Transcript Modeling Consumer Decision Making and Discrete Choice Behavior

Stochastic FrontierModels
Model Extensions
[Part 6] 1/37
Stochastic Frontier Models
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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
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Model Extensions
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Model Extensions

Simulation Based Estimators


Normal-Gamma Frontier Model
Bayesian Estimation of Stochastic Frontiers
A Discrete Outcomes Frontier
 Similar Model Structures
 Similar Estimation Methodologies
 Similar Results

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Model Extensions
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Functional Forms
Normal-half normal and normal-exponential: Restrictive functional
forms for the inefficiency distribution
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Model Extensions
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Normal-Truncated Normal
More flexible. Inconvenient, sometimes ill
behaved log-likelihood function.
MU=-.5
MU=0
Exponential
MU=+.5
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Model Extensions
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Normal-Gamma
Very flexible model. VERY difficult log likelihood function.
Bayesians love it. Conjugate functional forms for other model parts
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Model Extensions
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Normal-Gamma Model
P
f u (u i ) 
u
P 1
(P)
exp(  u i /  u ) u i
, u i  0, P  0,  u  0
E [ u ]   u P , S tandard deviation =  u
Ln L (    v   u ) =

N
i 1
P
  P ln  u  ln  ( P )  ln q ( P  1,  i )



2
2
 - i   v /  u 
i .
 1 v 
  ln  
 +
+ 

2



v
u 
 u 



r
q ( r ,  i ) = E  z | z > 0,  i  ,
z ~ N[-i + v2/u, v2].
q(r,εi) is extremely difficult to compute
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Model Extensions
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Normal-Gamma Frontier Model
G a m m a Fro n tie r M o de l
D e te rm in istic Fro n tie r
y = x'  - u
P
f(u ) = [  /  (P )]e
 u
u
P 1
, u  0
S to ch a stic Fro n tie r
y = x'  + v - u = x'  + 
2
f(v) = N [0 ,  ]

Lo gL= N [Pln +   ln  (P )]   i 1   i  ln 

1
2
N
+  i= 1 ln
N
2


0
z

0
 z  i 

dz
   
2
,  i    i  
1  z  i 

dz
   
P 1

 i

     
 

1
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Model Extensions
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Simulating the Log Likelihood
L n L S (     v   u ) =

N
i 1
2

 -  i   v2 /  u 
i
1  v 
  P ln  u  ln  ( P )+ 

ln

+



2  u 
v
u




P 1 
Q
1
  ln  1




(
F

(1

F
)

(


/

)
  q 1  i

v
iq
iq
i
v 

Q



i = yi - ’xi,
i = -i - v2/u,
= v, and
PL = (-i/)
Fq is a draw from the continuous uniform(0,1) distribution.



.



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Model Extensions
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Application to C&G Data
This is the standard data set for developing and testing Exponential,
Gamma, and Bayesian estimators.
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Model Extensions
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Application to C&G Data
Descriptive Statistics for JLMS Estimates of E[u|e]
Based on Maximum Likelihood Estimates of Stochastic
Frontier Models
Model
Mean
Std.Dev.
Minimum
Maximum
Normal
.1188
.0609
.0298
.3786
Exponential
.0974
.0764
.0228
.5139
Gamma
.0820
.0799
.0149
.5294
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Model Extensions
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Inefficiency Estimates
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Model Extensions
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Tsionas Fourier Approach to Gamma
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Model Extensions
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A 3 Parameter Gamma Model
P
f u (u i ) 
c u
(P)
cP  1
exp(  u i /  u ) u i
c
, u i  0, P  0
P roduces several interesting cases:
c= 1: G am m a
c= 2, P = 1/2: H alf norm al (standard frontier)
P = 1: W eibull
(G riffin and S teel, JP A , 29,1, 2008)
E stim ated by B ayes ian (M C M C ) m ethods using
W inB U G S , JP A , 27,3, 2007.)
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Model Extensions
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Functional Form

Truncated normal



Has the advantage of a place to put the z’s
Strong functional disadvantage – discontinuity.
Difficult log likelihood to maximize
Rayleigh model
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
Parameter affects both mean and variance
Convenient model for heterogeneity
Much simpler to manipulate than gamma.
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Model Extensions
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Stochastic Frontiers with a Rayleigh Distribution
Gholamreza Hajargasht, Department of Economics, University of
Melbourne, 2013
 u2 
f ( u )  2 exp 
, u0
2 
u
 2u 
u
E [u ]   u

