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Stochastic FrontierModels

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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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Main Issues in Panel Data Modeling

Issues Capturing time invariant effects Dealing with time variation in inefficiency Separating heterogeneity from Inefficiency Examining technical change and total factor

productivity growth Contrasts – Panel Data vs. Cross Section

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Technical Change Technical Change

LnOutputit = f(xit,zi,,t) + vit - uit.

LnCostit = c(xit,zi,,t) + vit + uit. Independent of other factors, TC = f(..)/t Change in output not explained by change in factors

or environment – shift in production or cost function Time shift the goal function.

Lnyit = xit + zi + t + vit - uit.

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Familiar RE and FE Models

Wisdom from the linear model FE: y(i,t) = f[x(i,t)] + a(i) + e(i,t)

What does a(i) capture? Nonorthogonality of a(i) and x(i,t) The LSDV estimator

RE: y(i,t) = f[x(i,t)] + u(i) + e(i,t) How does u(i) differ from a(i)? Generalized least squares and maximum likelihood

What are the time invariant effects?

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The Cross Section Departure Point: 1977

2

2

2

Aigner et al. (ALS) Stochastic Frontier Model

~ [0, ]

| | and ~ [0, ]

Jondrow et al. (JLMS) Inefficiency Estimator

( )ˆ [ | ]

1 ( )

,

i i i i

i v

i i i u

ii i i i

i

ui i i

y v u

v N

u U U N

u E u

v u

x

2 2, ,

iv u i

v

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A Frontier Model for Panel Data

y(i,t) = β’x(i,t) – u(i) + v(i,t) Effects model with time invariant inefficiency Same dichotomy between FE and RE –

correlation with x(i,t). FE case is completely unlike the assumption in the

cross section case

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The Panel Data Models Appear: 1981

2 2

1

Pitt and Lee Random Effects Approach: 1981

~ [0, ], | | and ~ [0, ]

Counterpart to Jondrow et al. (1982)

( / )ˆ [ | ,..., ]

1 ( / )

=

it it it i

it v i i i u

it it i

ii i i iT i

i

ii

y v u

v N u U U N

v u

u E u

T

x

2

2 ,

1 1

u u

vT T

Time fixed

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Estimating Technical Efficiency

2* 2 2

*2

1, , (1 )( ),

1u

i i i i i i vv iT

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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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Rayleigh vs. Half NormalSwiss Railway Data

2 2 22 *

2 2 2 2

* 2 2 *

* *

Technical Efficiency Estimator in the Rayleigh Model

, ,

( ) ( ) ( )

( ) ( )

u u v ii i i i

v i u v u

i i i ii

i i i

TT

TE

Rayleigh

Half Normal

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Reinterpreting the Within Estimator: 1984

2

Schmidt and Sickles Fixed Effects Approach: 1984

~ [0, ],

.

Counterpart to Jondrow et al. (198

it i it it

it v i

y v

v N

x

semiparametically specified

fixed mean, constant variance

2)

ˆ ˆˆ max ( )

(The cost of the semiparametric specification is the

location of the inefficiency distribution. The authors

also revisit Pitt and Lee to demonstrate.)

i i i iu Time fixed

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Schmidt and Sickles FE Model

lnyit = + β’xit + ai + vit

estimated by least squares (‘within’)ˆ ˆˆ > 0 (for production or profit)

ˆ ˆˆ - > 0 (for cost or distance)

Implications: One firm is perfectly efficient.

Either deterministic frontier, or firm

i j j i

i i j j

u =max( a )- a

u =a min( a )

s

are compared to other firms,

not an absolute standard of zero.

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Misgivings About Time Fixed Inefficiency: 1990-

20 1 2

2 1

Cornwell Schmidt and Sickles (1990)

ˆ ˆˆ, u max( )

Kumbhakar (1990)

[1 exp( )] | |

Battese and Coelli (1992, 1995)

exp[ ( )] | |, exp[ ( , , )] | |

Cuesta (

it i i i it it it

it i

it i it it i

t t

u bt ct U

u t T U u g t T U

z

2000)

exp[ ( )] | |, exp[ ( , , )] | |it i i it i it iu t T U u g t T U z

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Battese and Coelli Models

it

i

2

In half or truncated normal model

u(i,t) = |U(i)| exp[g(z , t, )]

U(i) ~ normal, mean =

variance =

Commonly used formulation. g(...) is flexible.

Time invariant

ui

random component is still a major

influence on the model. Results are still vastly

different from models in which the random part

varies with time.

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Variations on Battese and Coelli (There are many) Farsi, M. JPA, 30,2, 2008.

i

20

log (log )

= fixed effects

0~ ,

0

it it i i it it

it i i i

i

i

C t u v

u u t t

N

x z

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Time Invariant Heterogeneity

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Observable Heterogeneity

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Are the time varying inefficiency models more like time fixed or freely time varying?

