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2/70 Mundlak, Y., Empirical production function free of management bias. Journal of Farm Economics 43, (Wrote about (omitted) fixed effects.) History A 50 th Anniversary

3/70 William Greene New York University 22 nd International Panel Data Conference Fremantle, Australia, June, 2016 Distinguishing Heterogeneity from Inefficiency in a Panel Data Stochastic Frontier Model Introduction

4/70 Introduction

5/70 Distinguishing Heterogeneity from Inefficiency Why? How? Why? An Application World Health Report by WHO (2000) Heterogeneity and Inefficiency in a SF Model How? True Fixed Effects Model Fixed Effects and the Incidental Parameters Problem TFE Stochastic Frontier Estimators The Application A Random Effects Solution Agenda Introduction

6/70 No. 30: Composite Measure of Health System Performance No. 29: Disability Adjusted Life Expectancy Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

7/70 Composite Index of Health Care Outcomes Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

8/70 Composite Index of Performance Optimal weights? Maria Ana Lugo & Esfandiar Maasoumi, “Multidimensional Poverty Measures from an Information Theory Perspective,” Working Paper 85, ECINEQ, Society for the Study of Economic Inequality. Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

9/70 Measuring Performance via a Frontier Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions Worst Best

10/70 Model of Health System Outcomes Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

11/70 Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

12/70 Time fixed Reinterpreting the Within Estimator Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

13/70 Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

14/70 Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

15/70 Based on 1,000 repetitions Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

16/70 Performance Measured for 191 Countries Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

17/70 Performance of the 37 th Best Country Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

18/70 Observers in the U.S. were not amused. Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

19/70 This meeting took place at the ASSA convention in New Orleans in Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

20/70 Performance (observed ) Inputs (observed ) Unaccounted for (but observed 1997) heterogeneity Data for Measuring a Performance Frontier Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

21/70 SF Model for Health Care System Outcomes Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions Greene, W., “Distinguishing Between Heterogeneity and Inefficiency: Stochastic Frontier Analysis of the World Health Organization’s Panel Data on National Health Care Systems,” Health Economics, 13, 2004, pp

22/70 The Stochastic Frontier Model Inputs; Deterministic Agent specific frontier Technical inefficiency Stochastic Frontier Model Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

23/70 Efficiency Measurement with the Stochastic Frontier Model Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

24/70 Skew Normal Density Stochastic Frontier Log Likelihood Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

25/70 Time varying inefficiency Time fixed heterogeneity True Common Effects SF Model Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

26/70 Estimation of TFE and TRE Models Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

27/70 What is True About True Fixed Effects Model? HHG FENB “True” FENB Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

28/70  The WHO FE model blends heterogeneity and inefficiency and calls the result inefficiency.  The true fixed effects model is an attractive alternative specification.  The fixed effects MLE (may) have an Incidental Parameters Problem. What is the incidental parameters problem? How serious is it in general and in this context? Is there a consistent estimator available? True Fixed Effects Models Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

29/70 Poisson and Negative Binomial Regressions Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

30/70 Stata on the True Fixed Effects (Tobit) MLE Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

31/70 “The fixed effects logit estimator of  immediately gives us the effect of each element of x i on the log- odds ratio… Unfortunately, we cannot estimate the partial effects… unless we plug in a value for α i. Because the distribution of α i is unrestricted – in particular, E[α i ] is not necessarily zero – it is hard to know what to plug in for α i. In addition, we cannot estimate average partial effects, as doing so would require finding E[Λ(x it  + α i )], a task that apparently requires specifying a distribution for α i.” Heckman and MaCurdy (1980) proposed solution 1. Estimate  by conditional MLE 2. Estimate α i (  )for all agents for whom there is variation in y it. 3. Estimate average partial effects as usual Does this estimate the APE? Heckman & MaCurdy Sequential Estimator Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

32/70 Known Results for Unconditional MLE of the TFE Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

33/70 Recent Research Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

34/70 Received Wisdom for True Fixed Effects MLE The MLE converges to something, not necessarily , when T is fixed The MLE of  converges to  as T grows (and the MLE of  i converges to  i ) (Hahn and Newey 2004)) (Heckman and MaCurdy 1980)) Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

35/70 Heckman and MaCurdy, A Life Cycle Model of Female Labor Supply, Review of Economic Studies, 47, 1980, pp Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

36/70 Bias Reduction via Panel Jackknife (Hahn and Newey) Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

37/70 Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

38/70 Fixed Effects Sledge Hammer Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

39/70 Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

40/70 Appears: Slope Estimator is not biased Variance estimator is biased Note: Scale factor is used to compute partial effects. Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

41/70 These are % biases. E.g., with T = 8, the partial effects are off by about 1-2%. Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

42/70 Fernandez-Val Estimates of Partial Effects after Bias Correction Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

43/70  Where is the small T bias?  We are estimating inefficiency and ranks. The parameters are of secondary interest Both variances and slopes are used in the computations Deriving a systematic bias in the JLMS estimators is hopeless Deriving a systematic bias in the ranks of the JLMS estimators is hopeless Fixed Effects in the Stochastic Frontier Model Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

