Discrete Choice Models in Transport. - An Application to Gran Canaria- Tenerife Corridor

7/30/2019 Discrete Choice Models in Transport. - An Application to Gran Canaria- Tenerife Corridor

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Discrete Choice Models in Transport:

An application to Gran Canaria- Tenerife corridor

Jos Mara Grisola Santos

Universidad de Las Palmas de GC

Departamento de Anlisis Econmico Aplicado

Universidad de Las Palmas de GC

Campus Universitario de Tafira Baja

35017 Las Palmas de GC

Telfono: 928458195Fax: 928458183

E-mail: [email protected]

Abstract

Discrete choice models analyse individuals decisions when they face choices among several

alternatives. In the last decades these models have shown a notable improvement, with applications to

a wide variety of fields, especially in transport. This work uses discrete choice models to analyse thecorridor between two islands, Gran Canaria and Tenerife. This corridor, with more than two million of

annual trips, constitutes the most important transport demand of Canary Island and one of the most

important of Spain.

Between these two islands there are four available modes (plane, ferry, fast ferry and slow ferry) and,

over this scenario, a survey of Stated and Revealed Preferences (SP and RP) is carried out. Data is

used to estimate logit models and mixed logit models obtaining different values of time. Mixed Logit

is the most advance model of discrete choice. It gives a wide flexibility to the researcher and allows

for individual parameter estimation.

Results are able to reproduce partially, previous value of time estimated in the same market. The highvalue of time obtained, compared with the wage rate, suggest a re-valuation of public investment

assessments. In addition, the results permits understand recent changes in this market thanks to the

transference from in-vehicle time to access time, which is less valuable for travellers.

Keywords: Value of time, mixed logit (ML), discrete choice analysis, transport demand.


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1. INTRODUCTION

Canary Island has a population near of two million of inhabitants, 85% of those are concentrated in

Gran Canaria and Tenerife. The distance between these islands is around 300 kilometres, and due to its

economic importance, this is the most transited corridor of the island. Transport facilities in both

islands are modern and well developed. In maritime transport there are two routes: direct trip from Las

Palmas (in the north-east) to Tenerife or, much shorter, from Agaete which is situated in northwest of

Gran Canaria.

With 1.410.642 passengers trips in 2001, traffic between these islands has increased significantly

from 1993. Essentially, there are three modes and four companies:

Plane: there is a public air company,Binter.

JetFoil: served by a public enterprise Transmediterranea.

Ferry: there are three companies and two routes:

a) Route A, is the longest route, and goes from the main port of Las Palmas to Santa Cruz de Tenerife.

Takes about 3 hours and 30minutes. Ferry Transmediterranea andferry Armas use this route1.

b) Route B, which goes from Agaete to Santa Cruz de Tenerife. It takes one hour. The only operator is

Ferry Fred Olsen. An important part of travellers use their cars to drive from Las Palmas to Agaete.

The company also offers a free bus service. Furthermore, it is a sort ofmixed service car/bus-ferry.

Thanks to the liberalisation of the market, Ferry Fred Olsen started to offers its service in 1993. Theshorted trip and the flexibility of use the cars (also available in ferry Armas from Las Palmas) led this

company to a success. In terms of the whole market a notable increasing of trips and drop of prices

was observed: the new offer not only attracted passengers from other companies but expanded demand

in near of half million new passengers2.

Table 1.1: Modal split in 2001

Company Market shared

Ferry Fred Olsen 42.50%

Jet Foil 24.78%

Plane 24.26%Ferry Armas 8.46%

Source: Transport operators

Table 1.1 shows the current modal split. It is noticeable that Fred Olsen is the leader of market with

42,50% of all trips. Before its service started, the plane and jet foil shared the market with near to 50%

each.

1 Transmediterranea ferry occupies a marginal position between the two ferries. Less than ten passengers perday are transported everyday because it is devoted to freight transport. In order to simplify the exposition wewill not mention again ferry Transmediterranea, although it was considered during the survey in terms of

design (in the questions asked to respondents) but for budget reasons we refused to include its passengers . Itmust take into account that this work analyses only passengers demand and not freight transport.


