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Transportation Research Part B 70 (2014) 18–34
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Transportation Research Part B
journal homepage: www.elsevier .com/ locate / t rb
Understanding relative efficiency among airports: A generaldynamic model for distinguishing technical and allocativeefficiency
http://dx.doi.org/10.1016/j.trb.2014.07.0040191-2615/� 2014 Elsevier Ltd. All rights reserved.
⇑ Corresponding author. Tel.: +1 4135454192.E-mail addresses: [email protected] (A.G. Assaf), [email protected] (D. Gillen), [email protected] (E.G. Tsionas).
A. George Assaf a,⇑, David Gillen b, Efthymios G. Tsionas c
a Isenberg School of Management, University of Massachusetts-Amherst, United Statesb Sauder School of Business, University of British Columbia, Canadac Lancaster University Management School, United Kingdom
a r t i c l e i n f o
Article history:Received 21 December 2013Received in revised form 10 July 2014Accepted 28 July 2014
Keywords:Technical efficiencyAllocative efficiencyShort-runLong-runAirport ownership and regulation
a b s t r a c t
The paper introduces a new dynamic frontier model that is used to analyze the impact ofboth ownership and regulation on airport technical and allocative efficiencies. We differen-tiate between the short and long-term effects. Based on a large sample of internationalairports, we find in the short-run the majority of the improvements are from reducingtechnical inefficiency, which come for the most part from adjusting output, something thatcan be accomplished in the short-term. There are relatively small changes, in the short run,resulting from improving allocative efficiency. We find that adding economic regulationleads to a decrease in technical efficiency in the short-run. Quite different conclusions holdfor the long-term; there are improvements available from reducing allocative inefficiencyand comparable benefits are available from cutting technical inefficiency. In the long-runwe find that technical and allocative inefficiency decreases by moving away from govern-ment owned to fully privatized airports and moving away from rigid regulation.
� 2014 Elsevier Ltd. All rights reserved.
1. Introduction
Due to heavy competition, airports continue to face increased pressure to be more cost efficient and to serve new airlinebusiness models (Assaf and Gillen, 2012). Following airline deregulation, the increased competition among carriers led toreduced fares and market growth of both traffic and routes. The rapid growth in air travel coupled with government’s lackof resources meant airports had to acquire capital for expansion by other means. These financial pressures in turn led to are-evaluation of the exclusive ownership of airports by governments, or the lack of a commercial orientation, if theyremained under government ownership. The issue of shifting ownership, in turn, raised the issue of whether economic reg-ulation was desirable or necessary if governance or ownership form changed (Gillen, 2011). Capital markets required thatairports improve their cost efficiency and serve a rapidly changing market. The development of the low cost carrier (LCC)business model has been particularly important in bringing pressure to bear in order to improve cost efficiency.
Another factor driving cost efficiency at airports is airport competition. Such competition exists not only in the context ofinternational long haul connecting hubs like Heathrow, Schiphol and Frankfurt, but also for secondary airports that bid forairlines to provide service and to base aircraft at their airport. Istanbul has been growing with double-digit traffic growth but
A.G. Assaf et al. / Transportation Research Part B 70 (2014) 18–34 19
their ability to compete with other Middle East hubs will depend on their ability to deliver operational efficiencies. LCCs arefootloose and can easily move from one airport to another. Airports compete with each other because the owners, often gov-ernment, recognize their airports value in promoting economic growth in the region. Such growth depends largely onwhether the governments have adopted a form of ownership and price regulation, if imposed, to ensure high performance.
There has been a mix of different types of airport privatizations varying from complete privatization to a mix of publicand private ownership. In some jurisdictions airport privatization has been put in place along with some form of price reg-ulation. Some developed economies are reluctant to move away from public ownership or public control; for example, Fin-land, France, Italy, Spain, Sweden and the United States (U.S.). The choices of ownership type and whether to impose priceregulation are important because airports as part of the aviation supply chain will have some impact on the growth anddiversity of the economy. Aviation policy will determine the ownership/price regulation combination. The policy will affectthe ability of airports to attract needed capital for sustaining investment and to create network connectivity to supportgrowth of trade in goods, services and tourism. The differing types of regulation or ownership combinations will affect theseoutcomes (Parker, 1999; Oum et al., 2008).
A controversy however does exist with regard to which regulation/ownership combination is superior, where ‘superior’ ismeasured in terms of cost efficiency. It may be insufficient to simply measure the impact, or to disentangle the effect ofownership and regulation separately on airport performance. In order to determine which combination of ownership andprice regulation affects airport performance the different combinations must be examined (Assaf and Gillen, 2012).
Recent studies have highlighted two important gaps in the current literature regarding airport cost efficiency. Firstly,there have been few studies that analyze the combined impact of ownership and regulation on airport performance; moststudies have treated ownership and regulation separately, implicitly assuming impacts are linear additive. Secondly, therehas been a lack of the use of more precise econometric methodologies in assessing the joint effect of ownership and regu-lation on airport performance. Most studies used cost efficiency as the main metric for airport performance without decom-posing this metric into technical and allocative efficiency; technical inefficiency arises when, for a given set of inputs, outputis less than it could potentially be. Allocative inefficiency occurs when for a given output and set of input prices, the inputchoices are sub-optimal. The product of these two measures is cost efficiency. Thus cost inefficiency arises potentially fromboth too little output produced, and producing too little output with an inappropriate mix of inputs given input prices.Therefore, it is potentially possible that airport managers and air policy makers can identify strategies that could improveon either or both inefficiencies and be able to observe the changes in each over time, as different strategies and policiesare pursued.
The present study addresses two gaps in the literature offering four important contributions. Firstly, we analyze the com-bined impact of ownership and regulation on airport performance using an extensive panel data of airports from across theglobe. Secondly, instead of focusing on technical efficiency or cost efficiency as the main metrics for airport performance, wedecompose the overall cost efficiency into its technical and allocative components using flexible functional forms. Thirdly,and importantly, we introduce a new stochastic frontier model that allows both technical efficiency (TE) and allocative effi-ciency (AE) to follow a dynamic framework. The notion that efficiency improves in the long-run, in competitive markets, isalso quite prevalent. Hence, the use of a dynamic framework is a more realistic assumption. Fourthly, the use of dynamicsallows us to analyze the combined impact of ownership and regulation on performance in both the short-run and the long-run (steady state). We are therefore able to identify which of technical or allocative efficiency dominates in the short andlong-run. It also allows us to further validate our results and determine whether there are any long-run expected changesdue to the effects of the various ownership and regulation forms on airport performance.
