Stochastic Frontier Models - PowerPoint Presentation

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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 ApplicationsSlide2

Where to Next?

Heterogeneity: “Where do we put the z’s?”Other variables that affect production and inefficiencyEnter production frontier, inefficiency distribution, elsewhere?HeteroscedasticityAnother form of heterogeneity

Production “risk”Bayesian and simulation estimatorsThe stochastic frontier model with gamma inefficiencyBayesian treatments of the stochastic frontier modelPanel DataHeterogeneity vs. Inefficiency – can we distinguishModel forms: Is inefficiency persistent through time?ApplicationsSlide3
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Swiss Railway DataSlide5

Observable Heterogeneity

As opposed to unobservable heterogeneityObserve: 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?Slide6

Shifting the Outcome Function

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, u

2], i = 0 + 1zi,Slide7

Heterogeneous Mean in Airline Cost ModelSlide8

Estimated Economic EfficiencySlide9

How do the Zs affect inefficiency?Slide10

Effect of Zs on EfficiencySlide11

Swiss Railroads Cost FunctionSlide12

One Step or Two Step

2 Step: 1. Fit Half or truncated normal model, 2. Compute JLMS ui, regress ui on zi

Airline EXAMPLE: Fit model without POINTS, LOADFACTOR, STAGE1 Step: Include zi in the model, compute ui including zi Airline example: Include 3 variables Methodological issue: Left out variables in two step approach.Slide13

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 1.0Slide14

Application: WHO DataSlide15

Unobservable Heterogeneity

Parameters vary across firmsRandom variation (heterogeneity, not Bayesian)Variation partially explained by observable indicatorsContinuous variation – random parameter models: Considered with panel data models Latent class – discrete parameter variationSlide16

A Latent Class ModelSlide17

Latent Class Efficiency Studies

Battese and Coelli – growing in weather “regimes” for Indonesian rice farmersKumbhakar and Orea – cost structures for U.S. BanksGreene (Health Economics, 2005) – revisits WHO Year 2000 World Health ReportKumbhakar, Parmeter, Tsionas (JE, 2013) – U.S. Banks.Slide18

Latent Class Application

Estimates of Latent Class Model: Banking DataSlide19

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 managementImplication, no inefficiency: u = 0.Slide20
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Nursing Home Costs

44 Swiss nursing homes, 13 yearsCost, Pk, Pl, output, two environmental variablesEstimate cost functionEstimate inefficiencySlide23

Estimated Cost EfficiencySlide24

A Two Class Model

Class 1: With InefficiencylogC = f(output, input prices, environment) + vv + uuClass 2: Without Inefficiency

logC = f(output, input prices, environment) + vv u = 0Implement with a single zero restriction in a constrained (same cost function) two class modelParameterization: λ = u /v = 0 in class 2.Slide25

LogL= 464 with a common frontier model, 527 with two classesSlide26
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Heteroscedasticity in v and/or u

yi = ’xi + vi

- uiVar[vi | hi] = v2gv(hi,) = vi2 gv(hi,0) = 1,gv(hi,) = [exp(’hi

)]2Var[Ui | hi] = u2gu(hi,)= ui2 gu(hi,0) = 1,gu(hi,) = [exp(’hi)]2 Slide28

Heteroscedasticity Affects InefficiencySlide29
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A “Scaling” Truncation ModelSlide33

Application: WHO DataSlide34

Unobserved Endogenous Heterogeneity

Cost = C(p,y,Q), Q = qualityQuality is unobservedQuality is endogenous – correlated with unobservables that influence costEconometric Response: There exists a proxy that is also endogenousOmit the variable?Include the proxy?Question: Bias in estimated inefficiency (not interested in coefficients)Slide35

Simulation Experiment

Mutter, et al. (AHRQ), 2011Analysis of California nursing home dataEstimate model with a simulated data setCompare biases in sample average inefficiency compared to the exogenous caseEndogeneity is quantified in terms of correlation of Q(i) with u(i)Slide36

A Simulation Experiment

Conclusion: Omitted variable problem does not make the bias worse.Slide37

Sample Selection Modeling

Switching Models: y*|technology = bt’x + v –uFirm chooses technology = 0 or 1 based on c’z+ee is correlated with vSample Selection Model:

Choice of organic or inorganicAdoption of some technological innovationSlide38

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:Slide39

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.” Slide40

Two Step Selection

Heckman’s method is for linear equationsDoes not carry over to any nonlinear modelThe formal estimation procedure based on maximum likelihood estimation

Terza (1998) – general results for exponential models with extensions to other nonlinear modelsGreene (2006) – general template for nonlinear modelsGreene (2010) – specific result for stochastic frontiersSlide41

A Sample Selected SF Model

di = 1[

′zi + wi > 0], wi ~ N[0,12] yi = ′xi + i, i ~ N[0,

2] (yi,xi) observed only when di = 1. i = vi - ui ui = |u

Ui| = u |Ui

| where Ui ~ N[0,1

2] v

i = v

Vi where

V

i

~ N[0,1

2

].

(

w

i

,v

i

) ~ N

2

[(0,1), (1,



v

,



v

2

)]Slide42

Alternative Approach

Kumbhakar, Sipilainen, Tsionas (JPA, 2008)Slide43

Sample Selected SF ModelSlide44

Simulated Log Likelihood for a Stochastic Frontier Model

The simulation is over the inefficiency term.Slide45

2nd Step of the MSL ApproachSlide46

JLMS Estimator of uiSlide47

WHO Efficiency Estimates

OECD

Everyone ElseSlide48
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WHO Estimates vs. SF ModelSlide50
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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-UretaUniversity of ConnecticutDaniel SolisUniversity of MiamiWilliam GreeneStern School of BusinessNew York UniversitySlide52

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 3Slide53