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The availability of car in the complete trip makes this mode more attractive. The strong value of car

availability could explain the success of the new mode. Moreover, it seems that travellers prefer make

part of the stretch (Las Palmas to Agaete) in car than take a ferry straight from Las Palmas. The

particular perception of costs for car users is behind this behaviour.

In this work, an inter-island route in Canary Island will be analysed using discrete choice analysis.

This is a traffic corridor with a particular high density where 3 modes and 4 companies are competing.

Also is a very dynamic market, which has suffered sharp transformations in recent years due to the

liberalisation of maritime transports in the UE.

Our objective is model this demand and obtain and a variety of attribute values, specially the value of

time. With this purpose, a survey was carry out using stated preference and revealed preference

techniques in its design.

This paper is structured as follow: first, there is a brief revision of the theoretical issues that support

this work. The next section is devoted to the design of questionnaire. The fourth part is the stage of

modelling, where is specified the models to be used in the following part. The fifth section shows the

results in terms of value of time and the final section are the conclusions.

2. THEORETICAL FRAMEWORK

This section introduces the theoretical framework that supports the work, that is, the Random Utility

Theory. The purpose of this theory is modelling choices of individuals in different contexts. In

transport, we are interested in model the rational process of choice a mode j within a choice set ofAj

alternatives. Theory of Random Utility (see, for instance Ben-Akiva, 1985) postulates that the utility

function of an optionj for an individual n is determined by

jn jn jnU V = + (2.1)

In equation (2.1) we can distinguish a deterministic part called Vjn and a random component jn.

Residuals are identical and independent and identically distributed (IID). They represent both the

idiosyncrasies and specific preferences of each individual and the measurement errors. The

deterministic component, Vjq is a function of level of attributes of existing options x pondered by

coefficients . Thus,

=K

jkqKjjqV (2.2)

2 The interested reader about the effect of this liberalisation can read De Rus (1997)


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Give this framework, theory says that individual will select the alternative which maximise his utility.

Hence, individual q will select alternativej if:

,( )

jn in iU U A A q (2.3)

This leads to:

jn in iq jqV V (2.4)

Depending of distributions of disturbances two types of models arise: ifin is assumed to be normally

distributed a model probit is obtained. Under the assumption of logistically distributed disturbances

(Gumbel distribution) is obtained the logit model. The former is more complex and the later, simpler

and easier to use. We will use a logit in this work.

There are several kinds of logit models. Here, we will develop two of the most popular specifications:

Multinomial Logit Model (MNL) and Hierarchical Logit Model (HL). In MNL model if residuals are

distributed IID Gumbel it can be proved (Ortzar and Willunsen, 2001) that the probability that

individual q chose alternative i equals:

exp( )

exp( )j

iniq

jn

A Aq

VP

V

=

(2.5)

Where is a parameter related to the common standard deviation of the Gumbel distribution. In

practise, it cannot be estimated separately from parameters k. If there is correlation between

alternatives (i.e. some alternatives are more similar than others) or taste variation among individuals,

the MNL is not appropriate. In these cases is more adequate using HL. In a HL model the utility of the

composite alternatives is represented by:

zEMUVI += (2.6)

Again, (2.6) has two components. The first term, EMU, is the expected maximum utility of the

alternatives of the nest. EMU is derived from the following expression:

=j

j)Wexp(logEMU (2.7)

In (2.7) Wjis the utility function of alternative j where all common attributes z of the nest have been

taking out. Thus, the second term z is the vector of common attributes of the nest and parameters .The estimation process of these models focuses in obtaining estimation of the parameters *k in the

utility function (2.2). The method used is maximisation of likelihood (ML). Since we observe choices

from individuals, consider for example that individual 1 selects alternative 2, and individuals 2 selects

alternative 4, and so on. The Likelihood function is the result of the product of each probability. Thus,

"322412 PPP)(L = (2.8)


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Hence, it is necessary to find a specification, which can be maximised. After several transformations

(see Ortuzar and Willunsen, 2001) and taking logarithms it is possible to obtain the log of likelihood

(2.8) which is the function to be maximised.