The model we propose builds on Kumbhakar and Tsionas (2005) who solved so-called Greene’s problem in this context byusing a static cost function—share equations framework. Our model relies on the cost function and assume that technicalinefficiency and price distortions (that give rise to allocative inefficiency) follow a vector autoregressive scheme which,we believe, is quite flexible. Previous dynamic models of technical inefficiency that do not account of Greene’s probleminclude Tsionas (2005) and Emvalomatis et al. (2011). We develop our model using the Bayesian approach, which recentlygained increased popularity in the transportation literature (Farooq et al., 2013; Parry and Hazelton, 2013; Kobayashi et al.,2012; Martin and Voltes-Dorta, 2011; Yan et al., 2009).
The results demonstrate the value of estimating technical and allocative efficiencies separately, and to distinguish theshort and long-run differences in possible efficiency gains. In the short-run the majority of any efficiency gains are fromreducing technical inefficiency. This will come for the most part from adjusting output, something that can be accomplishedin the short-term, recognizing that airports that rely heavily on airside output, rather than on both airside and landside(retail) output, will have fewer degrees of freedom. There are relatively small gains to be had from improving allocativeefficiency since the source of the efficiency gains lie in adjusting the input mix or improving productivity, something moredifficult to do in the short-term. Quite different conclusions hold for the long-term; there are gains to be had from improvingallocative efficiency, and comparable gains are possible from improving technical efficiency for a smaller subset of airportsthat are either partially, or wholly, owned by government.
The modeling also shows that the gains available in the short-run are from reducing technical inefficiency. This is anoutput fix and airport managers should focus their efforts and strategies on adjusting output, and place relatively lessemphasis on adjusting inputs to improve efficiency. In the long-run, airport managers will have to optimize both inputmix and output mix.
20 A.G. Assaf et al. / Transportation Research Part B 70 (2014) 18–34
2. Airport ownership and regulation
Over the last decade a considerable literature has developed examining the evolution of the shift of airport ownership andprice regulation. This was part of a larger literature that had raised questions concerning financing the replacement of aginginfrastructure and expanding existing infrastructure particularly in the aviation sector (see Gillen, 2011). The crumblinginfrastructure, the need to meet rapidly growing demand and the lack of resources available from traditional governmentsources meant a new approach to ownership and management was needed to attract capital and to provide efficient man-agement. There was ‘‘little questioning of whether airports were operating in an institutional setting which gave them theincentive to produce and price efficiently. It was presumed that publicly and locally owned airports would keep prices closeto costs, set price structures efficiently, provide the range of services that users were willing to pay for, and keep costs to aminimum’’ (Gillen, 2011, p. 3). This turned out to be wrong by a large margin. (see Armstrong et al., 1994; Morrison andWinston, 2008, and Starkie and Yarrow, 2008a,b).
Governments around the world took a number of different approaches to evolving airport ownership and under somecircumstances imposing price regulation. In the U.S. despite airline deregulation and decades of private utility ownership,airports remained in government hands with a zero profit mandate. Europe took a divergent path with the U.K. opting forprivatization and price regulation at four major airports while Continental Europe either maintained government ownershipor had partial privatization but always with some form of price regulation. Canada adopted a not-for-profit model of localmarket privatization and Australia and New Zealand initially imposed price regulation with privatization but have subse-quently chosen price monitoring with privatization.
Today airport privatization has taken place in Europe, Asia, Australia and New Zealand, Latin America and the Caribbeanwhere major airports have been privatized. A growing phenomenon in the airport industry has been the development of air-port groups where several airports are owned by a consortium of domestic and foreign investors.1 The largest airport priv-atizations in 2011–12 have been in Europe and South America. Several planned airport privatizations in Portugal, Spain andGreece were put on hold due to the depressed state of these European economies, still recovering from the financial crises.In South America, Brazil began by offering long-term concession contracts for the three largest airports in the country, andin late 2012, offered concessions at two more key airports.2 India also privatized airports in Bangalore, New Delhi and Mumbai.
Price regulation has not always been imposed with privatization but this is the exception. Continental Europe for examplehas taken the position that airports are monopolies and will abuse their monopoly power regardless of incentives associatedwith revenues from complementary non-aviation revenues (Starkie, 2001). Price regulation has taken a range of formsincluding traditional rate of return regulation long associated with regulating public utilities or incentive based price-capregulation, either single or dual till.3 In India the government has chosen to impose price-cap regulation with any whole orpartial privatization. There have been cases of both single and dual till regulation being imposed. The airport concessions thathave been let in Brazil face price cap regulation, which is a hybrid between single and dual till (ICAO, 2013).
With pending privatizations or concessions in the Middle East (e.g., Saudi Arabia), Columbia, Croatia, and eventuallySpain, Portugal, and Greece, and with current privatized airports needing capital, the impact of ownership type and the formof economic regulation on airport economic performance needs to be understood. Assaf and Gillen, 2012 investigated thequestion of how the combination of ownership and regulation affected economic efficiency. However, what is needed is adeeper understanding of why airports with differing ownership/economic regulation characteristics may differ in terms oftheir cost efficiency, whether technical or allocative efficiency contributes more or less to economic efficiency, and how air-ports have improved (or not) their technical and allocative efficiency over time.
3. Literature on ownership, regulation and airport performance
3.1. Background
The relationship between ownership and economic regulation on firm performance has been extensively tested in thebusiness and economic literature (Assaf and Gillen, 2012; Liebert and Niemeier, 2013). Arguments from agency and publicchoice theories indicate that privatization should have a positive impact on the profitability and efficiency of firms (Cuervoand Villalonga, 2000). The theoretical belief is that privatization should lead to higher managerial incentives, and should cre-ate more effective monitoring systems (Gupta, 2005). Privatization is also expected to ‘‘trigger a change in the goals of thefirm and in the bargaining power of the different actors in the political market, increasing the search for efficiency and reduc-ing social considerations’’ (Cuervo and Villalonga, 2000, p. 582).
Empirical evidence, however, has not provided a conclusive answer. Though the majority of studies seem to indicate thatprivatization improves performance measured in terms of cost efficiency, some research reported mixed conclusions(Villalonga, 2000; Martin and Parker, 1997). The literature importantly distinguishes between fully private and partially-
1 Poole (2013) points out that among the largest airports in the world (measured by revenue) 36 are fully or partially owned by private investors.2 See discussion in Poole (2013).3 In rate of return regulation, an appointed regulator imposes an allowed rate of return on a rate base of the airport. Price cap regulation is a form of incentive
based economic regulation, which sets a limit on the maximum allowed price increase for regulated prices. Under single till all revenues are considered insetting the price cap whereas under dual till only airside revenues are included.
