Efficiency Analysis Working Papers

WP-EA-03

Questions to Airport Benchmarkers -

Some Theoretical and Practical Aspects Learned

from Benchmarking Other Sectors

Christian von Hirschhausen and Astrid Cullmann

Reprint from

Presentation at the German Aviation Research Society Workshop in Vienna (November 2005)

Dresden University of Technology DREWAG-Chair for Energy Economics

EE²

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Questions to Airport Benchmarkers –Some Theoretical and Practical Aspects Learned from

Benchmarking Other Industries

Christian von Hirschhausen and Astrid Cullmann

Chair of Energy Economics and Public Sector Management, Dresden University of Technology, and DIW Berlin

Agenda

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1. The Issue: Efficiency Measurement of Airports

2. Options in Nonparametric Approaches

3. Recent Developments in Parametric Approaches

4. Conclusions

Overview of Benchmarking Techniques

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Benchmarking

PartialApproaches

(one-dimensional)

Multi-dimensional Approaches

Frontier Approaches Average Approaches

PerformanceIndicators

Parametric Parametric InducedApproachNon-Parametric

DataEnvelopment

Analysis(DEA)

StochasticFrontierAnalysis

(SFA)

OrdinarayLeast Squares

(OLS)

Total FactorProductivity

(TFP)

StochasticDEA

(SDEA)

CorrectedOrdinary

Least Squares(COLS)

ModifiedOrdinary

Least Squares(MOLS)

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How Should We Measure Airport Efficiency?Model Specification - Pels (2000)

ATM MODEL APM MODEL

OUTPUT INPUT OUTPUT INPUT

Number of runways Check in Desks

Aircraft parking positions Baggage Claim

Number of remote aircraftparking positions

Airport surface area

ATM

Air PassengerMovement

Air Transport Movement

Stochastic Frontier Analysis(Pels (2000), 41)

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PARAMETES ATM PARAMETERS APM

α_constant 0.713 (0.083)* β_constant 0.213 (0.061)*α _96 0.670 (0.032)* β _96 0.016 (0.041)α _97 0.154 (0.061)* β _97 0.050 (0,051)α _area 0.403 (0.059)* β _ATM 0.848 (0.096)*α _runways 0.002 (0.115) β _check-in desks 0.490 (0.160)*α _positions 0.268 (0.211)* β _baggage claims -0.129 (0.191)α _remote 0.280 (0.055)* β ²_ATM -0.586 (0.162)*α ²_area -2.207 (0.458) β ²_check-ins-0.851 (0.722)α ²_runways -0.456 (0.077)* β ²_baggage claims -0.905 (0.268)*α ²_positions -0.606 (0.130)* β _ATM*check-ins 0.353 (0.477)α ²_remote -0.308 (0.137)* β _ATM*bag.claims 0.209 (0.469)α _area*runways 0.591 (0.068)* β _check-ins*bag.cl 0.436 (0.561)α _area*positions 1.208 (0.043)* δ _constant 0.815 (0.562)α _area*remote -0.090 (0.012)* δ _time restriction -0.592 (0.222)*α _runways*position -0.218 (0.157) δ _load factors -1.454 (0.199)*α _runways*remote -0.286 (0.128)* σ² 0.377 (0.064)*α _positions*remote 0.343 (0.152)* γ 0.999(0.5E-7)*δ_slot coordination -0.278 (0.726)δ _time restriction -2.363 (1.447)σ²U+σ²V 0.734 (0.106)*γ =σ²U/(σ²U+σ²V) 0.999 (0.6 E-05)*

-Translog Production function

- Battese and Coelli Specifictation (1995), the inefficiency effects are expressed as an explicit function of a vector of firm specific variables and a random error

Data Envelopment Analysis: A Variety of Results(Pels (2000), 52-53)

