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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
- 22 -EE²
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
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'
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)
- 24 -EE²
- 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
- 28 -EE²
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.