Stochastic Frontier Models

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Stochastic FrontierModels

Model Extensions

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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Model Extensions

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

Functional Forms Normal-half normal and normal-exponential: Restrictive functional

forms for the inefficiency distribution

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Model Extensions

Normal-Truncated NormalMore flexible. Inconvenient, sometimes ill behaved log-likelihood function.

MU=-.5

MU=+.5

MU=0

Exponential

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Model Extensions

Normal-GammaVery flexible model. VERY difficult log likelihood function.Bayesians love it. Conjugate functional forms for other model parts

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Model Extensions

Normal-Gamma Model1( ) exp( / ) , 0, 0, 0

( )

[ ] , Standard deviation =

PPu

u i i u i i u

u u

f u u u u PP

E u P P

2 21

ln ln ( ) ln ( 1, )

Ln ( ) = .- /1+ ln + 2

u iN

v u i v i v u i

u v u

P P q P

L

i( , ) | > 0, ,riq r = E z z z ~ N[-i + v

2/u, v2].

q(r,εi) is extremely difficult to compute

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Model Extensions

Normal-Gamma Frontier Model

P u P 1

2

21i2

Gamma Frontier ModelDeterministic Frontier y = x' - u f(u) = [ / (P)]e u , u 0Stochastic Frontier y = x' + v - u = x' + f(v) = N[0, ]

LogL=N[Pln + ln (P)] ln

N ii 1

P 1 i0N 2

i ii=1i

0

z1z dz + ln , z1 dz

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Model Extensions

Simulating the Log Likelihood

2 2

111

1

- /1ln ln ( )+ ln + 2

Ln ( ) = .1ln ( (1 ) ( / )

v i v u iu

N u v uS v u i

PQi v iq iq i vq

P PL

F FQ

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

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

Application to C&G Data

Model Mean Std.Dev. Minimum MaximumNormal .1188 .0609 .0298 .3786

Exponential .0974 .0764 .0228 .5139Gamma .0820 .0799 .0149 .5294

Descriptive Statistics for JLMS Estimates of E[u|e] Based on Maximum Likelihood Estimates of Stochastic Frontier Models

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Model Extensions

Inefficiency Estimates

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Model Extensions

Tsionas Fourier Approach to Gamma

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Model Extensions

A 3 Parameter Gamma Model1( ) exp( / ) , 0, 0

( )Produces several interesting cases:c=1: Gammac=2, P=1/2: Half normal (standard frontier)P=1: Weibull(Griffin and Steel, JPA, 29,1, 2008)Estimated by Bayes

Pc cPu

u i i u i icf u u u u P

P

ian (MCMC) methods usingWinBUGS, JPA, 27,3, 2007.)

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Model Extensions

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

Parameter affects both mean and variance Convenient model for heterogeneity Much simpler to manipulate than gamma.

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Model Extensions

2

2 2

2

( ) exp , 02

[ ]2

4[ ]2

exp( )

u u

u

u

ui u

u uf u u

E u

Var u

ih

Stochastic Frontiers with a Rayleigh DistributionGholamreza Hajargasht, Department of Economics, University of Melbourne, 2013

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Model Extensions

2 2

2 22

2 2

2*

2 2

2 2 2

* *

, ~ [0, ], ~ ( )Log density of

,

1 1log ( ) log log log2 2

1 +log ( ) ( )2

i i i i i vi i ui

i i i

vi uii

vi ui

ui i ii i

vi ui i

i vi ui i

i i i i i

y v u v N u Rayleighv u

f

x

22*

2

( )12

i

vi

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Model Extensions

Rayleigh

Half Normal

Exponential

Gamma

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Model Extensions

Rayleigh vs. Half Normal

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Model Extensions

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Model Extensions

Spatial Autoregression in a Linear Model

2

1

1 1

1

2 -1

+ . E[ | ] Var[ | ]=

[ ] ( ) [ ] [ ]E[ | ] [ ]Var[ | ] [( ) ( )]Estimators: Various f

y = Wy Xβ εε X =0, ε X I

y = I W Xβ ε= I W Xβ I W ε

y X = I W Xβy X = I W I W

orms of generalized least squares. Maximum likelihood | ~ Normal[ , ]ε 0

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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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Model Extensions

1

Yet to be developed: True stochastic frontier with efficiency spillovers.

1,..., ; 1,...,

nit i it it it ij jtj

y v u w u

i n t T

x

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Model Extensions

Discrete Outcome Stochastic Frontier

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Model Extensions

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Model Extensions

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Model Extensions

Chanchala Ganjay GadgeCONTRIBUTIONS 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

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

Bayesian Formulation of SF Model

2

N

2i=1

1ln +2

Ln ( ; ) = -(( ) / )ln +

vu

uv u

i i v u i i

v u

L datav u v u

Normal – Exponential Model

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Model Extensions

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

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

Priors over Parameters

v v

P 1

u

Diffuse priors are assumed for all of thesep( , ) Uniform over the real "line" so p(..)=1p(1/ ) Gamma(1/ | ,P )

= exp (1/ ) (1/ ) , 1/ 0(P )

p( ) exp( )

vv

u

v vPv

v v v vvPv

u uuP

1, 0.vPu u

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Model Extensions

Priors for Inefficiencies

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Model Extensions

Posterior

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Model Extensions

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Model Extensions

Gibbs Sampling: Conditional Posteriors

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Model Extensions

Bayesian Normal-Gamma Model Tsionas (2002)

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

Bayesian and Classical Results

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Model Extensions

Methodological Comparison Bayesian vs. Classical

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)

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Stochastic Frontier Models