A.G. Assaf et al. / Transportation Research Part B 70 (2014) 18–34 21
private firms, as these two ownership types do not behave in the same way, and hence their performance implications arealso different (Gupta, 2005). In the case of airports, the fact that a number of airports operate under a variety of ownershipregimes (private and public), with some price regulated and some not, creates greater complexity in analyzing the role ofownership. Research from the business and airport literature did not also necessarily indicate that mixed ownership resultsin better performance (Gupta, 2005; Oum et al., 2008).
With most airports being regulated, a further challenging aspect is that regulation can also interfere with the role ofownership on performance. The role of regulation on airport performance, for instance, is not well understood and has beensubject to strong debate. Recently, several economists have questioned whether price regulation is necessary (Forsyth, 2002;Starkie and Yarrow, 2000; Starkie, 2001; Tretheway, 2001). Gillen (2011), for example, argued that airports do not necessar-ily have market power nor do they need to be regulated, as the presence of economic rent may arise from both market powerand scarcity rent. Gillen (2011, p. 5) also ‘‘argued that price regulation could stifle innovation and dynamic efficiency even inthe context of RPI-X price regulation, which was designed to reinforce the incentives for efficiency and innovation’’. Otherstudies have also discussed the role of regulation within specific countries/contexts. Beesley (1999) discussed the limitationsof price-cap regulation in the context of Heathrow airport. Tretheway (2001) indicated that the rate of return of regulation(ROR) increases management complexity and is expensive to operationalize. Kunz and Niemeier (2000) also re-confirmedthese claims in the context of Germany. Starkie (2007, 2011) has consistently questioned the role of regulation, as airportsdo not usually behave in a monopolistic fashion.4
3.2. Current gaps
With the complexities discussed, it should be no surprise that research on the relationship between ownership and reg-ulation and airport performance has attracted increased attention in the literature over the last decade. Studies have applieda variety of methodologies including advanced approaches such as Data Envelopment Analysis (DEA) and stochastic frontiercost functions (SF), and have used a number of metrics to analyze airport performance such as productivity or cost efficiency.Known as the frontier methods, both DEA and SF are considered more comprehensive than simple performance indicators asthey can easily accommodate for multiple input and outputs. The difference between the two is that DEA uses a linear pro-gramming approach, while SF uses a parametric approach. In the context of panel data, SF is usually preferred as can betteraccount for the panel structure of the data.5 In Section 3 we only focus on research that has analyzed the effect of the variousforms of ownership and regulation on airport performance and we note that few studies considered both factors acting simul-taneously. Table 1 provides a listing and summary of the current studies based on the following criteria: methods and variablesused to measure airport performance, model assumptions (dynamic vs. non- dynamic), focus (ownership, regulation, or both)and the performance metrics they use.
The following can be observed from Table 1: first, most studies used the frontier methods to measure airportperformance. Both DEA and SF have been equally popular. However, DEA has been more popular when technical efficiencyor productivity was used as the performance metric, and SF was more popular in the context of cost efficiency. Second, moststudies analyzed the ownership and regulation separately. Only two studies have looked at the combined impact of bothownership and regulation on airport performance. The first, by Adler and Liebert (2010), used technical efficiency as itsperformance metric, and the second, by Assaf and Gillen (2012), focused on cost efficiency. Third, we can observe that noneof these studies was formulated in a dynamic fashion. While some studies included a time variant assumption using asystematic function of time (Oum et al., 2008; Assaf and Gillen, 2012), none was formulated in a dynamic framework. Fourth,allocative efficiency was completely ignored in all of the studies. Research has mostly focused on a single metric ofperformance. None of the existing studies looked at cost and technical efficiencies jointly, using one frontier model. Fifth,most research focused on airports from individual countries; only a few studies analyzed ownership/regulation using anextensive sample that covered a range of international airports.
Our careful analysis of each of these studies indicates that the findings are not necessarily consistent. It is however dif-ficult to draw conclusions as some studies have focused on individual countries and some only on one type of ownership(e.g., private vs. public) or regulation. For studies that focused on multiple types of ownership such as Oum et al. (2008),the findings indicated that fully private airports perform better than partially private or government airports. This findingwas not however completely confirmed by Assaf and Gillen (2012, p. 194) who ‘‘indicated that the most efficient airportsare those that are fully private and face light-handed regulation, and those that are government owned and have noregulation’’.
The limitations and gaps identified in the current literature support the need for this research. We offer additionalinsights on the effect of ownership and regulation on airport performance by filling in the gaps highlighted in Table 1. Thiswill include introducing a model that is formulated in a dynamic fashion and will also measure the impact of ownership andregulation on both technical and allocative efficiencies. This will provide a richer understanding of the impact of ownershipand regulation on cost efficiency. The use of dynamics is essential, and will ‘‘capture how well airports have been able toadjust to the wide array of changes that have taken place in the aviation industry’’ (Gillen, 2011). With this model, we will
4 For a more detailed discussion on this topic refer to Starkie (2007) and Gillen (2011).5 For a comprehensive treatment of the two methodologies and their use in the airport industry, refer to Gillen and Lall (1997), Tsionas (2005), Barros (2008),
Brissimis et al. (2010) and Suzuki et al. (2010).
Table 1Review of the extant literature on ownership, regulation and airport performance.