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ATM ATM APM APM

Technical Scale Technical Scale

LYS 0.368 0.768 0.513 0.970

MAN 0.794 0.821 0.774 0.990

MRS 0.699 0.702 1 0.970

MUC 1 0.942 0.757 0.994

MXP 0.955 0.316 0.842 0.771

NUE 1 0.506 0.917 0.494

OTP 0.734 0.199 0.949 0.401

ORY 0.601 0.921 0.927 1

PRG 0.541 0.673 0.672 0.785

STN 0.614 0.595 0.711 0.748

STO 0.999 0.821 0.896 0.988

STU 1 1 0.923 0.836

SXF 0.504 0.275 1 0.507

TRN 1 0.272 0.750 0.601

TXL 0.516 0.883 0.687 1

VIE 0.518 0.998 0.798 0.911ZRH 1 0.983 0.986 0.988

ATM ATM APM APM

Technical Scale Technical Scale

AMS 0.804 0.767 0.788 1

BLL 1 0.515 0.971 0.461

BRU 0.755 0.760 1 1

CDG 1 1 0.695 0.991

CPH 1 1 1 1

DUB 0.442 0.830 0.932 0.963

FAO 1 0.294 1 0.996

FCO 0.880 0.850 1 0.995

FRA 1 0.909 0.809 0.998

GOT 1 0.493 0.972 0.593

GVA 0.993 0.669 0.448 0.951

HAJ 0.819 0.689 0.674 0.928

GAM 0.650 0.864 0.643 1

LGW 1 1 0.954 1

LHR 1 1 1 1

LIN 1 1 1 1

LIS 0.760 0.612 0.707 0.886

Some Empirical Application

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Airports Methodologies

Gillen and Lall (1999) 22 in USA 1989-1993

Data Envelopment Analysis, Malmquist Indice, Tobit Regression

Tolofari, Ashford and Caves (1990)

7 in UK 1975-1987

Translog Cost Function

Martin and Roman (2001)

37 in Spain 1997

Static Data Envelopment Analysis

Pels (2000) 34 in Europe Data Envelopment Analysis,

Stochastic Frontier AnalysisHooper and Henscher

(1997)6 in

Australia 1989-1993

Tornqvist Indice

Holvad and Graham (2000)

25 in Europe, 12 in

Australia 1993

Data Envelopment Analysis and Free Disposal Hull

Agenda

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1. The Issue: Efficiency Measurement of Airports

2. Options in Nonparametric Approaches

3. Recent Developments in Parametric Approaches

4. Conclusions

Free Disposal Hull Estimator

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-Deprins et al. (1984) proposed measuring efficiency for a given unit (x,y) relative to the boundary of the free disposal hull of the sample, the smallest free disposable set.

- Restricts the dominance comparison of the units’ inputs and outputs to be with respect to other production units

→ FDH excludes linear combinations of production units from the analysis

DEA Frontier FDH Frontier

Bootstrapping

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- Literature survey:Ferrier and Hirschberg (1997, 1999)Simar and Wilson (1999, 2000)- Focus on Simar and Wilson (2000), Statistical Inference in Nonparametric Frontier Models: The State of the Art, Journal of productivity Analysis, 13, 49-78.

“Many have claimed that the FDH and DEA techniques are non-statistical, as opposed to econometric approaches where particular parametric expressions are posited to model the frontier”Simar and Wilson define a statistical model to determine the statistical properties of the nonparametric estimators→ Statistical inference is now possible→ Allow correction for the bias of the efficiency estimators and estimation of confidence intervals for the efficiency measures

Introduction

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Bootstrapping … Introduced by Efron, 1979

… Is a resampling method

… Is a tool to determine the statistical properties of estimates and test statistics

… Distinguish between parametric and non parametric bootstrapping

…Another similar resampling methodology is the Jackknife method, introduced by Quenouille, 1949

Sensitivity Analysis with Bootstrapping

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- Approximation of the sampling distribution in order to correct for small sample bias and construct confidence intervals

→ Bootstrapping involves the repeated simulation of the data generating process and the application of the original estimator to each simulated sample so that the resulting estimators mimic the sampling distribution of the original estimator

- Practical aspects: Software package FEAR 1.0 (Frontier Efficiency Analysis with R); implementation of the bootstrap methods described by Simar and Wilson (1998, 2000)

- Two important Bootstrap Algorithms for DEA and FDH Efficiency Scores:

Simar and Wilson (1998) – restrictive conditional bootstrap

Simar and Wilson (2000) – allow for heterogeneity in the structure of efficiency

Agenda

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1. The Issue: Efficiency Measurement of Airports

2. Options in Nonparametric Approaches

3. Recent Developments in Parametric Approaches

4. Conclusions

Short Introduction into Panel Data Analysis

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- Use panel data to view the unobserved factors affecting the dependent variable: those that are constant and those that vary over time.

