years.

10. Duality Between Production and Cost

11. Implied Cost Frontier Function

12. Stochastic Cost Frontier

13. Cobb-Douglas Cost Frontier

14. Translog Cost Frontier

15. Restricted Translog Cost Function

16. Cost Application to C&G Data

17. Estimates of Economic Efficiency

18. Duality – Production vs. Cost

19. Multiple Output Cost Frontier

20. Banking Application

21. Economic Efficiency

22. Allocative Inefficiency and Economic Inefficiency

23. Cost Structure – Demand System

24. Cost Frontier Model

25. The Greene Problem • Factor shares are derived from the cost function by differentiation. • Where does ek come from? • Any nonzero value of ek, which can be positive or negative, must translate into higher costs. Thus, u must be a function of e1,…,eK such that ∂u/∂ek > 0 • Noone had derived a complete, internally consistent equation system  the Greene problem. • Solution: Kumbhakar in several papers. (E.g., JE 1997) • Very complicated – near to impractical • Apparently of relatively limited interest to practitioners

26. A Less Direct Solution(Sauer,Frohberg JPA, 27,1, 2/07) • Symmetric generalized McFadden cost function – quadratic in levels • System of demands, xw/y = * + v, E[v]=0. • Average input demand functions are estimated to avoid the ‘Greene problem.’ Corrected wrt a group of firms in the sample. • Not directly a demand system • Errors are decoupled from cost by the ‘averaging.’ • Application to rural water suppliers in Germany

27. Modeling Heterogeneity

28. 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.

30. Observable Heterogeneity • As opposed to unobservable heterogeneity • Observe: Y or C (outcome) and X or w (inputs or input prices) • Firm characteristics z. Not production or cost, characterize the production process. • Enter the production or cost function? • Enter the inefficiency distribution? How?

31. Shifting the Outcome Function Firm specific heterogeneity can also be incorporated into the inefficiency model as follows: This modifies the mean of the truncated normal distribution yi = xi + vi - ui vi ~ N[0,v2] ui = |Ui| where Ui ~ N[i, u2], i = 0 + 1zi,

32. Heterogeneous Mean

33. Estimated Economic Efficiency

34. One Step or Two Step 2 Step: Fit Half or truncated normal model, compute JLMS ui, regress ui on zi Airline EXAMPLE: Fit model without POINTS, LOADFACTOR, STAGE 1 Step: Include zi in the model, compute ui including zi Airline example: Include 3 variables Methodological issue: Left out variables in two step approach.

35. One vs. Two Step 0.8 0.9 1.0 Efficiency computed without load factor, stage length and points served. Efficiency computed with load factor, stage length and points served.

36. Application: WHO Data

37. Unobservable Heterogeneity • Parameters vary across firms • Random variation (heterogeneity, not Bayesian) • Variation partially explained by observable indicators • Continuous variation – random parameter models: Considered with panel data models • Latent class – discrete parameter variation

38. A Latent Class Model

39. Latent Class Efficiency Studies • Battese and Coelli – growing in weather “regimes” for Indonesian rice farmers • Kumbhakar and Orea – cost structures for U.S. Banks • Greene (Health Economics, 2005) – revisits WHO Year 2000 World Health Report • Kumbhakar, Parmeter, Tsionas (JE, 2013) – U.S. Banks.

40. Latent Class Application

41. Inefficiency? • Not all agree with the presence (or identifiability) of “inefficiency” in market outcomes data. • Variation around the common production structure may all be nonsystematic and not controlled by management • Implication, no inefficiency: u = 0.

43. Nursing Home Costs • 44 Swiss nursing homes, 13 years • Cost, Pk, Pl, output, two environmental variables • Estimate cost function • Estimate inefficiency

44. Estimated Cost Efficiency

45. A Two Class Model • Class 1: With Inefficiency • logC = f(output, input prices, environment) + vv + uu • Class 2: Without Inefficiency • logC = f(output, input prices, environment) + vv • u = 0 • Implement with a single zero restriction in a constrained (same cost function) two class model • Parameterization: λ = u /v = 0 in class 2.

46. LogL= 464 with a common frontier model, 527 with two classes

49. Heteroscedasticity in v and/or u Var[vi | hi] = v2gv(hi,) = vi2 gv(hi,0) = 1, gv(hi,) = [exp(’hi)]2 Var[Ui | hi] = u2gu(hi,)= ui2 gu(hi,0) = 1, gu(hi,) = [exp(’hi)]2

50. Application: WHO Data