Study Method Inputs Outputs Input prices Ownership Regulation Dynamic Associations
Oum et al. (2004) Variable factor productivitybased on Data EnvelopmentAnalysis
1. Labor2. Soft cost
1. Passengers2. Aircraft
movements3. Non-aeronautical
revenue
– No Yes No Regulation ? Productivity
Oum et al. (2006) Variable factor productivitybased on Data EnvelopmentAnalysis
1. Labor2. Soft cost
1. Passengers2. Aircraft
movements3. Non-aeronautical
revenue
– Yes No No Ownership ? Productivity
Lin and Hong (2006) Data Envelopment Analysis 1. Employees2. Number of parking
spaces3. Number of runways4. Number of check-in
counters5. Number of aprons6. Terminal area7. Number of boarding
gages8. Number of baggage col-
lection belts
1. Passengers2. Cargo3. Aircraft
Movements
– Yes No No Ownership ? Technicalefficiency
Oum et al. (2008) Stochastic frontier 1. Employees2. Non-labor variable cost3. Number of runways4. Terminal size
1. Passengers2. Aircraft
movements3. Non-aeronautical
revenue
1. Wage rate2. Non-labor
input price
Yes No No Ownership ? Costefficiency
Marques and Barros (2010) Stochastic frontier 1. Employees2. Capital premises
1. Aircraft2. Passengers
1. Price of labour Yes Yes No Ownership ? Costefficiency
Martín et al., (2009) Stochastic frontier 1. Capital2. Material3. Employees
1. Air trafficmovements
2. Work load units
1. Price of capital2. Price of
material3. Price of
personnel
No Yes No Regulation ? Costefficiency
Ablanedo-Rosas andGemoets (2010)
Data Envelopment Analysis 1. Number of operationsper hour
2. Number of passengersper hour
1. Passengers2. Aircraft
movements3. Cargo
– Yes No No Ownership ? Technicalefficiency
Curi et al. (2010) Data Envelopment Analysis 1. Labor cost2. Capital invested3. Operational costs
1. Planes2. Passengers3. Cargo4. Aeronautical
Sales5. Handling receipts6. Commercial sales
– Yes No No Ownership ? Technicalefficiency
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Table 1 (continued)
Study Method Inputs Outputs Input prices Ownership Regulation Dynamic Associations
Adler and Liebert (2010) Data Envelopment Analysis 1. Staff costs2. Other operating costs3. Declared runway
capacity
1. Passengers2. Cargo3. Aircraft
movements4. Non-aeronautical
revenue
– Yes Yes No Ownership ? TechnicalefficiencyRegulation ? Technicalefficiency
Curi et al. (2010) Data Envelopment Analysis 1. Employees2. Number of runways3. Apron size
1. Passengers2. Cargo3. Aircraft
movements
– Yes No No Ownership ? Technicalefficiency
Gitto and Mancuso (2012) Data Envelopment Analysis 1. Labor cost2. Capital invested3. Soft costs
1. Cargo2. Movement3. Passengers4. Aeronautical
revenue5. Non-aeronautical
revenue
– Yes No No Ownership ? Technicalefficiency
Assaf and Gillen (2012) Stochastic frontier 1. Employees2. Non-labor variable cost3. Number of runways4. Terminal size
1. Passengers2. Aircraft
movements3. Non-aeronautical
revenue
1. Wage rate2. Non-labor
input price
Yes Yes No Ownership ? CostefficiencyRegulation ? Costefficiency
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24 A.G. Assaf et al. / Transportation Research Part B 70 (2014) 18–34
distinguish between short-run and long-run technical and allocative efficiencies. The model is developed using the stochasticfrontier methodology as described in detail in the next section.
4. The econometric model
4.1. Economic foundations
Theoretically consistent models that allow for the estimation of both technical and allocative inefficiencies (the so calledGreene’s problem) have been proposed by Kumbhakar (1997) and were estimated in Kumbhakar and Tsionas (2005). Theirmodels introduced important innovations and provide consistent estimation of both technical and allocative efficiency andserves as the benchmark in this paper.6 Consider that shadow and observed input prices are connected via the relationship:w�j ¼ wjenj , where j = 1, . . . , J, J is the number of inputs, and n2, . . . , nJ are random variables also known as price distortions orallocative inefficiencies.7 Following Kumbhakar (1997) and Kumbhakar and Tsionas (2005) a system of equations, whichaddress the issue of consistent treatment of technical and allocative inefficiency, is as follows:
6 Sin7 We
ln Ca ¼ ln Co þ ln Cal þ uþ vSa
j ¼ Soj þ gj; j ¼ 1; . . . ; J:
ð1Þ
Here, ln Ca and Saj represent actual (observed) log cost and cost shares respectively. Also, u P 0 represents technical ineffi-
ciency, which can be interpreted as the percentage increase in cost due to technical inefficiency, v is a standard error term,and error terms gj are quite complicated functions that are defined in what follows. Moreover,
ln Co ¼ ao þXJ
j¼1
aj ln wj þ cy ln yþ 12cyy ln yð Þ2 þ 1
2
XJ
j¼1
XJ
k¼1
bjk ln wj ln wk þXJ
j¼1
cjy ln wj ln y
which is the usual translog form with y as an output, and without any distortions,
Soj ¼ aj þ
XJ
k¼1
bjk ln wk þ cjy ln y;
are the associated share equations in the absence of input price distortions,
ln Cal ¼ ln GþXJ
j¼1
ajnj þ cy ln yþ 12cyyðln yÞ2 þ 1
2
XJ
j¼1
XJ
k¼1
bjknjnk þXJ
j¼1
cjynj ln y
is the percentage increase in cost due to allocative inefficiency,
G ¼XJ
j¼1
S�j expð�njÞ; and gj ¼So
j 1� Genj
� �þ Aj
Genj;Aj ¼ S�j � So
j ¼ Soj ¼
XJ
k¼1
bjknk
where
S�j ¼@ ln C�
@ ln w�j¼ aj þ
XJ
k¼1
bjk ln w�k þ cjy ln y; j ¼ 1; . . . ; J
Here, S�j represents shadow cost shares-derived by Shephard’s lemma applied to the shadow cost function C⁄ whose func-tional form is:
ln C� ¼ ao þXJ
j¼1
aj ln w�j þ cy ln yþ 12cyyðln yÞ2 þ 1
2
XJ
j¼1
XJ
k¼1
bjk ln w�j ln w�k þXJ
j¼1
cjy ln w�j ln y
where w� ¼ w1;w�2; . . . ;w�J� �
¼ w1en2 ; . . . ;wJenJ� �
. The presence of complicated error terms gj in the basic representation (1)makes estimation of this model quite challenging. Following Kumbhakar and Tsionas (2005) we modify (1) as follows:
ln Ca ¼ ln Co þ ln Cal þ uþ vSa
j ¼ Soj þ gj þ v j; j ¼ 1; . . . ; J;
ð2Þ
where v and vj are statistical error terms distributed according to NJ+1(O, R). In (2) the gj s can then be treated as randomeffects which depend on price distortions (n) but v and vj do not. This provides the means to perform full Bayes inferenceof the model using MCMC, organized around the Gibbs sampler, with data augmentation.
ce the model has been presented in detail in these papers, we only state the pertinent results.can assume n1 = 0 without loss of generality.