- Including variable (unobserved heterogeneity) which captures all unobserved, time constant factors that affect dependent variable: unobserved effect, fixed effect; idiosyncratic error, time varying error.

- First differentiating

- Fixed effects estimation

- Random effects estimation

Panel data analysis (Wooldridge 2003)

Two – Periods

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Use panel data to view the unobserved factors affecting the dependent variable:

1) Effects that are constant over time

2) Those that vary over time

With i = cross section and t = time period

captures all unobserved time constant factors that affect dependent variable. Unobserved effect, fixed effect, firm heterogeneity.

Unobserved effects model, fixed effects model.

error term, idiosyncratic error, unobserved factors that change over time and affect dependent variable.

itiittit uaxdy ++++= 100 2 βδβ

ia

itu

How Should We Estimate Parameter of Interest?

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1) OLS Regression

To produce a consistent estimator of beta we have to assume that the unobserved effect is uncorrelated with the explanatory variables. If this is not the case:

Heterogeneity bias: standard errors are not correct.

Reason for collecting panel data: to allow for the unobserved effect, correlated with explanatory variables.

2) First differenced Estimator

are uncorrelated

All explanatory variables have to change over time.

Homoskedasticity assumption

The error term has to follow a random walk when differentiating with more than two time periods.

iii uxy ∆+∆+=∆ 10 βδii ux ∆∆ ,

Advanced Panel Data Models

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3) Fixed effects estimator:

Uses transformation to remove the unobserved effect prior to estimation.

Any time-constant explanatory variables are removed along with the unobserved effect.

4) Random effects estimator:

Attractive when we think that the unobserved effect is uncorrelated with all explanatory variables.

Condition: good controls in our equation, we believe that any leftover neglected heterogeneity only induces serial correlation in the composite error term.

Estimation of Random effects model by Generalized Least Squares.

Fixed Effects Estimation

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Fixed effect transformation (=within transformation) of

For each i average this equation over time

Subtraction results in time demeaned data, unobserved effect hasdisappeared.

A pooled OLS estimator that is based on the time demeaned variables is called: the fixed effects estimator!

Explanatory variables have to be strict exogen, than the fixed estimator is unbiased.

Other important assumption: homoskedastic and serially uncorrelated error.

itiitit uaxy ++= 1β

iiii uaxy ++= 1β

Random Effects Estimation

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Unobserved effect model:

with the composite error

Suppose that is uncorrelated, then using a transformation to eliminate the unobserved effects results in inefficient estimators.

Random effects model when we assume

Beta can be consistently estimated using a single cross-section, but disregards much useful information.

With panel estimates we have to keep in mind the serial correlation of the composite error. OLS standard errors ignore this correlation.

GLS solve the serial correlation problem!

ititit vxy ++= 10 ββ

ia

0),( =iitj axCov

22

2

),(ua

aisit vvCorr

σσσ+

=

itiit uav +=

GLS Transformation

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Aim = Eliminates serial correlation

Generalized Least Squares (GLS) estimator:

An estimator that accounts for a known structure of the error variance (heteroskedasticity), serial correlation pattern in the errors, or both, via a transformation of the original model.

Deriving the GLS transformation that eliminates serial correlation see Wooldridge (2002)

Transformation of the original model:

FEE → subtracts the time averages

REE →subtracts a fraction of that time average

21

22

2

][1au

u

Tσσσλ+

−=

)()()1( 10 iitiitiit vvxxyy λλβλβλ −+−+−=−

Estimation of Lambda

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Never known in practice, can always be estimated:

The feasible GLS estimator that uses the estimated lambda is called the random effects estimator:

- Consistent

- And asymptotically normally distributed

2/122

)]}/(1/[1{1∧∧∧

+−= uaT σσλ

∧

+=

∧−

==

−∧

∑∑∑+−−= is

T

tsit

T

t

N

ia vvkTNT

1

1

11

12

)]1(2/)1([σ

Effects Models for Stochastic Frontiers

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Stochastic Frontier Model:

S (+1 production or profit, -1 cost)

Time varying part, functions of input quantities or output and input prices; time trend to account for technical change

Time invariant component, observable heterogeneity , not related to the production structure, captures firm specific or unit specific effects

Measures of firm efficiency or inefficiency: Jondrow et al. (1982)conditional estimator of used for estimation of

],0[~

],,0[~

,...1,...1

'),(

2

2

'

uititit

vit

ititiitititiitit

NwhereUUu

N

TtNi

SuzxSuzxfy

σ

συ

υµβυ

=

==

−++=−+=

itx'β

iz'µ

itu itu ][ ititit uEu ε=∧

Fixed and Random Effects

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Fixed effects estimator:- Is distribution free, only statement of the conditional mean- Lose the individual identity of the estimated inefficiency- Effects can only be estimated relative to the best- Time invariant effects treated ambiguously in this frameworkRandom effects model:- Tighter parameterization allows direct individual specific estimates of the

inefficiency term- Rests on the strong assumption that the effects are time invariant and

uncorrelated with the variables included in the model – unreasonable! Share two shortcoming:1) Each assumes that the inefficiency is time invariant, or in the models

above obey the same trajectory2) Latent cross firm heterogeneity is not related to inefficiency. (forced into

the firm specific term

Greene (2005)

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- Stochastic frontier models with panel data relied on traditional fixed (Schmidt and Sickles, 1984) and random effects models (Pitt and Lee,1981)

Two problems of the approaches:

1) Conventional panel data approaches assume that technical or costinefficiency is time invariant

2) Fixed and random effects estimators force time invariant cross unit heterogeneity into the same term that is being used to capture the inefficiency

Greene propose extensions of the conventional stochastic frontier models:

I) The fixed effects model that employs the nonlinear specification

II) The random effects model, reformulated as a special case of the random parameters model

itiittit uaxdy ++++= 100 2 βδβ

True Fixed and Random Effects Formulation

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I) True Fixed Effect Formulation:

Retains the distributional assumptions of the stochastic frontier model, allows for freely time varying inefficiency, allows the heterogeneity term to be correlated with the included variables

-Trivial extension of the basic stochastic frontier models: replace the overall constant term with a complete set of firm dummy variables and estimate it by the now conventional means.

II) True Random Effect Formulation

-Stochastic frontier model with a random (across firms) constant term, retains the essential characteristics of the stochastic frontier model, while relaxing the problematic assumptions discussed earlier

- Time invariant term interpreted as firm specific heterogeneity, not inefficiency! Shift all the invariant content of into a heterogeneity term

- Solved Identification Problem

itititiit Suxy −++= υβα '

itititiiit Suxy −+++= υβϖα ')(

Agenda

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1. The Issue: Efficiency Measurement of Airports

2. Options in Nonparametric Approaches

3. Recent Developments in Parametric Approaches

4. Conclusions

Conclusions

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We just recall two traditional issues in efficiency measurement, to enrich the discussion of how to fill the GAP:

1) Methodology first, what model to use, etc.?

2) Data requirements/availability second, and the ways of going about

Literature

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Deprins, Simar and Tulkens (1984): Measuring labor inefficiency in post offices, in The Performance of Public Enterprises: Concepts and measurement Amsterdam, 243-267.

Greene W. (2005): Fixed and Random Effects in Stochastic Frontier Models, Journal of Productivity Analysis, 23, 7-23.

Pels, A.J.H. (2000): Airport Economics and Policy- Efficiency Competition, and Interaction with Airlines, Tinbergen Institute Research Series Nr. 222, 27-54.

Simar, L. and P.W. Wilson (1998): Sensitivity analysis of Efficiency scores: How to bootstrap in non-parametric frontier models, Management Science, 44, 49-61.

Simar L. and P. W. Wilson (2000): A General Methodology for Bootstrapping in nonparametric frontier models, Journal of Applied Statistics, 27, 779-802.

Simar, L. and P.W. Wilson (2000): Statistical inference in nonparametric frontiermodels, Journal of Productivity Analysis 13, 49-78.