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. 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.

3. The Normal-Half Normal Model

4. 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.

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

6. 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).

7. 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.

8. Duality Between Production and Cost

9. Implied Cost Frontier Function

10. Stochastic Cost Frontier

11. Cobb-Douglas Cost Frontier

12. Translog Cost Frontier

13. Restricted Translog Cost Function

14. Cost Application to C&G Data

15. Cost Application to C&G Data

16. Estimates of Economic Efficiency

17. Duality – Production vs. Cost

18. Multiple Output Cost Frontier

19. Banking Application

20. Economic Efficiency

21. Allocative Inefficiency and Economic Inefficiency

22. Cost Structure – Demand System

23. Cost Frontier Model

24. 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 • Requires data on input shares typically not available

25. 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