A.G. Assaf et al. / Transportation Research Part B 70 (2014) 18–34 25
4.2. Econometric formulation
As an extension to the model in (1), we propose the following:
8 It re9 Not
log uit
nit
� �¼ g0 þ C
log ui;t�1
ni;t�1
� �þ
e1;it
e�1;it
� �ð3Þ
In the case of M = 3 inputs we would have the following model structure:
log uit ¼ g01 þ C11 log ui;t�1 þ C12f1;i;t�1 þ C13f2;i;t�1 þ z0itk1 þ e1;t ;
n2;it ¼ g01 þ C21 log ui;t�1 þ C22f1;i;t�1 þ C23f2;i;t�1 þ z0itk2 þ e2;t ;
n3;it ¼ g01 þ C31 log ui;t�1 þ C32f1;i;t�1 þ C33f2;i;t�1 þ z0itk3 þ e3;t ;
In this model, inefficiency and price distortions are related to a common vector of covariates (zit) and, conditional on thatvector, they follow a vector autoregressive (VAR) scheme. This again follows Kumbhakar and Tsionas (2005) closely in termsof modeling the price distortion parameters (n), but in terms of technical inefficiency, we use here a dynamic inefficiencymodel. Details about our Bayesian model development can be obtained from the authors upon request. For details aboutthe sensitivity of the model to various priors, refer to technical Appendix A.
5. Data and model specification
We use several sources to collect the data for this study including the ATRS Global Benchmarking reports (e.g. Air Trans-port Research Society 2008), annual reports of various airports, the Australian Competition and Consumer Commission, andthe Federal Aviation Administration. The final sample consisted of 71 international airports located in Europe, North Amer-ica, and Australia, for the years 2003–2008 (71�6 = 426 observations).
To estimate the cost function, we need information of total cost, input prices, and outputs. Following Table 1 and otherrelated studies in the literature (e.g. Fung et al., 2008; Pathomsiri et al., 2008), we identified the following four outputs: one,international passengers, two, other passengers, three, aircraft movements, and four, revenue from non-aeronauticalservices. These outputs, in particular passenger and movements are common in the literature (Barros and Dieke, 2007;Assaf, 2009, 2010). We differentiated between domestic and international passengers to capture the impact that interna-tional operations would have on airport costs International services are provided the vast majority of the time by widebodyaircraft. Finally, despite being ignored in several studies, the ‘‘non-aeronautical revenues’’8 is also a key output (Oum et al.,2008), and is becoming a major source of revenue for airports, accounting in some cases for up to 60% of total revenues.
For inputs, we select the following two variable inputs: one, labor, measured by the number of employees who workdirectly for an airport operator, and two, non-labor input that would include materials and contracted services. We measurethe price of labor input as the average salary per employee (including benefits). The non-labor variable input includesnumerous items and activities it would be very difficult to form a price index which would be consistent across differentairports in different countries and over time. Therefore, we used the Purchasing Power Parity (PPP) as a proxy for thenon-labor input price (Oum et al., 2008). PPP are calculated using multilateral, symmetric and transitive price indices. Wealso included two fixed capital inputs (the number of runways, and the total size of the passenger terminal area measuredin square meters). Hence, as our capital inputs are fixed, we label from here the cost function we estimate as a variable costfunction. Finally, we also added two control variables to capture specific effects; first, if an airport is a gateway is will have alarge number of international traffic and numerous international airlines. This will impose some higher costs on the airport,thus we added ‘gateway dummy’; second, some airports in the sample are used by FedEX, UPS and DHL large integratedcourier companies and some gateway airports are also cargo hubs. To capture any outstanding impacts on these airportswe have added a ‘cargo hub’ dummy variable. Table 2 provides descriptive statistics for all input and output variables.
Finally, in order to analyze the impact of the various ownership/regulation forms on airport efficiency, we classified theairports in our sample into the following ownership/regulation dummy variable categories:
a. Airports that are government owned and RORb. Airports that are government owned and single-till regulatedc. Airports that are government owned and do not follow any type of regulationd. Airports that are partially private and single-till regulatede. Airports that are partially private and dual-till regulatedf. Airports that are fully private and follow a price monitoring systemg. Airports that are fully private and single-till regulated9
fers to revenues generated from concessions, parking and numerous other services.e that UK airports have now moved away from regulation. However during our sample period designated UK airports were still regulated.
Table 2Summary statistics of input and output variables.
Mean Standard deviation Min Max Geometric mean
No of international passengers (in million) 9.78 12.16 4.21 42.10 3.23No of domestic passenger (in million) 14.75 15.41 5.79 61.11 7.43No of movements (in 000’s) 322.33 211.56 113.01 893.78 263.49Non-aeronautical revenues (in 000’s $) 187.34 208.62 12.15 212.40 106.52Number of employees (in 000’s) 1.28 1.37 0.71 7.03 7.63Non-labor costs (in 000’s US$) 115.84 49.40 11.00 323.00 105.37Number of runways 3.12 1.33 1.00 7.00 2.85Terminal size 192.11 137.50 14.58 592.58 143.23
26 A.G. Assaf et al. / Transportation Research Part B 70 (2014) 18–34
6. Model estimation
The variable cost function is modeled using a translog form consisting of the variable cost function and the cost shareequations. The formulation leads to a nonlinear random effects system model in a panel setup where technical and allocativeinefficiencies are separated from random noise terms, as explained in Section 4 above. The translog model can be written asfollows:
10 The11 For12 Cos
ln C�it yit;wit ; kit; t; controlsð Þ ¼
a0 þX4
j¼1
cjyjit þX2
j¼1
aj ln wjit þX2
j¼1
kjkjit
þ 12
X4
j¼1
X4
n¼1
hjn ln yjit ln ynit þX4
j¼1
X2
n¼1
/jn ln yjit ln wnit
þX4
j¼1
X2
n¼1
1jn ln yjit ln knit
þ 12
X2
j¼1
X2
n¼1
bjn ln wjit ln wnit þX3
j¼1
X2
n¼1
gjn ln wjit ln knit
þ 12
X2
j¼1
X2
n¼1
qjn ln kjit ln knit þX2
j¼1
djcontrolsjit
where yjit represent the outputs (international passengers, other passengers, movements, non-aeronautical revenues), wjit
represent the input prices (labor, PPP), kjit represent the fixed capital measures (number of runways, size of the passengerterminal area), and controls represent the control variables which include the two dummy variables representing gatewayand cargo hub airports.