2
2  4   
V ar [ u ]   u 

 2 
 ui   u exp(  h i )
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Model Extensions
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y i   x i  v i  u i , v i ~ N [0,  vi ], u i ~ R ayleigh (  ui )
2
2
L og density of  i  v i  u i
 vi  ui
2
i 
2
2
 vi   ui
2
2
  ui  i
2
i 

2
vi

2
ui
log f (  i )  
,  
*
i
1
2
log 
i
i
2
vi
 log 
2
ui

1
2
log  i
+ log   i  (  i )   i  (  i )  
*
*
2
1
2
 i
*

2

1 (i )
2  vi
2
2
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Model Extensions
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Exponential
Gamma
Rayleigh
Half Normal
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Model Extensions
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Rayleigh vs. Half Normal
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Model Extensions
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Discrete Outcome Stochastic Frontier
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Model Extensions
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Model Extensions
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Model Extensions
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Chanchala Ganjay Gadge
CONTRIBUTIONS TO THE INFERENCE ON
STOCHASTIC FRONTIER MODELS
DEPARTMENT OF STATISTICS AND CENTER
FOR ADVANCED STUDIES,
UNIVERSITY OF PUNE
PUNE-411007, INDIA
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Model Extensions
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Model Extensions
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Model Extensions
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Model Extensions
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Bayesian Estimation
Short history – first developed post 1995
 Range of applications




Largely replicated existing classical methods
Recent applications have extended received
approaches
Common features of the applications
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Model Extensions
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Bayesian Formulation of SF Model
Normal – Exponential Model
N
L n L ( data ;     v   u ) =

i= 1
2




1
  ln  u +  v  
2 u 


 -(( v i  u i )   v2 /  u ) 
vi  u i

ln  
+



u
v










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Model Extensions
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Bayesian Approach
vi – ui = yi -  - ’xi.
Estimation proceeds (in principle) by specifying priors
over  = (,,v,u), then deriving inferences from
the joint posterior p(|data). In general, the joint
posterior for this model cannot be derived in closed
form, so direct analysis is not feasible. Using Gibbs
sampling, and known conditional posteriors, it is
possible use Markov Chain Monte Carlo (MCMC)
methods to sample from the marginal posteriors and
use that device to learn about the parameters and
inefficiencies. In particular, for the model parameters,
we are interested in estimating E[|data],
Var[|data] and, perhaps even more fully
characterizing the density f(|data).
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Model Extensions
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On Estimating Inefficiency
One might, ex post, estimate E[ui|data] however,
it is more natural in this setting to include
(u1,...,uN) with , and estimate the conditional
means with those of the other parameters. The
method is known as data augmentation.
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Model Extensions
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Priors over Parameters
D iffu se prio rs a re a ssu m e d fo r a ll o f th e se
p(  ,  )  U n ifo rm o ve r th e re a l "lin e " so p(..)= 1
p(1 /  v )  G a m m a(1 /  v | v , Pv )
P
=
v v
 (Pv )
e xp   v (1 /  v )  (1 /  v )
Pv  1
, 1 / v  0
P
p(  u )

v u
 (Pu )
e xp    u  u   u
Pv  1
, u  0.
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Model Extensions
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Priors for Inefficiencies
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Model Extensions
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Posterior
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Model Extensions
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Model Extensions
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Gibbs Sampling: Conditional Posteriors
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Model Extensions
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Bayesian Normal-Gamma Model

Tsionas (2002)
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



Erlang form – Integer P
“Random parameters”
Applied to C&G (Cross Section)
Average efficiency 0.999
River Huang (2004)


Fully general
Applied (as usual) to C&G
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Model Extensions
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Bayesian and Classical Results
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Model Extensions
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Methodological Comparison

Bayesian vs. Classical

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


Interpretation
Practical results: Bernstein – von Mises Theorem in the
presence of diffuse priors
Kim and Schmidt comparison (JPA, 2000)
Important difference – tight priors over ui in this context.
Conclusions


Not much change in existing results
Extensions to new models (e.g., 3 parameter gamma)