True Random Effects in Stochastic Frontier Models William

True Random Effects in Stochastic Frontier Models William Greene New York University North American Productivity Workshop Ottawa, June 6, 2014 1/78

Agenda Skew normality – Adelchi Azzalini 2/78 Stochastic frontier model Panel Data: Time invariant inefficiency models Panel Data: Time varying inefficiency models Panel Data: True random effects models Applications of true random effects Spatial effects in a stochastic frontier model Persistent and transient inefficiency in Swiss railroads A panel data sample selection corrected stochastic frontier model

Skew Normality 3/78

The Stochastic Frontier Model 4/78

Log Likelihood Skew Normal Density 5/78

Birnbaum (1950) Wrote About Skew Normality Effect of Linear Truncation on a Multinormal Population 6/78

Weinstein (1964) Found f( ) Query 2: The Sum of Values from a Normal and a Truncated Normal Distribution See, also, Nelson (Technometrics, 1964), Roberts (JASA, 1966) 7/78

O’Hagan and Leonard (1976) Found Something Like f( ) Resembles f( ) Bayes Estimation Subject to Uncertainty About Parameter Constraints 8/78

ALS (1977) Discovered How to Make Great Use of f( ) See, also, Forsund and Hjalmarsson (1974), Battese and Corra (1976) Poirier, … Timmer, … several others. 9/78

Azzalini (1985) Figured Out f( ) And Noticed the Connection to ALS © 2014 10/78

http: //azzalini. stat. unipd. it/SN/abstracts. html#sn 99 ALS 11/78

http: //azzalini. stat. unipd. it/SN/ 12/78

A Useful FAQ About the Skew Normal 13/78

Random Number Generator 14/78

How Many Applications of SF Are There? 15/78

W. D. Walls (2006) On Skewness in the Movies 16/78 Cites Azzalini.

SNARCH Model for Financial Crises (2013) “The skew-normal distribution developed by Sahu et al. (2003)…” Does not know Azzalini. 17/78

A Skew Normal Mixed Logit Model (2010) Greene (2010, knows Azzalini and ALS), Bhat (2011, knows not Azzalini … or ALS) 18/78

Skew Normal Applications Foundation: An Entire Field n Stochastic Frontier Model Occasional Modeling Strategy Culture: Skewed Distribution of Movie Revenues Finance: Crisis and Contagion Choice Modeling: The Mixed Logit Model How can these people find each other? Where else do applications appear? 19/78

Stochastic Frontier 20/78

The Cross Section Departure Point: 1977 21/78

The Panel Data Models Appear: 1981 Time fixed 22/78

Reinterpreting the Within Estimator: 1984 Time fixed 23/78

Misgivings About Time Fixed Inefficiency: 1990 - 24/78

Are the systematically time varying models more like time fixed or freely time varying? 25/78

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Skepticism About Time Varying Inefficiency Models: Greene (2004) 27/78

True Random Effects 28/78

True Random and Fixed Effects: 2004 Time varying Time fixed 29/78

Estimation of TFE and TRE Models: 2004 30/78

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The Most Famous Frontier Study Ever 33/78

The Famous WHO Model log. COMP= + 1 log. Per. Capita. Health. Expenditure + 2 log. Years. Educ + 3 Log 2 Years. Educ + = v - u Schmidt/Sickles FEM 191 Countries. 140 of them observed 1993 -1997. 34/78

The Notorious WHO Results 35/78

August 12, 2012 37 No, it doesn’t. 36/78

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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. 959 -980. 38/78

Three Extensions of the True Random Effects Model 39/78

Spatial Stochastic Frontier Models: Accounting for Unobserved Local Determinants of Inefficiency: A. M. Schmidt, A. R. B. Morris, S. M. Helfand, T. C. O. Fonseca, Journal of Productivity Analysis, 31, 2009, pp. 101 -112 Simply redefines the random effect to be a ‘region effect. ’ Just a reinterpretation of the ‘group. ’ No spatial decay with distance. True REM does not “perform” as well as several other specifications. (“Performance” has nothing to do with the frontier model. ) 40/78

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A Stochastic Frontier Model with Short. Run and Long-Run Inefficiency: Colombi, R. , Kumbhakar, S. , Martini, G. , Vittadini, G. University of Bergamo, WP, 2011, JPA 2014, forthcoming. 42/78

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“From the sampling theory perspective, the application of the model is computationally prohibitive when T is large. This is because the likelihood function depends on a (T+1)-dimensional integral of the normal distribution. ” [Tsionas and Kumbhakar (2012, p. 6)] 44/78

Tsionas, G. and Kumbhakar, S. Firm Heterogeneity, Persistent and Transient Technical Inefficiency: A Generalized True Random Effects Model Journal of Applied Econometrics. Published online, November, 2012. Extremely involved Bayesian MCMC procedure. Efficiency components estimated by data augmentation. 45/78

Kumbhakar, Lien, Hardaker Technical Efficiency in Competing Panel Data Models: A Study of Norwegian Grain Farming, JPA, Published online, September, 2012. Three steps based on GLS: (1) RE/FGLS to estimate ( , ) (2) Decompose time varying residuals using Mo. M and SF. (3) Decompose estimates of time invariant residuals. 46/78

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Estimating Efficiency in the CSN Model 48/78

WHO Results: 2014 49/78

Computation of the GTRE Model is Actually Fast and Easy 247 Farms, 6 years. 100 Halton draws. Computation time: 35 seconds including computing efficiencies. 50/78