We used 40,000 MCMC iterations, and our posterior results are based on the last 10,000 draws. Convergence assessmentis based on Geweke’s (1992) convergence diagnostics.10 To ensure that the cost function satisfies the theoretical expectation,we checked monotonicity and concavity requirements. Fig. 1 reports the elasticities with respect to the variable input price, andthe three outputs. We can see that the total cost is increasing with the input prices, and the outputs, hence satisfying the mono-tonicity property. To test whether the variable cost function is concave relative to the input prices, we derived the Hessianmatrix of the cost frontier relative to the input prices. As shown from the density in Fig. 2, the determinant of the Hessian matrixis negative, indicating that the concavity requirement has been met. For more details on these issues refer to Coelli et al.(1998)11.
We present in Table 3 the measures of the short run impacts of each of the ownership and regulation dummies on tech-nical and allocative inefficiencies; we use as the base case an airport that is privately owned and has price monitoring as aform of price regulation; examples of such airports are Australian and New Zealand airports. This configuration was selectedas the base case since it is generally found to be to have the highest cost efficiency (see, Assaf and Gillen, 2012; Oum et al.,2008).12 The values contained in Table 3 are interpreted as, what would happen to cost efficiency, and its components technicaland allocative efficiency, with a move to an ownership/regulation configuration, different from the base case.
Examining the short-run results for technical inefficiency in Table 3, we can see that a move from the base case to anyother ownership/price regulation configuration results in an increase in technical inefficiency except for the instance of pri-vate ownership and single till regulation. In this case the one thing that differs from the base case is the form of price reg-ulation. This exception is not an unexpected result, earlier work by Assaf et al. (2012), found that under single till, airports
se are available from the authors upon request.space limitation, we do not present the complete results of the translog estimation but these can be obtained from the authors upon request.t efficiency is technical efficiency times allocative efficiency.
Labor Price Passengers
Aircraft Movements Revenue from non-aeronautical service
01
23
4
0 .1 .2 .3 .4 .5
0.5
11
.52
2.5
0 .2 .4 .6 .8 1
0.5
11
.5
0 .5 1 1.5 2
0.5
11
.52
2.5
0 .5 1 1.5
Fig. 1. Monotonicity conditions.
0.5
11
.52
-1.5 -1 -.5 0
Fig. 2. Distribution of determinant of Hessian for concavity.
A.G. Assaf et al. / Transportation Research Part B 70 (2014) 18–34 27
were cost efficient, which is what price cap regulation was designed to do, to incentivize airports to reduce costs in order toincrease profit rather than to simply raise prices to improve profits.13 This form of price regulation improves technical effi-ciency because the airport is incentivized to expand output to achieve cost economies due to fixed asset utilization and thiscoupled with prices that are regulated increases profit.14
In every other case a move away from the base case results in an increase in technical inefficiency or an output level thatdiffers from that which would occur under the base case. There is a range of results for changes to technical inefficiency if
13 It should be kept in mind that we are measuring cost efficiency not highest profits. Privately owned airports have an incentive to maximize profits and inthe absence of price regulation their profits would be higher but so would their costs.
14 Heathrow airport fits the category of private ownership and single till and it is slot-constrained meaning the assets are fully utilized.
Table 4Long-run estimates from differences in ownership and regulation from base case.
Mean St. dev Coeff. variation
Impact on technical inefficiencyGovernment & single till 0.092 0.014 0.152Fully private & single till 0.165 0.012 0.073Partially private & dual till 0.051 0.009 0.176Partially private & single till 0.085 0.012 0.141Government & no regulation 0.155 0.013 0.084Government & ROR 0.181 0.007 0.039
Impact on allocative inefficiencyGovernment & single till 0.136 0.014 0.103Fully private & single till 0.119 0.011 0.092Partially private & dual till 0.025 0.007 0.280Partially private & single till 0.151 0.031 0.205Government & no regulation 0.087 0.014 0.161Government & ROR 0.163 0.032 0.196
Table 3Short-run estimates from differences in ownership and regulation from base case.
Mean St. dev Coeff. variation
Impact on technical inefficiencyGovernment & single till 0.201 0.265 1.321Fully private & single till �0.886 0.326 �0.370Partially private & dual till 2.496 0.356 0.141Partially private & single till 0.797 0.329 0.551Government & no regulation 2.076 0.165 0.080Government & ROR 0.046 0.193 4.201
Impact on allocative inefficiencyGovernment & single till �0.112 0.064 �0.571Fully private & single till 0.413 0.171 0.411Partially private & dual till 0.471 0.109 0.230Partially private & single till 0.566 0.304 0.541Government & no regulation 0.356 0.095 0.270Government & ROR 0.465 0.101 0.221
28 A.G. Assaf et al. / Transportation Research Part B 70 (2014) 18–34
either or both ownership form and price regulation changes from the base case. Two interesting cases are a move to partialprivatization results in increases in technical inefficiency but the increase is greater under dual till than single till. With par-tial privatization the private owner is most always in a minority position. With partial privatization the objective function forthe airport would move from profit maximization to a mix of government objectives such as promoting economic develop-ment plus an acceptable level of profit.15 Under single till the price regulation is stricter and would provide more cost disci-pline. The second outcome of interest is a shift to government ownership no regulation; this would describe U.S. airports. A shiftto this ownership/regulation combination from the base case increases technical inefficiency considerably. U.S. airports cannotmove profits out of the airport and cities, counties or a combination of the two own most airports. The incentive is to have alocal view and promote connectivity using the airport. There may be overinvestment in capacity with the view that thepotential for more flights (output) is always better but actual output is less than capacity; that is excess capacity is viewedas a positive for potential flight growth rather than a waste of resources.
Thus the conclusion is that adding economic regulation generally leads to reduced technical efficiency, too little output isproduced, and this outcome is exacerbated under government ownership. Imposing economic regulation will, in the short-run, change incentives. It may be that economic regulation reduces the return from expanding output, given the currentinput structure. A move away from private ownership changes incentives and the airports’ objective function.
Examining the lower part of Table 3 we can see the results for changes in allocative efficiency; that is, cost deviations dueto an inefficient input mix of labor, service contracts and other inputs (which in the model are held constant) given what isproduced. What we see is quite different from the outcomes for technical efficiency. The move away from the base caseresults in a small increase in allocative inefficiency in each case except government and single till. The results differ in a rel-atively small way whichever ownership/price regulation regime is chosen. Notably the move to government ownership andsingle till price regulation reduces allocative inefficiency in a small way. Single till disciplines cost efficiency under govern-ment ownership but under private or partially private ownership increases allocative inefficiency. The evidence, Gillen
15 For example, Government may be more interested in using the airport as an economic development tool and promoting access to more distant markets atthe expense of local markets; having an international airport creates greater visibility and prominence.
