[part 5] 1/53 stochastic frontiermodels heterogeneity stochastic frontier models william greene...

[Part 5] 1/53

Stochastic FrontierModelsHeterogeneity

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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Where to Next? Heterogeneity: “Where do we put the z’s?”

Other variables that affect production and inefficiency Enter production frontier, inefficiency distribution, elsewhere?

Heteroscedasticity Another form of heterogeneity Production “risk”

Bayesian and simulation estimators The stochastic frontier model with gamma inefficiency Bayesian treatments of the stochastic frontier model

Panel Data Heterogeneity vs. Inefficiency – can we distinguish Model forms: Is inefficiency persistent through time?

Applications

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Swiss Railway Data

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Observable Heterogeneity

As opposed to unobservable heterogeneity

Observe: Y or C (outcome) and X or w (inputs or input prices)

Firm characteristics or environmental variables. Not production or cost, characterize the production process. Enter the production or cost function? Enter the inefficiency distribution? How?

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Shifting the Outcome Function

ln f( , ) ( , ) ( )x zit it it it ity g h t v u

Firm specific heterogeneity can also be incorporated into the inefficiency model as follows: This modifies the mean of the truncated normal distribution

yi = xi + vi - ui

vi ~ N[0,v2]

ui = |Ui| where Ui ~ N[i, u2],

i = 0 + 1zi,

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Heterogeneous Mean in Airline Cost Model

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Estimated Economic Efficiency

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How do the Zs affect inefficiency?

2

2

For a Normal-Half Normal Production Frontier Estimator

( )[ | ] ,

1 ( )

[ | ] [ | ]

[ | ]1 coefficient

1

i ii i i i

i

i i i i i

i

i i

E u

E u E u

z z

E u

z

= {+} coefficient

[ | ]For a Cost Frontier, and {-} coefficienti i i

i

E u

z

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Effect of Zs on Efficiency

2


( )[ | ] ,

1 ( )

[ | ]{+} coefficient

ˆ ˆe uˆ ˆ ˆUsing e = Exp(-u), Exp(-u) {-} coefficient

For a

i ii i i i

i

i i

E u

E u

z

z z

eCost Frontier, {+} coefficient

z

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Swiss Railroads Cost Function

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One Step or Two Step2 Step: 1. Fit Half or truncated normal model,

2. Compute JLMS ui, regress ui on zi

Airline EXAMPLE: Fit model without POINTS, LOADFACTOR, STAGE

1 Step: Include zi in the model, compute ui including zi

Airline example: Include 3 variables

Methodological issue: Left out variables in two step approach.

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One vs. Two Step

Efficiency computed without load factor, stage length and points served.

Efficiency computed with load factor, stage length and points served.

0.8 0.9 1.0

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Application: WHO Data

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Unobservable Heterogeneity Parameters vary across firms

Random variation (heterogeneity, not Bayesian) Variation partially explained by observable indicators

Continuous variation – random parameter models: Considered with panel data models

Latent class – discrete parameter variation

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A Latent Class Model

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Latent Class Efficiency Studies Battese and Coelli – growing in weather

“regimes” for Indonesian rice farmers Kumbhakar and Orea – cost structures for U.S.

Banks Greene (Health Economics, 2005) – revisits

WHO Year 2000 World Health Report Kumbhakar, Parmeter, Tsionas (JE, 2013) – U.S.

Banks.

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Latent Class Application

Estimates of Latent Class Model: Banking Data

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Inefficiency?

Not all agree with the presence (or identifiability) of “inefficiency” in market outcomes data.

Variation around the common production structure may all be nonsystematic and not controlled by management

Implication, no inefficiency: u = 0.

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Nursing Home Costs

44 Swiss nursing homes, 13 years Cost, Pk, Pl, output, two environmental

variables Estimate cost function Estimate inefficiency

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Estimated Cost Efficiency

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A Two Class Model

Class 1: With Inefficiency logC = f(output, input prices, environment) + vv + uu

Class 2: Without Inefficiency logC = f(output, input prices, environment) + vv

u = 0

Implement with a single zero restriction in a constrained (same cost function) two class model

Parameterization: λ = u /v = 0 in class 2.

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LogL= 464 with a common frontier model, 527 with two classes

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Heteroscedasticity in v and/or u

yi = ’xi + vi - ui

Var[vi | hi] = v2gv(hi,) = vi

2

gv(hi,0) = 1,

gv(hi,) = [exp(’hi)]2

Var[Ui | hi] = u2gu(hi,)= ui

2

gu(hi,0) = 1,

gu(hi,) = [exp(’hi)]2

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2

2 2


( )[ | ] ,

1 ( )

exp( )

exp( )

[ exp( )] [ exp( )]

Differentiation for the delta

i i i i ii i i i

i i i

u ii

v i

i v i u i

E u

h

h

h h

method is hopelessly complicated.

We do it numerically.

Heteroscedasticity Affects Inefficiency

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A “Scaling” Truncation Model

i i i

i

0

1

2

u ( , u * where f(u *) does not involve

Scales both mean and variance of u

Ln ( , , , , ) = -(N/2) ln 2 - ln + ln ( / ) +

1 ln

2

i i

N

i i uii

i i i

i i i

h

L

z z

1

2

exp( ),

exp( ),

exp( ),

/ ,

N i ii

i

i i

ui u i

vi v i

i ui vi

i vi

z

z

z

2ui

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Application: WHO Data

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Unobserved Endogenous Heterogeneity Cost = C(p,y,Q), Q = quality

Quality is unobserved Quality is endogenous – correlated with

unobservables that influence cost Econometric Response: There exists a proxy

that is also endogenous Omit the variable? Include the proxy?

