Date post: | 18-Dec-2015 |
Category: |
Documents |
Upload: | derick-nicholson |
View: | 219 times |
Download: | 3 times |
[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
[Part 3] 2/49
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)
[Part 3] 3/49
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.
[Part 3] 4/49
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
[Part 3] 5/49
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)
[Part 3] 6/49
Stochastic FrontierModels
Stochastic Frontier Model
Example: Dairy Farms
[Part 3] 7/49
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
[Part 3] 8/49
Stochastic FrontierModels
Stochastic Frontier Model
Normal-Half Normal Variable
[Part 3] 9/49
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
[Part 3] 10/49
Stochastic FrontierModels
Stochastic Frontier Model
Standard Form: The Skew Normal Distribution
[Part 3] 11/49
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
[Part 3] 12/49
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
[Part 3] 13/49
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.
[Part 3] 14/49
Stochastic FrontierModels
Stochastic Frontier Model
Airlines Data – 256 Observations
[Part 3] 15/49
Stochastic FrontierModels
Stochastic Frontier Model
Least Squares Regression
[Part 3] 16/49
Stochastic FrontierModels
Stochastic Frontier Model
[Part 3] 17/49
Stochastic FrontierModels
Stochastic Frontier Model
Alternative Models:Half Normal and Exponential
[Part 3] 18/49
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
[Part 3] 19/49
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
[Part 3] 20/49
Stochastic FrontierModels
Stochastic Frontier Model
Truncated Normal Model: mu=.5
[Part 3] 21/49
Stochastic FrontierModels
Stochastic Frontier Model
Effect of Differing Truncation Points
From Coelli, Frontier4.1 (page 16)
[Part 3] 22/49
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
[Part 3] 23/49
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.)
[Part 3] 24/49
Stochastic FrontierModels
Stochastic Frontier Model
Now Include LM in the Production Model
[Part 3] 25/49
Stochastic FrontierModels
Stochastic Frontier Model
[Part 3] 26/49
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.
[Part 3] 27/49
Stochastic FrontierModels
Stochastic Frontier Model
[Part 3] 28/49
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.
[Part 3] 29/49
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
[Part 3] 30/49
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
[Part 3] 31/49
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.
[Part 3] 32/49
Stochastic FrontierModels
Stochastic Frontier Model
Application: Electricity Generation
[Part 3] 33/49
Stochastic FrontierModels
Stochastic Frontier Model
Estimated Translog Production Frontiers
[Part 3] 34/49
Stochastic FrontierModels
Stochastic Frontier Model
Inefficiency Estimates
[Part 3] 35/49
Stochastic FrontierModels
Stochastic Frontier Model
Inefficiency Estimates
[Part 3] 36/49
Stochastic FrontierModels
Stochastic Frontier Model
Estimated Inefficiency Distribution
[Part 3] 37/49
Stochastic FrontierModels
Stochastic Frontier Model
Estimated Efficiency
[Part 3] 38/49
Stochastic FrontierModels
Stochastic Frontier Model
Confidence Region
Horrace, W. and Schmidt, P., Confidence Intervals for Efficiency Estimates, JPA, 1996.
[Part 3] 39/49
Stochastic FrontierModels
Stochastic Frontier Model
Application (Based on Electricity Costs)
[Part 3] 40/49
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.
[Part 3] 41/49
Stochastic FrontierModels
Stochastic Frontier Model
Airlines Application
[Part 3] 42/49
Stochastic FrontierModels
Stochastic Frontier Model
Efficiency Distributions
[Part 3] 43/49
Stochastic FrontierModels
Stochastic Frontier Model
Nonparametric Methods - DEA
[Part 3] 44/49
Stochastic FrontierModels
Stochastic Frontier Model
DEA is done using linear programming
[Part 3] 45/49
Stochastic FrontierModels
Stochastic Frontier Model
[Part 3] 46/49
Stochastic FrontierModels
Stochastic Frontier Model
Methodological Problems with DEA
Measurement error Outliers Specification errors The overall problem with the
deterministic frontier approach
[Part 3] 47/49
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
[Part 3] 48/49
Stochastic FrontierModels
Stochastic Frontier Model
[Part 3] 49/49
Stochastic FrontierModels
Stochastic Frontier Model
Comparing the Two Methods.