A.G. Assaf et al. / Transportation Research Part B 70 (2014) 18–34 29
(2011) is that under private ownership and single till quality may be sacrificed to reduce costs; for example, contracted ser-vices may be reduced resulting in more allocative inefficiency.
The minor changes in short run allocative inefficiency may be explained by the fact the airport production technology forproducing flights and passenger trips is relatively standard so there would be less latitude in deviating from the standardmethod in which airports are set up; the input mix is well established and expands in a relatively narrow range of run-way/terminal area ratio.16 We do not have a price for capital in the model and thus the measure of the allocative inefficiencydistortion includes only labor and contracted services. It is the inefficient mix of these two inputs that is captured in the mea-sures of allocative inefficiency in the model.
Table 4 contains the results calculated for long-run changes in inefficiency from moving to the base case, private owner-ship and price monitoring economic regulation. The long run is considered a steady state and reflects the likely values thatwould occur for a given ownership/regulation combination in the long run. The impacts are separated for technical and allo-cative inefficiency. To calculate these long-run effects of changing ownership on technical and allocative inefficiency we tookthe following steps:
i) Suppose a particular element of the zit variables changes from 0 to 1 or 1 to 0 depending on the case we want to exam-ine. Suppose, without loss of generality, that the first element of this vector changes.
ii) As MCMC draws for all parameters and latent variables (including technical inefficiency and price distortions) areavailable, we find the steady state solution for uit and nit in (3) for each MCMC draw. We compute ln Cal from (2b)and uit from (3) using both zit=0 and zit ¼ 1. These expressions correspond to the long-run or steady-state valuesfor technical and allocative inefficiency.
iii) We record the difference in both technical and allocative inefficiency. This difference is both airport-specific and time-specific.
iv) In standard Rao–Blackwell fashion the differences are averaged across MCMC draws. The average represents the pos-terior mean of change in technical and allocative inefficiency under a different ownership/regulation scheme17.
The data used to estimate the model ranged from 2003 through 2008, a period over which airport management would beable to adjust inputs to reduce costs and adjust output and only labor and contracted services are included as the adjustableinputs. From Table 4 what is immediately evident are the smaller magnitudes in general relative to Table 4, which containthe short-run impacts. The long-run results show all combinations of ownership/price regulation are inferior to the base caseof private ownership with price monitoring. Secondly, the differences between technical and allocative inefficiency resultsare much closer in the long-run than was the case for the short-run results.18 Thirdly, the coefficient of variation is for themost part smaller for both sets of results, technical and allocative inefficiency than was the case for the short run results.
In the long-run airports have the ability to adjust both their output as well as their input mix. Our results show that gen-erally reductions in allocative inefficiency are higher by moving away from rigid regulation or providing market disciplinethrough full privatization. Rigid regulation affects investment incentives and the labor-contracted services input mix willhave the largest gains from privatization. However, given privatization is in place, move away from rigid regulation. The out-come for technical inefficiency in the long run exhibits a similar tendency. Specifically, technical inefficiency decreases bymoving away from government owned to fully privatized airports and moving away from rigid regulation.
7. Summary and conclusions
The literature on airport economics has shown that ownership form and economic regulation are important as discussedin Section 3 above. The previous airport economics literature has failed, in three ways, to provide intelligence regarding air-port efficiency performance. First, either ownership or regulation was considered, but not both. Second, there was not a dis-tinction made between technical and allocative efficiency, which is important since each require different strategies forimprovement. Third, there were no dynamics therefore improvements in performance over time was not considered or mod-eled and the impact of introduced changes to regulations, for example, not assessed. This paper considers all three aspectsthat were missing from the literature.
We investigate the impact of both ownership and regulation on airport technical and allocative efficiency. We differen-tiate between the short and long-term effects. Based on a large sample of international airports, we find that adding eco-nomic regulation leads to a decrease in technical efficiency in the short-run. A shift from fully private to partially privateownership always decreases technical efficiency. The results from short-run allocative efficiency showed mixed conclusions.
16 A simple regression of the number of runways on a terminal area found a linear relationship, and each new runway results also in an addition of 50,000square feet of terminal space.
17 Next we describe how long- run effects can be computed when the weakly exogenous variables are not necessarily dummy variables. A particular elementof zit changes from its current value to zit + h where h is set to 0.001 times the minimum sample value and we follow steps (ii) through (iv) above and thedifference in technical and allocative inefficiency is divided by h. This procedure does not require computing explicitly derivatives of steady-state technical andallocative inefficiencies with respect to the elements of zit.
18 Of course in the long run you have two degrees of freedom, changing output and changing the input mix whereas in the short run it is output that ischanged.
30 A.G. Assaf et al. / Transportation Research Part B 70 (2014) 18–34
In the long-run we find that technical and allocative inefficiency decreases by moving away from government owned to fullyprivatized airports and moving away from rigid regulation. The differences between technical and allocative inefficiencyresults are much closer in the long-run than was the case for the short-run results.
The value of separating efficiency into the components of technical and allocative and to distinguish the short and long-run differences in possible inefficiency changes has been clearly demonstrated. In the short-run the majority of the improve-ments are from reducing technical inefficiency. This will come for the most part from adjusting output, something that canbe accomplished in the short-term, recognizing that airports that rely heavily on airside output will have fewer degrees offreedom. There are relatively small changes, in the short-run, resulting from improving allocative efficiency since the sourceof change lies in adjusting the input mix of airport labor and contracted services or improving productivity, something moredifficult to do in the short-term. Quite different conclusions hold for the long-term; there are improvements available fromreducing allocative inefficiency and comparable benefits are available from cutting technical inefficiency for a smaller subsetof airports where government is a partial or whole owner, or where there is rigid price regulation. The modeling shows thereductions in inefficiency are available in the short-run from an output fix and airport managers should focus their efforts onoutput and focus relatively less on inputs. In the long-run, airport managers have to adjust, and optimize across both inputmix and output.
Future research might consider extending the current study both in terms of methodology and content. For example, itmight be worth developing the proposed model using more flexible functional forms such the Fourier. Extending the timeperiod and scope of the study would permit the introduction of variables that capture changes in airline business models andstrategies; for example, do the different alliance hubs have lower or higher costs. Two important areas for research in airporteconomics are to obtain a measure of the price of capital for airports and to estimate a complete cost function in order tovalidate the current findings and to better capture differences in airside and non-airside contributions to costs. With amulti-product firm with highly complementary but distinct outputs of airside operations and terminal revenues with alsomarkedly dedicated inputs, how are cost efficiencies measured and how do they interact?
