[part 3] 1/49 stochastic frontiermodels stochastic frontier model stochastic frontier models william...

[Part 3] 1/49

Stochastic FrontierModels

Stochastic Frontier Model

Stochastic Frontier ModelsWilliam Greene

Stern School of Business

New York University

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

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Stochastic FrontierModels

Stochastic Frontier Model

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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Stochastic FrontierModels

Stochastic Frontier Model

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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Stochastic FrontierModels

Stochastic Frontier Model

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:

3

1

1 ˆˆ( - [ ])N

N

3 i ii

= Em

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Stochastic FrontierModels

Stochastic Frontier Model

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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Stochastic FrontierModels

Stochastic Frontier Model

Example: Dairy Farms

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Stochastic FrontierModels

Stochastic Frontier Model

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

x

x

1 1normal ( ) ,0

(0)i

i iu u

uf u v

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Stochastic FrontierModels

Stochastic Frontier Model

Normal-Half Normal Variable

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Stochastic FrontierModels

Stochastic Frontier Model

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 u

Var

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Stochastic FrontierModels

Stochastic Frontier Model

Standard Form: The Skew Normal Distribution

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Stochastic FrontierModels

Stochastic Frontier Model

Battese Coelli Parameterization

2 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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Stochastic FrontierModels

Stochastic Frontier Model

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 32 u v 3 u

- 2 2 4 = + = 1 - m m

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Stochastic FrontierModels

Stochastic Frontier Model

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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Stochastic FrontierModels

Stochastic Frontier Model

Airlines Data – 256 Observations

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Stochastic FrontierModels

Stochastic Frontier Model

Least Squares Regression

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Stochastic FrontierModels

Stochastic Frontier Model

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Stochastic FrontierModels

Stochastic Frontier Model

Alternative Models:Half Normal and Exponential

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Stochastic FrontierModels

Stochastic Frontier Model

Normal-Exponential Likelihood

2 2n

ui=1

Ln ( ; ) =

(( ) / ( )1-ln ln

2

v u

u i i v u i i

v v u

L data

v u v u

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Stochastic FrontierModels

Stochastic Frontier Model

Normal-Truncated Normal2

2

2

2

1

2

~ [0, ]

~ [ , ], | |

Nonzero mean for

log log log2 2log2

1 log

2

where 1

i v

i u i i

i

u

N i i

i

u

v N

U N u U

U

NL

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Stochastic FrontierModels

Stochastic Frontier Model

Truncated Normal Model: mu=.5

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Stochastic FrontierModels

Stochastic Frontier Model

Effect of Differing Truncation Points

From Coelli, Frontier4.1 (page 16)

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Stochastic FrontierModels

Stochastic Frontier Model

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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Stochastic FrontierModels

Stochastic Frontier Model

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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Stochastic FrontierModels

Stochastic Frontier Model

Now Include LM in the Production Model

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Stochastic FrontierModels

Stochastic Frontier Model

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Stochastic FrontierModels

Stochastic Frontier Model

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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Stochastic FrontierModels

Stochastic Frontier Model

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Stochastic FrontierModels

Stochastic Frontier Model

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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Stochastic FrontierModels

Stochastic Frontier Model

Fundamental Tool - JLMS

2

( )[ | ] ,

1 ( )i i

i i i ii

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( )1

i i ii i i i

i

yE u

x

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Stochastic FrontierModels

Stochastic Frontier Model

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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Stochastic FrontierModels

Stochastic Frontier Model

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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Stochastic FrontierModels

Stochastic Frontier Model

Application: Electricity Generation

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Stochastic FrontierModels

Stochastic Frontier Model

Estimated Translog Production Frontiers

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Stochastic FrontierModels

Stochastic Frontier Model

Inefficiency Estimates

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Stochastic FrontierModels

Stochastic Frontier Model

Inefficiency Estimates

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Stochastic FrontierModels

Stochastic Frontier Model

Estimated Inefficiency Distribution

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Stochastic FrontierModels

Stochastic Frontier Model

Estimated Efficiency

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Stochastic FrontierModels

Stochastic Frontier Model

Confidence Region

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

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Stochastic FrontierModels

Stochastic Frontier Model

Application (Based on Electricity Costs)

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Stochastic FrontierModels

Stochastic Frontier Model

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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Stochastic FrontierModels

Stochastic Frontier Model

Airlines Application

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Stochastic FrontierModels

Stochastic Frontier Model

Efficiency Distributions

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Stochastic FrontierModels

Stochastic Frontier Model

Nonparametric Methods - DEA

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Stochastic FrontierModels

Stochastic Frontier Model

DEA is done using linear programming

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Stochastic FrontierModels

Stochastic Frontier Model

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Stochastic FrontierModels

Stochastic Frontier Model

Methodological Problems with DEA

Measurement error Outliers Specification errors The overall problem with the

deterministic frontier approach

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Stochastic FrontierModels

Stochastic Frontier Model

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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Stochastic FrontierModels

Stochastic Frontier Model

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Stochastic FrontierModels

Stochastic Frontier Model

Comparing the Two Methods.