A Pooled Model

Battese and Coelli (1992) exp[ ( )] | |

Pitt and Lee (1981) | |

Where is Battese and Coelli?

Closer to

it it it it

it i

it it it i

y v u

u t T U

y v U

x

x

the pooled model or to Pitt and Lee?

Greene (2004): Much closer to the Pitt and Lee model

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2 2

In these models with time varying inefficiency,

( , ) | |

~ [0, ] and ~ [0, ],

where does unobserved time invariant

heterogeneity end up?

In the inefficiency! Even with t

it it it i it i

it v it u

y v g t U

v N U N

x z

he extensions.

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Greene, W., Distinguishing Between Heterogeneity and Inefficiency: Stochastic Frontier Analysis of the World Health Organization’s Panel Data on National Health Care Systems, Health Economics, 13, 2004, pp. 959-980.

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True Random and Fixed Effects: 2004

2 2

True Random and Fixed Effects Approach: 2004

~ [0, ], | | and ~ [0, ]

Unobserved time invariant heterogeneity,

not unobserved time invariant inefficiency

Jo

it i it it it

it v it it it u

i

y v u

v N u U U N

x

2

2 2

ndrow et al. (JLMS) Inefficiency Estimator

( )[ | ]

1 ( )

, , ,

itit it it

it

u itit it it v u i

v

E u

v u

Time varying

Time fixed

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The True RE Model is an RP Model

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Skepticism About Time Varying Inefficiency Models: Greene (2004)

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Estimation of TFE and TRE Models: 2004

2 2

True Fixed Effects: MLE

~ [0, ], | | and ~ [0, ]

Unobserved time invariant heterogeneity,

not unobserved time invariant inefficiency

it i it it it

it v it it it u

i

y v u

v N u U U N

x

2 2 2

Just add firm dummy variables to the SF model (!)

True Random Effects: Maximum Simulated Likelihood (RPM)

( )

~ [0, ], | | and ~ [0, ], ~ [0, ]

it i it it it

it v it it it u i w

y w v u

v N u U U N w N

x

Unobserved time invariant heterogeneity,

not unobserved time invariant inefficiency

Random parameters stochastic frontier model

i

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A True FE Model

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Schmidt et al. (2011) – Results on TFE Problem of TFE model – incidental parameters problem. Where is the bias? Estimator of u

Is there a solution? Not based on OLS Chen, Schmidt, Wang: MLE for data in group mean deviation

form

[ ] ( ) ( )

Trades fixed effects problem for the problem of obtaining

the distribution of the deviations of the one sided terms.

Derives a "within MLE" estimator.

it i it i it i it iy y v v u u x x

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1

Log likelihood function for stochastic frontier model

2log log

log ( , , , ) = ( )

log

i i

N

ii i

y

Ly

x

x

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1 1 1

for stochastic frontier model

with a time invariant random constant term. (TRE model)

2 ( )

1log ( , , , , ) = log

( (

it w ir it

N R TSw i r t

it w

y w

LR y w

Simulated log likelihood fun t

x

c ion

) )

draws from N[0,1].

ir it

irw

x

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TRE SF Model for 247 Spanish Dairy Farms

u

v

w

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Moving Heterogeneity Out of Inefficiency

World Health Organization study of life expectancy (DALE) and composite health care delivery (COMP)

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Generalized True Random Effects Stochastic Frontier Model

Short run (transient) random components

Time varying normal - half normal SF

Long run

it i i it it it

it it

y A B v u

v u

x

(permanent) random components

Time fixed normal - half normal SFi iA B

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2

Generalized True Random Effects Stochastic Frontier Model

( | |)

Short run (time varying, transient) random components

~ [0, ], | | and ~ [0,

it w i i it it it

it v it it it u

y w e v u

v N u U U N

x

2 ],

Long run (time invariant) random components

~ [0,1], ~ [0,1]

The random constant term in this model has a closed skew

normal distribution, instead of the usual normal distribution.

i iw N e N

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A Stochastic Frontier Model with Short-Run and Long-Run Inefficiency:Colombi, R., Kumbhakar, S., Martini, G., Vittadini, G.University of Bergamo, WP, 2011

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1

Colombi et al. Classical Maximum Likelihood Estimator

log ( , )log

log ( ( , )) log 2

(...) T-variate normal pdf.

(..., )) ( 1) Multivariate normal int

N T i i T

iq i i T

T

q

Lnq

T

y X 1 AVA

R y X 1

egral.

Too time consuming and complicated. Estimated in two steps.

First step a variant of least squares for .

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Tsionas, G. and Kumbhakar, S.

