1. Efficiency Measurement William Greene Stern School of Business New York University

2. Session 5 Modeling Heterogeneity

3. Production Function Model with Inefficiency

4. 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.

5. The Normal-Half Normal Model

6. 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.

7. Normal-Truncated Normal

8. Truncated Normal Model

9. 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.

10. Estimated Translog Production Frontiers

11. Inefficiency Estimates

12. Estimated Inefficiency Distribution

13. Cost Inefficiency y* = f(x)  C* = g(y*,w) (Samuelson – Shephard duality results) Cost inefficiency: If y < f(x), then C must be greater than g(y,w). Implies the idea of a cost frontier. lnC = lng(y,w) + u, u > 0.

14. Stochastic Cost Frontier

15. Estimates of Economic Efficiency

16. Duality – Production vs. Cost

17. Where to Next?  Heterogeneity: “Where do we put the z’s?” Other variables that affect production and inefficiency Enter production frontier, inefficiency distribution, elsewhere?  Heteroscedasticity Another form of heterogeneity Production “risk”  Bayesian and simulation estimators The stochastic frontier model with gamma inefficiency Bayesian treatments of the stochastic frontier model  Panel Data Heterogeneity vs. Inefficiency – can we distinguish Model forms: Is inefficiency persistent through time?  Applications

19. 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?

20. 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 y i = x i + v i - u i v i ~ N[0, v 2 ] u i = |U i | where U i ~ N[ i,  u 2 ],  i = 0 +  1 z i,

21. Heterogeneous Mean

22. Estimated Economic Efficiency

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

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

25. Application: WHO Data

26. 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

27. A Latent Class Model

28. 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.

29. Latent Class Application

30. 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.

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

33. Estimated Cost Efficiency

34. A Two Class Model  Class 1: With Inefficiency logC = f(output, input prices, environment) +  v v +  u u  Class 2: Without Inefficiency logC = f(output, input prices, environment) +  v v  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.

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

38. Heteroscedasticity in v and/or u Var[v i | h i ] =  v 2 g v (h i,) =  vi 2 g v (h i,0) = 1, g v (h i,) = [exp( ’ h i )] 2 Var[U i | h i ] =  u 2 gu(hi,)=  ui 2 g u (h i,0) = 1, g u (h i,) = [exp( ’ h i )] 2

39. Application: WHO Data

40. A “Scaling” Model

41. Unobserved Endogenous Heterogeneity  Cost = C(p,y,Q), Q = quality Quality is unobserved Quality is endogenous – correlated with unobservables that influence cost  Econometric Response: There exists a proxy that is also endogenous Omit the variable? Include the proxy?  Question: Bias in estimated inefficiency (not interested in coefficients)

42. Simulation Experiment  Mutter, et al. (AHRQ), 2011  Analysis of California nursing home data  Estimate model with a simulated data set  Compare biases in sample average inefficiency compared to the exogenous case  Endogeneity is quantified in terms of correlation of Q(i) with u(i)

43. A Simulation Experiment Conclusion: Omitted variable problem does not make the bias worse.