Stochastic Frontier Models - PowerPoint Presentation

Slide1

Stochastic Frontier Models

William GreeneStern School of BusinessNew York University

0 Introduction1 Efficiency Measurement2 Frontier Functions3 Stochastic Frontiers4 Production and Cost5 Heterogeneity6 Model Extensions7 Panel Data8 ApplicationsSlide2

Stochastic Frontier Models

Motivation:Factors not under control of the firmMeasurement errorDifferential rates of adoption of technologyFrontier

is randomly placed by the whole collection of stochastic elements which might enter the model outside the control of the firm. Aigner, Lovell, Schmidt (1977), Meeusen, van den Broeck (1977), Battese, Corra (1977)Slide3

The Stochastic Frontier Model

u

i > 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.Slide4

Least Squares Estimation

Average inefficiency is embodied in the third moment of the disturbance εi = vi - u

i. 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:Slide5

Application to Spanish Dairy Farms

Input

Units

MeanStd. Dev.

Minimum

Maximum

Milk

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 4.0LandHectares of land devoted to pasture and crops. 12.99 6.17 2.0 45.1FeedTotal amount of feedstuffs fed to dairy cows (tons) 57,94147,9813,924.14 376,732

N = 247 farms, T = 6 years (1993-1998)Slide6

Example: Dairy FarmsSlide7

The Normal-Half Normal ModelSlide8

Normal-Half Normal VariableSlide9

The Skew Normal VariableSlide10

Standard Form: The Skew Normal DistributionSlide11

Battese Coelli ParameterizationSlide12

Estimation: Least Squares/MoM

OLS estimator of β is consistentE[ui] = (2/π)1/2

σu, so OLS constant estimates α+ (2/π)1/2σuSecond and third moments of OLS residuals estimateUse [a,b,m2,m3] to estimate [,,u, v]Slide13

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.Slide14

Airlines Data – 256 ObservationsSlide15

Least Squares RegressionSlide16
Slide17

Alternative Models:Half Normal and ExponentialSlide18

Normal-Exponential LikelihoodSlide19

Normal-Truncated NormalSlide20

Truncated Normal Model: mu=.5Slide21

Effect of Differing Truncation Points

From Coelli, Frontier4.1 (page 16)Slide22

Other Models

Other Parametric Models (we will examine several later in the course)Semiparametric and nonparametric – the recent outer reaches of the theoretical literatureOther variations including heterogeneity in the frontier function and in the distribution of inefficiencySlide23

A Possible Problem with theMethod of Moments

Estimator of σu is [m3/-.21801]1/3

Theoretical m3 is < 0Sample m3 may be > 0. If so, no solution for σu . (Negative to 1/3 power.)Slide24

Now Include LM in the Production ModelSlide25
Slide26

Test for Inefficiency?

Base test on u = 0 <=>  = 0Standard test proceduresLikelihood ratioWaldLagrange

Nonstandard testing situation: Variance = 0 on the boundary of the parameter spaceStandard chi squared distribution does not apply.Slide27
Slide28

Estimating ui

No direct estimate of uiData 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]This is E[ui|yi, xi]vi – ui is estimable with ei = yi – b’x

i.Slide29

Fundamental Tool - JLMS

We can insert our maximum likelihood estimates of all parameters.

Note: This estimates E[u|vi – ui], not ui.Slide30

Other Distributions

Slide31

Technical EfficiencySlide32

Application: Electricity GenerationSlide33

Estimated Translog Production FrontiersSlide34

Inefficiency EstimatesSlide35

Inefficiency EstimatesSlide36

Estimated Inefficiency DistributionSlide37

Estimated EfficiencySlide38

Confidence Region

Horrace, W. and Schmidt, P., Confidence Intervals for Efficiency Estimates, JPA, 1996.Slide39

Application (Based on Electricity Costs)Slide40

A Semiparametric Approach

Y = g(x,z) + v - u [Normal-Half Normal](1) Locally linear nonparametric regression estimates g(x,z)(2) Use residuals from nonparametric regression to estimate variance parameters using MLE(3) Use estimated variance parameters and residuals to estimate technical efficiency.Slide41

Airlines ApplicationSlide42

Efficiency DistributionsSlide43

Nonparametric Methods - DEASlide44

DEA is done using linear programmingSlide45
Slide46

Methodological Problems with DEA

Measurement errorOutliersSpecification errorsThe overall problem with the deterministic frontier approachSlide47

DEA and SFA: Same Answer?

Christensen and Greene dataN=123 minus 6 tiny firmsX = capital, labor, fuelY = millions of KWHCobb-Douglas Production Function vs. DEASlide48
Slide49

Comparing the Two Methods.