Firm Heterogeneity, Persistent and Transient Technical Inefficiency: A Generalized True Random Effects ModelJournal of Applied Econometrics. Published online, November, 2012.Forthcoming.

Extremely involved Bayesian MCMC procedure. Efficiency components estimated by data augmentation.

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Kumbhakar, Lien, Hardaker

Technical Efficiency in Competing Panel Data Models: A Study of Norwegian Grain Farming, JPA, Published online, September, 2012.

Three steps based on GLS: (1) RE/FGLS to estimate (,) (2) Decompose time varying residuals using MoM and SF. (3) Decompose estimates of time invariant residuals.

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1 1 1

Maximum Simulated Full Information log likelihood function for the

"generalized true random effects stochastic frontier model"

( | |)2,

1logL , = log

,

it w ir ir

TN RS

i r t

w

y w U

R

( ( | |) )

draws from N[0,1]

|U | absolute values of draws from N[0,1]

it

it w ir ir it

ir

ir

y w U

w

x

x

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Estimating Efficiency in the CSN Model

1 12

1

Moment Generating Function for the Multivariate CSN Distribution

( , )E[exp( ) | ] exp

( , )

(..., ) Multivariate normal cdf. Parts defined in Colombi et al.

Computed using

T ii i i

T i

Rr tt u y t Rr t t

Rr

1

GHK simulator.

1 0 0

0 1 0, = , , ...,

0 0 1

i

ii

iT

e

u

u

u t

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247 Farms, 6 years.100 Halton draws.Computation time: 35 seconds including computing efficiencies.

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Estimated efficiency for farms 1-10 of 247.

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Kumbhakar, Lien, Hardaker suggest

Time invariant efficiency timesOverall Technical Efficiency =

Time varying efficiency

.8022 .8337 .8691 .8688 .8725 .8745

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Cost Efficiency of Swiss Railway Companies: Model Specification

C = f ( Y1, Y2, PL , PC , PE , N, DA )

43

C = Total costs

Y1 = Passenger-km

Y2 = Freight ton-km

PL = Price of labor (wage per FTE)

PC = Price of capital (capital costs / total number of seats)

PE = Price of electricity

N = Network length

DA = Dummy variable for companies also operating alpine

lines

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Data

50 railway companies, Period 1985 to 1997

Unbalanced panel with number of periods (Ti) varying from 1 to 13 and with 45 companies with 12 or 13 years, resulting in 605 observations

Data source: Swiss federal transport office

Data set available at http://people.stern.nyu.edu/wgreene

Data set used in: Farsi, Filippini, Greene (2005), Efficiency and measurement in network industries: application to the Swiss railway companies, Journal of Regulatory Economics

44

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Model Specifications: Special Cases and Extensions

Pitt and Lee

True Random Effects

Extended True Random Effects

Mundlak correction for the REM, group means of time varying variables

Extended True Random Effects with Heteroscedasticity in vit: v,it=vexp(’zit)

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Efficiency Estimates

46

TRE Models Move Heterogeneity Out of the Inefficiency Estimate

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2. Cost Efficiency of Norwegian Electricity Distribution Companies:

Model Specification

C = f ( Y, CU, NL, PL , PC )

47

C = Total costs of the distribution activity

Y = Output (total energy delivered in kWh)

CU = Number of customers

NL = Network length in km

PL = Price of labor (wage per FTE)

PC = Price of capital (capital costs / transformer

capacity)

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Data

111 Norwegian electricity distribution utilities

Period 1998 – 2002

Balanced panel with 555 observations

Data source: Norwegian electricity regulatory authority (Unpublished)

48

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Mundlak Specification

Suggests ei or wi may be correlated with the inputs.

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Efficiency Estimates

50

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Appendix A: Implementation Customized version of NLOGIT 5/LIMDEP 10.

Both instructions exist in current version. Modifications were:

For the generalized TRE, allow the random constant term in the TRE model to have a second random component that has a half normal distribution.

For the selection model, allow products of groups of observations to appear as the contribution to the simulated log likelihood

Now available from the author as an update to LIMDEP or NLOGIT. To be released with the next version.

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DISTANCE FUNCTION

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A Distance Function Approach

http://www.young-demography.org/docs/08_kriese_efficiency.pdf

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Kriese Study of Municipalities

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TOTAL FACTOR PRODUCTIVITY

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Factor Productivity Growth Change in output attributable to change in factors,

holding the technology constant Malmquist index of change in technical efficiency

TE(t+1|t) = technical efficiency in period t+1 based on factor usage in period t+1 in comparison to firms using factors and producing output in period t.

Index measures the change in productivity

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TFP measurement using DEA

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Total Factor Productivity GrowthSpanish Dairy Farms

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Malmquist Index of Factor Productivity Growth Spanish Dairy Farms