MSL Estimation 51/78

Why is the MSL method so computationally efficient compared to classical FIML and Bayesian MCMC for this model? Conditioned on the permanent effects, the group observations are independent. The joint conditional distribution is simple and easy to compute, in closed form. The full likelihood is obtained by integrating over only one dimension. (This was discovered by Butler and Moffitt in 1982. ) Neither of the other methods takes advantage of this result. Both integrate over T+1 dimensions. 52/78

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Equivalent Log Likelihood – Identical Outcome One Dimensional Integration over δi T+1 Dimensional Integration over Rei. 54/78

Simulated [over (w, h)] Log Likelihood Very Fast – with T=13, one minute or so 55/78

Also Simulated Log Likelihood GHK simulator is used to approximate the T+1 variate normal integrals. Very Slow – Huge amount of unnecessary computation. 56/78

Does the simulation chatter degrade the econometric efficiency of the MSL estimator? Hajivassiliou, V. , “Some practical issues in maximum simulated likelihood, ” Simulation-based Inference in Econometrics: Methods and Applications, Mariano, R. , Weeks, M. and Schuerman, T. , Cambridge University Press, 2008 Speculated that Asy. Var[estimator] = V + (1/R)C The contribution of the chatter would be of second or third order. R is typically in the hundreds or thousands. No other evidence on this subject. 57/78

An Experiment Pooled Spanish Dairy Farms Data Stochastic frontier using FIML. Random constant term linear regression with constant term equal to - |w|, w~ N[0, 1] This is equivalent to the stochastic frontier model. Maximum simulated likelihood 500 random draws for the simulation for the base case. Uses Mersenne Twister for the RNG 50 repetitions of estimation based on 500 random draws to suggest variation due to simulation chatter. 58/78

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Simulation Noise in Standard Errors of Coefficients Chatter. 00543. 00590. 00042. 00119 60/78

Is It Really Simulation? Halton or Sobol sequences Not random – far more stable than random draws, by a factor of about 10. There is no simulation chatter View the same as numerical quadrature There may be some approximation error. How would we know? 61/78

Sample Selection 62/78

TECHNICAL EFFICIENCY ANALYSIS CORRECTING FOR BIASES FROM OBSERVED AND UNOBSERVED VARIABLES: AN APPLICATION TO A NATURAL RESOURCE MANAGEMENT PROJECT Empirical Economics: Volume 43, Issue 1 (2012), Pages 55 -72 Boris Bravo-Ureta University of Connecticut Daniel Solis University of Miami William Greene New York University 63/78

The MARENA Program in Honduras Several programs have been implemented to address resource degradation while also seeking to improve productivity, managerial performance and reduce poverty (and in some cases make up for lack of public support). One such effort is the Programa Multifase de Manejo de Recursos Naturales en Cuencas Prioritarias or MARENA in Honduras focusing on small scale hillside farmers. 64/78

Expected Impact Evaluation 65/78

Methods A matched group of beneficiaries and control farmers is determined using Propensity Score Matching techniques to mitigate biases that would stem from selection on observed variables. In addition, we deal with possible self-selection on unobservables arising from unobserved variables using a selectivity correction model for stochastic frontiers introduced by Greene (2010). 66/78

A Sample Selected SF Model di = 1[ ′zi + hi > 0], hi ~ N[0, 12] yi = + ′xi + i, i ~ N[0, 2] (yi, xi) observed only when di = 1. i = v i - ui ui = u|Ui| where Ui ~ N[0, 12] vi = v. Vi where Vi ~ N[0, 12]. (hi, vi) ~ N 2[(0, 1), (1, v 2)] 67/78

Simulated log. L for the Standard SF Model This is simply a linear regression with a random constant term, αi = α - σu |Ui | 68/78

Likelihood For a Sample Selected SF Model 69/78

Simulated Log Likelihood for a Selectivity Corrected Stochastic Frontier Model The simulation is over the inefficiency term. 70/78

JLMS Estimator of ui 71/78

Closed Form for the Selection Model The selection model can be estimated without simulation “The stochastic frontier model with correction for sample selection revisited. ” Lai, Hung-pin. Forthcoming, JPA Based on closed skew normal distribution Similar to Maddala’s 1982 result for the linear selection model. See slide 42. Not more computationally efficient. Statistical properties identical. Suggested possibility that simulation chatter is an element of inefficiency in the maximum simulated likelihood estimator. 72/78

Closed Form vs. Simulation Spanish Dairy Farms: Selection based on being farm #1 -125. 6 periods The theory works. 73/78

Variables Used in the Analysis Production Participation 74/78

Findings from the First Wave 75/78

A Panel Data Model Selection takes place only at the baseline. There is no attrition. 76/78

Simulated Log Likelihood 77/78

Main Empirical Conclusions from Waves 0 and 1 p p 78/78 Benefit group is more efficient in both years The gap is wider in the second year Both means increase from year 0 to year 1 Both variances decline from year 0 to year 1

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Summary The skew normal distribution Two useful models for panel data (and one potentially useful model pending development) Extension of TRE model that allows both transient and persistent random variation and inefficiency Sample selection corrected stochastic frontier Spatial autocorrelation stochastic frontier model Methods: Maximum simulated likelihood as an alternative to received brute force methods 80/78 Simpler Faster Accurate Simulation “chatter” is a red herring – use Halton sequences