Question: Bias in estimated inefficiency (not interested in coefficients)

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Simulation Experiment

Mutter, et al. (AHRQ), 2011 Analysis of California nursing home data Estimate model with a simulated data set Compare biases in sample average inefficiency

compared to the exogenous case Endogeneity is quantified in terms of correlation

of Q(i) with u(i)

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A Simulation Experiment

Mean Inefficiency vs. Gamma

GAM M A

.216

.283

.350

.417

.484

.150

.20 .40 .60 .80 1.00.00

QS_ INCL QS_ EXCL

Ine

fficie

ncy

Conclusion: Omitted variable problem does not make the bias worse.

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Sample Selection Modeling

Switching Models: y*|technology = bt’x + v –u Firm chooses technology = 0 or 1

based on c’z+e e is correlated with v

Sample Selection Model: Choice of organic or inorganic Adoption of some technological innovation

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Early Applications

Heshmati A. (1997), “Estimating Panel Models with Selectivity Bias: An Application to Swedish Agriculture”, International Review of Economics and Business 44(4), 893-924.

Heshmati, Kumbhakar and Hjalmarsson Estimating Technical Efficiency, Productivity Growth and Selectivity Bias Using Rotating Panel Data: An Application to Swedish Agriculture

Sanzidur Rahman Manchester WP, 2002: Resource use efficiency with self-selectivity: an application of a switching regression framework to stochastic frontier models:

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Sample Selection in Stochastic Frontier Estimation

• Bradford et al. (ReStat, 2000):“... the patients in this sample were not randomly assigned to each treatment group. Statistically, this implies that the data are subject to sample selection bias. Therefore, we utilize a standard Heckman two-stage sample-selection process, creating an inverse Mill’s ratio from a first-stage probit estimator of the likelihood of CABG or PTCA. This correction variable is included in the frontier estimate....”

• Sipiläinen and Oude Lansink (2005) “Possible selection bias between organic and conventional production can be taken into account [by] applying Heckman’s (1979) two step procedure.”

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Two Step Selection

• Heckman’s method is for linear equations• Does not carry over to any nonlinear model• The formal estimation procedure based on

maximum likelihood estimation– Terza (1998) – general results for exponential models

with extensions to other nonlinear models– Greene (2006) – general template for nonlinear models– Greene (2010) – specific result for stochastic frontiers

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A Sample Selected SF Model

di = 1[ z′ i + wi > 0], wi ~ N[0,12]

yi = x′ i + i, i ~ N[0,2]

(yi,xi) observed only when di = 1.

i = vi - ui

ui = |uUi| = u |Ui| where Ui ~ N[0,12]

vi = vVi where Vi ~ N[0,12].

(wi,vi) ~ N2[(0,1), (1, v, v2)]

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Alternative ApproachKumbhakar, Sipilainen, Tsionas (JPA, 2008)

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Sample Selected SF Model

2 212

2

exp ( | |) / )

2( | ,| |, , )

( | |) /

1

(1 ) ( )

i i u i v

v

i i i i i i

i i u i i

i i

y x U

f y U d dy x U

d

x zz

z

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Simulated Log Likelihood for a Stochastic Frontier Model

2 212

1 12

exp ( | |) / )

2

1 ( | |) /log ( , , , , ) log 1

(1 ) ( )

i i u ir v

i

v

N Ri i u ir iS u v i r

i i

y Ud

y ULR

d

x

x z

z

The simulation is over the inefficiency term.

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2nd Step of the MSL Approach

2 212

, 1 1 2

exp ( | |) / )

2

( | |) /1log ( , , , ) log

1

(1 ) ( )

ˆwhere =

i i u ir v

i

v

N R i i u ir v iS C u v i r

i i

i i

y Ud

y U aL

R

d a

a

x

x

z

2 212

, 1 1

2

exp ( | |) / )

21log ( , , , ) log

( | |) /

1

i

i i u ir v

vR

S C u v d r

i i u ir v i

y U

LR y U a

x

x

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JLMS Estimator of ui

2 212

1

2

1

1

ˆ ˆ ˆexp ( | |) / )

ˆ 21ˆˆˆ ˆ ˆ( | |) /

ˆ1

1 ˆ

1 ˆˆ ˆ( | |)

ˆEstimator of E[u | ]

ˆ

A method that does not

x

x

i i u ir v

R vi r

i i u ir v i

R

irr

R

i u ir irr

ii i

i

y U

BR y U a

fR

A U fR

A

B

require simulation: Hung-pin Lai (Academica Sinica)

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WHO Efficiency Estimates

OECD

Everyone Else

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WHO Estimates vs. SF Model

Efficiencies from Selection SF Model vs. WHO Estimates

EFFSEL

.20

.40

.60

.80

1.00

.00.700 .750 .800 .850 .900 .950 1.000.650

WH

OE

FF

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Sample Selection in a Stochastic Frontier Model

TECHNICAL EFFICIENCY ANALYSIS CORRECTING FOR BIASES FROM OBSERVED AND UNOBSERVED

VARIABLES: AN APPLICATION TO A NATURAL RESOURCE MANAGEMENT PROJECT

Boris Bravo-Ureta

University of Connecticut

Daniel Solis

University of Miami

William Greene

Stern School of Business

New York University

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Component II - Module 3 focused on promoting investments in sustainable production systems with a budget of US $7.6 million (Bravo-Ureta, 2009).

The major activities undertaken with beneficiaries: training in business management and sustainable farming practices; and the provision of funds to co-finance investment activities through local rural savings associations (cajas rurales).

Component II - Module 3

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[part 5] 1/53 stochastic frontiermodels heterogeneity stochastic frontier models william greene...

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