stochastic frontier models

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Stochastic FrontierModels

Stochastic Frontier Model

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 Applications

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Stochastic Frontier Models Motivation:

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.

Aigner, Lovell, Schmidt (1977), Meeusen, van den Broeck (1977), Battese, Corra (1977)

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The Stochastic Frontier Model ( )

ln +

= + .

iviii

i i ii

i i

= fy eTE = + v uy

+

x

x

x

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.

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Least Squares EstimationAverage 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:

3

1

1 ˆˆ( - [ ])N

N

3 i ii

= Em

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Application to Spanish Dairy Farms

Input Units Mean Std. 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 4.0

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 3,924.14

376,732

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

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Example: Dairy Farms

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The Normal-Half Normal Model

2

2

ln

1Normal component: ~ [0, ]; ( ) , .

Half normal component: | |, ~ [0, ]

1 Underlying normal: ( ) ,

Half

i i i i

i i

ii v i i

v v

i i i u

ii i

u u

y v u

vv N f v v

u U U N

Uf U v

xx

1 1normal ( ) ,0(0)

ii i

u u

uf u v

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Normal-Half Normal Variable

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The Skew Normal Variable

2

2

2 2

| | where ~ [0,1]

2 2[ ] ; [ ]

[( 2) / ][ ][ ] [( 2) / ]

u

u u

u

v u

u U U N

E u Var u

Var uVar

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Standard Form: The Skew Normal Distribution

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Battese Coelli Parameterization2 2

2 2

2 22 2 2

2 2 2

, ~ [0, ], ~ [0, ]Aigner, Lovell, Schmidt

0, = 0

Coelli, Battese and Coelli

0 1, 0; = 1

v u

uv u

v

uv u

v u

v u v N u N

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Estimation: Least Squares/MoM OLS estimator of β is consistent E[ui] = (2/π)1/2σu, so OLS constant estimates

α+ (2/π)1/2σu

Second and third moments of OLS residuals estimate

Use [a,b,m2,m3] to estimate [,,u, v]

and 0 2 2 3

2 u v 3 u- 2 2 4 = + = 1 - m m

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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.

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Airlines Data – 256 Observations

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Least Squares Regression

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Alternative Models:Half Normal and Exponential

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Normal-Exponential Likelihood

2 2n

ui=1

Ln ( ; ) =

(( ) / ( )1-ln ln2

v u

u i i v u i i

v v u

L data

v u v u

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Normal-Truncated Normal2

2

2

2

1

2

~ [0, ]

~ [ , ], | | Nonzero mean for

log log log2 2log2

1 log2

where 1

i v

i u i i

i

u

N i ii

u

v N

U N u UU

NL

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Truncated Normal Model: mu=.5

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Effect of Differing Truncation Points

From Coelli, Frontier4.1 (page 16)

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Other Models Other Parametric Models (we will examine

several later in the course) Semiparametric and nonparametric – the recent

outer reaches of the theoretical literature Other variations including heterogeneity in the

frontier function and in the distribution of inefficiency

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A Possible Problem with theMethod of Moments

Estimator of σu is [m3/-.21801]1/3

Theoretical m3 is < 0

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

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Now Include LM in the Production Model

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Test for Inefficiency? Base test on u = 0 <=> = 0 Standard test procedures

Likelihood ratio Wald Lagrange

Nonstandard testing situation: Variance = 0 on the boundary of the parameter

space Standard chi squared distribution does not apply.

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Estimating ui

No direct estimate of ui

Data 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’xi.

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Fundamental Tool - JLMS

2

( )[ | ] , 1 ( )

i ii i i i

i

E u

We can insert our maximum likelihood estimates of all parameters.Note: This estimates E[u|vi – ui], not ui.

2

ˆ ˆˆ ˆˆ ( ) ( )ˆ ˆ ˆˆ[ | ] , ˆ ˆ ˆ( )1i i i

i i i ii

yE u

x

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Other Distributions

2 2

2

2

( / )| = + , = - /

( / )

i u vi

vii it i v i i v u

vi

zE u z z

z

For the Normal- Truncated Normal Model

For the Normal-Exponential Model

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Technical Efficiency

* 2** * *

**

2 2* 2 2 2 u v

i u * 2

[( / ) ][exp( ) | ] exp

[( / )] 2

where = + / and

ii i i

i

i

E u

For the Normal- Truncated Normal Model

For the normal-half normal model, = 0.

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Application: Electricity Generation

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Estimated Translog Production Frontiers

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Inefficiency Estimates

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Inefficiency Estimates

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Estimated Inefficiency Distribution

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Estimated Efficiency

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Confidence Region

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

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Application (Based on Electricity Costs)

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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.

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Airlines Application

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Efficiency Distributions

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Nonparametric Methods - DEA

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DEA is done using linear programming

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Methodological Problems with DEA Measurement error Outliers Specification errors The overall problem with the

deterministic frontier approach

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DEA and SFA: Same Answer?

Christensen and Greene data N=123 minus 6 tiny firms X = capital, labor, fuel Y = millions of KWH

Cobb-Douglas Production Function vs. DEA

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Comparing the Two Methods.

stochastic frontier models

Documents

ols estimates

skew normal distribution

ols constant estimates

skewness of ols residuals

moments of ols residuals

symmetric distribution

evidence of inefficiency

evi ui