Stochastic Frontier Models Stochastic Frontier Model Part 3

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 1/49 Stochastic Frontier Models 0 1 2 3 4 5 6 7 8 Introduction Efficiency Measurement Frontier Functions Stochastic Frontiers Production and Cost Heterogeneity Model Extensions Panel Data Applications William Greene Stern School of Business New York University

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 2/49 Stochastic Frontier Models p Motivation: n n n p Factors not under control of the firm Measurement error Differential rates of adoption of technology Frontier is randomly placed by the whole collection of stochastic elements which might enter the model outside the control of the firm. p Aigner, Lovell, Schmidt (1977), Meeusen, van den Broeck (1977), Battese, Corra (1977)

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 3/49 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.

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 4/49 Least Squares Estimation Average inefficiency is embodied in the third moment of the disturbance εi = vi - ui. So long as E[vi - ui] is constant, the OLS estimates of the slope parameters of the frontier function are unbiased and consistent. (The constant term estimates α-E[ui]. The average inefficiency present in the distribution is reflected in the asymmetry of the distribution, which can be estimated using the OLS residuals:

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 5/49 Application to Spanish Dairy Farms N = 247 farms, T = 6 years (1993 -1998) Input Units Mean Std. Dev. Minimum Maximum Milk production (liters) 131, 108 92, 539 14, 110 727, 281 Cows # of milking cows 2. 12 11. 27 4. 5 82. 3 Labor # man-equivalent units 1. 67 0. 55 1. 0 4. 0 Land Hectares of land devoted to pasture and crops. 12. 99 6. 17 2. 0 45. 1 Feed Total amount of feedstuffs fed to dairy cows (tons) 57, 941 47, 981 3, 924. 1 4 376, 732

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Example: Dairy Farms 6/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] The Normal-Half Normal Model 7/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Normal-Half Normal Variable 8/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] The Skew Normal Variable 9/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Standard Form: The Skew Normal Distribution 10/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Battese Coelli Parameterization 11/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 12/49 Estimation: Least Squares/Mo. M OLS estimator of β is consistent p E[ui] = (2/π)1/2σu, so OLS constant estimates α+ (2/π)1/2σu p Second and third moments of OLS residuals estimate p p Use [a, b, m 2, m 3] to estimate [ , , u, v]

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 13/49 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.

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Airlines Data – 256 Observations 14/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Least Squares Regression 15/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 16/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Alternative Models: Half Normal and Exponential 17/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Normal-Exponential Likelihood 18/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Normal-Truncated Normal 19/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Truncated Normal Model: mu=. 5 20/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 21/49 Effect of Differing Truncation Points From Coelli, Frontier 4. 1 (page 16)

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 22/49 Other Models Other Parametric Models (we will examine several later in the course) p Semiparametric and nonparametric – the recent outer reaches of theoretical literature p Other variations including heterogeneity in the frontier function and in the distribution of inefficiency p

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 23/49 A Possible Problem with the Method of Moments p Estimator of σu is [m 3/-. 21801]1/3 p Theoretical m 3 is < 0 p Sample m 3 may be > 0. If so, no solution for σu. (Negative to 1/3 power. )

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 24/49 Now Include LM in the Production Model

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 25/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 26/49 Test for Inefficiency? Base test on u = 0 <=> = 0 p Standard test procedures p n n n p Likelihood ratio Wald Lagrange Nonstandard testing situation: n n Variance = 0 on the boundary of the parameter space Standard chi squared distribution does not apply.

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 27/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Estimating ui No direct estimate of ui p Data permit estimation of yi – β’xi. Can this be used? p n n n p εi = yi – β’xi = vi – ui Indirect estimate of ui, using E[ui|vi – ui] This is E[ui|yi, xi] vi – ui is estimable with ei = yi – b’xi. 28/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 29/49 Fundamental Tool - JLMS We can insert our maximum likelihood estimates of all parameters. Note: This estimates E[u|vi – ui], not ui.

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Other Distributions 30/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Technical Efficiency 31/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Application: Electricity Generation 32/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 33/49 Estimated Translog Production Frontiers

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Inefficiency Estimates 34/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Inefficiency Estimates 35/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 36/49 Estimated Inefficiency Distribution

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Estimated Efficiency 37/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Confidence Region Horrace, W. and Schmidt, P. , Confidence Intervals for Efficiency Estimates, JPA, 1996. 38/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 39/49 Application (Based on Electricity Costs)

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 40/49 A Semiparametric Approach Y = g(x, z) + v - u [Normal-Half Normal] p (1) Locally linear nonparametric regression estimates g(x, z) p (2) Use residuals from nonparametric regression to estimate variance parameters using MLE p (3) Use estimated variance parameters and residuals to estimate technical efficiency. p

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Airlines Application 41/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Efficiency Distributions 42/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Nonparametric Methods - DEA 43/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 44/49 DEA is done using linear programming

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Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 46/49 Methodological Problems with DEA Measurement error p Outliers p Specification errors p The overall problem with the deterministic frontier approach p

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 47/49 DEA and SFA: Same Answer? p Christensen and Greene data n n n p N=123 minus 6 tiny firms X = capital, labor, fuel Y = millions of KWH Cobb-Douglas Production Function vs. DEA

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] 48/49

Stochastic Frontier. Models Stochastic Frontier Model [Part 3] Comparing the Two Methods. 49/49