44/70 “The fixed effects logit estimator of  immediately gives us the effect of each element of x i on the log- odds ratio… Unfortunately, we cannot estimate the partial effects… unless we plug in a value for α i. Because the distribution of α i is unrestricted – in particular, E[α i ] is not necessarily zero – it is hard to know what to plug in for α i. In addition, we cannot estimate average partial effects, as doing so would require finding E[Λ(x it  + α i )], a task that apparently requires specifying a distribution for α i.” Estimation of Technical Efficiency  The fixed effects are not just nuisance parameters.  They are needed for the JLMS estimator.  Each TFE estimator includes estimators of  i, ,.  Here, we do know what to plug in for  i. Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

45/70  Generic Hahn and Newey Jackknife (2004)  Semiparametric Stochastic Frontier Amsler and Schmidt (2014)  TFE Stochastic Frontier Chen, Schmidt and Wang (2014) Belotti and Ilardi (2014) Ke, Lo and Tsay (2014) Wang and Ho (2010) Wikstrom (2016) Solutions to the IP Problem Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

46/70 Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

47/70 Amsler and Schmidt Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

48/70 Amsler and Schmidt Inefficiency and heterogeneity are identified by what they are correlated with. Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

49/70 ● By what measure is the assumption of normality in the binary choice (probit) model so strong that a manifestly incoherent model, the linear probability model, is preferable? ● In what way could the normality assumption be violated so severely that a linear probability model would be a preferable approach? Speaking of Strong Assumptions … Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

50/70 “This article discusses a remedy to the … problem of estimating time invariant … variables in FE models with unit effects.” Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

51/70 Step 3 magically reproduces the results of 1 and 2 (now with smaller standard errors) and reveals the coefficients on Z. How did they do this? Data from Cornwell and Rupert, Journal of Applied Econometrics, 3, 1988, pp Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

52/70 The Sleight of Hand Is Exposed Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

53/70 Vol. 19, No. 2, Spring 2011 Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

54/70 Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

55/70 Exact MLE for within transformed stochastic frontier model (same for pooled or true random effects) Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

56/70 Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

57/70 An Analytic Approximation Approach for Estimating the True Panel Stochastic Frontier Models Ke, P., Lo, T., and Tsay, W. Academica Sinica and National Dong Hwa University, Taiwan April, Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

58/70 Exact MLE for First differences transformed stochastic frontier model (based on the “scaling” specification) Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

59/70 Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

60/70 Moment Equations for the Method of Moments Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

61/70 Compare MoM and TFE Features Estimation of parameters Estimation of technical efficiency Estimation of ranks of technical efficiency Lab Experiment y = a i + bx i,t + v i,t – u i,t a i, x i,t, u i,t fixed Simulate over v i,t Explore the Behavior of FE Estimators Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

62/70 Monte Carlo Experiment Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

63/70 2.7% Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

64/70 Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

65/70 Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

66/70 N = 100, T = 7 RANKCORR= RANKCTFE= RANKCMOM= Cor.Mat.| EUIT TECHEFF TECHMOM EUIT| TECHEFF| TECHMOM| N = 100, T = 10 RANKCORR= RANKCTFE= RANKCMOM= Cor.Mat.| EUIT TECHEFF TECHMOM EUIT| TECHEFF| TECHMOM| N = 100, T = 15 RANKCORR= RANKCTFE= RANKCMOM= Cor.Mat.| EUIT TECHEFF TECHMOM EUIT| TECHEFF| TECHMOM| N = 100, T = 20 RANKCORR= RANKCTFE= RANKCMOM= Cor.Mat.| EUIT TECHEFF TECHMOM EUIT| TECHEFF| TECHMOM| Estimated Technical Efficiency, EUIT = exp[-u(i,t)] RankCorr = Rank correlation of TFE estimates with MoM Estimates RankCtfe = Rank correlation of TFE estimates with true values RankCmom = Rank correlation of MoM estimates with true values Correlations are Pearson correlations of estimated values Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

67/70 Technical Efficiency, TFE vs. True Values, N=100, T=7 and 20 Technical Efficiency, MoM vs. True Values, N=100, T=7 and 20 Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

68/70  The unconditional estimator rarely works at all when T < 6. The FE’s soak up all the variation in v it. MoM frequently works in these cases.  Estimators of the parameters are not biased similarly Estimator of  appears to be consistent Estimate of σ u appears to be trivially positively biased Estimator of σ v appears to consistently underestimate σ v When estimation fails, the estimator of σ v collapses  TFE appears to be a better predictor of technical efficiency even with small biases in the parameters  Two good alternatives Wickstrom Method of Moments True random effects with Mundlak adjustment Results for the TFE Stochastic Frontier Model Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

69/70  Received results focus on additive bias in coefficient estimators Received firm theory is about scaling, not additive bias  Linear FEM  Logit FEM  IP problem seems to relate to variance parameters, not location.  Biases are mixed when dependent variables are continuous  Biases in derived results such as partial effects and predictions are unclear The Incidental Parameters Problem Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions

70/70 Distinguishing Heterogeneity from Inefficiency Application True Fixed Effects SFM Incidental Parameters Problem Consistent Estimation Conclusions