Appendix A.
Prior sensitivity analysis
Suppose h is the vector of structural parameters and z ¼ u; nð Þ denotes the latent variables of the problem. Given a base-line prior Po(h), MCMC draws can be obtained from the posterior kernel Po(h|Y) and Po(z|Y). The posterior meansEoðhjYÞ ¼
RhpoðhjYÞdh and E0ðzjYÞ ¼
RzpoðzjYÞdh (or variances and standard deviations) can be easily computed. The question
is how these moments change when the prior departs from the baseline prior, Po(h).Let L(h;Y) denote the likelihood and Po(h,z|Y) the joint posterior of structural parameters and latent variables under the
new prior, P(h). Clearly,
EðhjYÞ ¼R
hpðhjYÞdhRpðhjYÞdh
¼R
h PðhÞPoðhÞ Lðh; YÞPoðhÞdhR PðhÞ
PoðhÞ Lðh; YÞPoðhÞdh’ S�1PS
s¼1hðsÞws
S�1PSs¼1ws
;
where Ws ¼ PðhðsÞÞPoðhðsÞÞ
, and {h(s), s = 1, . . . , S} is a sample from the posterior kernel PoðhjYÞ / Lðh; YÞPoðhÞ. Next, we computeposterior expectations of the latent variables. We have
EðzjYÞ ¼R
zpðzjYÞdzRPðzjYÞdz
¼R
zpðzjh; YÞPðhjYÞdzRpðzjh; YÞPðhjYÞdz
¼R
zpðzjh;YÞLðhjYÞpðhÞdzRpðzjh;YÞLðhjYÞpðhÞdz
¼R
zpðzjh;YÞ pðhÞpoðhÞ
LðhjYÞpohdzRpðzjh;YÞ pðhÞ
poh LðhjYÞpohdz’ S�1PS
s¼1wszðsÞ
S�1PSs¼1ws
where ws ¼ pðhðsÞÞpoðhðsÞÞ
, and {h(s), z(s), s = 1, . . . , S} is a sample from the posterior kernel Poðh; zjYÞ / Lðh; z; YÞpðzjhÞPoðhÞ. Since this
sample is available from MCMC sampling of the baseline model, computation of the weights is straightforward. Thus theapproximation to the posterior expectation under the new prior is straightforward.
The condition is to choose priors of h that have (at most) thinner tails relative to the baseline prior. This condition is toorestrictive only when the baseline prior is rather informative. With informative but highly diffuse priors numerical problemscannot arise (for example, weights that are close to zero).
To compute approximations to the Bayes factor under the new prior consider the marginal likelihood
lðYÞ ¼Z
Pðh; zjYÞdhdz ¼X
Pðzjh;YÞLðh; YÞpðhÞdhdz;
from which
lðYÞ ¼Z
Pðzjh;YÞLðh; YÞ pðhÞpoðhÞ
poðhÞdhdz ¼Z
Pðzjh;YÞ pðhÞpoðhÞ
poðhjYÞdhdz
-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
relative difference of posterior means
Posterior means of structural parameters (differences from baseline)
-0.06 -0.04 -0.02 0 0.02 0.04 0.060
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
relative difference of posterior s.d.
Posterior deviations of structural parameters (relative differences from baseline)
Fig. B1. Relative differences of posterior means and posterior s.d. of structural parameters from baseline prior.
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.20
0.05
0.1
0.15
0.2
0.25
0.3
0.35
relative difffferenence of poposterior meaneans
Posterior meaeans of latenent t variabables (r(relative differenences from babaseliline)
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.150
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
relative difference of poposterior s.d.
Posterior s.d. . of latenent t variabables (r(relative differences from babaseliline)
Fig. B2. Relative differences of posterior means and posterior s.d. of latent variables from baseline prior.
A.G. Assaf et al. / Transportation Research Part B 70 (2014) 18–34 31
which implies lðYÞ ’ S�1PSs¼1ws, a quantity that has been computed in the approximation of the posterior mean. The Bayes
factor in favor of Po(h|Y) and against P(h|Y) is simply BF ¼ loðYÞlðYÞ if the prior odds ratio is 1. To compute the baseline marginal
likelihood loY we rely on the identity loðYÞ ¼ pðhjYÞLðh;YÞpoðhÞ
, from which loðYÞ ¼ pðhjYÞLðh;YÞpoðhÞ
, where we take h to be the posterior mean
from the baseline model. The denominator is trivial to compute. For the numerator we use the Laplace approximation:log pðhjYÞ ¼ � 1
2 logjPj, under approximate posterior normality, where
Pis the posterior covariance matrix of h, viz.
-6 -4 -2 0 2 4 6 8 10 12
x 10-3
0
0.05
0.1
0.15
0.2
0.25
relative difference
Posterior means of steady states (rel. differences from baseline)
-0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.0150
0.05
0.1
0.15
0.2
0.25
0.3
0.35
relative difference
Posterior means of IRF (rel. differences from baseline)
Fig. B3. Relative difference of posterior means of impulse response functions from baseline prior.
-2 -1.5 -1 -0.5 0 0.5 1 1.5 20
0.05
0.1
0.15
0.2
0.25
difference in log Bayes factors
Fig. B4. Bayes factors against competing models from different priors.
32 A.G. Assaf et al. / Transportation Research Part B 70 (2014) 18–34
P¼ S�1PS
s¼1ðhðsÞ � hÞðhðsÞ � hÞ0, h ¼ S�1PS
s¼1hðsÞ. For numerical stability reasons we consider only the draws for which
ðhðsÞ � hÞ0P�1ðhðsÞ � hÞ 6 v2
0:95;dimðhÞ, where v20:95;dimðhÞ denotes the upper 95% critical value of the chi-square distribution with
degrees of freedom dim(h).Based on these approximations we can examine posterior sensitivity to the prior assumptions using a vast number of
prior distributions. Specifically, the functional forms of the priors are as described in the main text but their parametersare varied according to univariate uniform distributions in the interval [�10, 10] for parameters that can take values inthe real line and [10�6, 20] for positive parameters (like standard deviations, etc). We use 10,000 such priors for the struc-tural parameters h, and we examine
(i) sensitivity of posterior means of h and posterior standard deviations,(ii) sensitivity of posterior means and posterior standard deviations of the resulting latent variables, generated from
p(z|h, Y).
A.G. Assaf et al. / Transportation Research Part B 70 (2014) 18–34 33
The results are presented in Figs. B1–B4 below and show that posterior moments do not differ greatly relative to the base-line specification (prior A).
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