=q j

jqjq Plogg)(Llog (2.9)

Once the set of *k parameters have been estimated, the next step is use the (2.5) to obtain

probabilities of each alternative. In the case of HL it will be necessary to calculate first the marginal

probability of each option inside the nest and, after that, multiply for the probability of the nest.

Despite of its popularity MNL has many shortcomings due to the fact of its assumptions. On of the

most important disadvantages is the well kwon paradox pointed out by Debreu (1960) of red bus/ blue

bus: if there is a new model introduced in the market, the ratio of probabilities of previous models

does not change. Also, MNL is not able to represent the variety of tastes of consumers because it

assumes a fixed structure of parameters. In addition, MNL presents problems of estimation in case of

repeated choices which is the case of SP.

Mixed Logit (ML) is a more general model which avoids all problems we have explained of logit and

probit. Thus, ML contains a wide flexibility due to the fact that parameters vary among costumers. An

excellent explanation of this model is found it in Train (2003). Earlier applications can be found in

Ben-Akiva et al (1993) but it was recently, with the advances in software for simulation, when ML has

become in the most popular model for discrete choices.

ML is a general model: modeller does not know nand so, the probability that individual n chooses

option j is a conditional probability in . Assuming that = b

)()( bPPP njnj == (2.10)

Conditional probability Pnj is just the simple logit. In the case of fixed parameters, ML collapses into

MNL. Ifn is a discrete variable, Pnj would be the sum of all probabilities conditioned to each n

weighted with every probabilityn=bm. This is called the latent classes model:

=

=M

m

nnjmnj PsP1

)(. (2.11)

If we considern continuous, it is necessary to use an integral where probability is weighted with a

density function f() which is the most used expression of ML and the one we developed in this

article.

dfe

eP

j

xb

xb

njnim

nim

)('

'

= (2.12)


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Now, the modeller has to estimate two sets of parameters: mean b and co-variance matrix W. Bothe

can be denominated . Since the researcher can select whatever distribution for these parameters, the

distribution selection is one of the most relevant issues in the estimation procedure. Most popular

distributions are fixed, normal (which allows for a complete variation in the parameters), uniform,

triangular and lognormal. This last distribution might be a solution for incorrect sign in those

parameters whose signs are previously known. However, lognormal distribution also produces

difficulties in the estimation (Hensher and Green, 2003).

There are two procedures for estimation in ML: classis and Bayesian:

Classic estimation, which is used in this paper, consists in a maximisation of a log likelihood using

simulation procedures. In a first stage the method involves the following steps: (1)Given , take draws

from the distribution f(/) (2) Calculate simple logit Lni for each draw (3)After several repetitions

average the results. This average is a unbiased estimator ofPni

=

=R

r

r

nini LR

P1

)(1

(2.13)

These simulated probabilities are inserted in the log likelihood function

= =

=N

n

J

i

njnnj PLdSLL1 1

(2.14)

Maximising (2.14) estimatoris obtained.

Bayesian estimation does not need to maximise any function. Its results are based in Bayes theorem

which postulates a relationship between a prior distribution (a previous knowledge about the

phenomena) and a posterior distribution. This relationship will be proportional like:

)()/()()/( kYLYLYk = (2.15)

Where k() is a prior distribution; k(|Y) is the posterior distribution; L(Y) is the probability to obtain

the observed choices in the sample and L(Y/) is the probability of these choices conditional on .

Then, it is possible to derive:

)(

)()/()/(

YL

kYLYk

= (2.16)

From (2.16) the researcher will have to estimate which can be expressed as the mean of posterior

distribution

= dk )( (2.17)

3. DATA ANALYSIS AND QUESTIONARY DESIGN

In this section we describe the process of data collection and the design of the questionnaire and SP.


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In the design of questionnaire, around 30 questions were included in the questionnaires. These

questions were about origin and destination, frequency and motive of trip, costs of current mode,

perceived times and costs of other modes available, an SP exercise and, finally, socio-economic

questions as age, sex, household size and composition, job and income level. A survey of 420

travellers were carried out with this survey.

Regarding to SP, a ranking design was chosen for this work for operational reasons. Main issues to

consider in the SP design are the levels of attributes, and structure of competition. By considering

these issues, a master plan for number and levels of attributes is designed. In this early stage we collect

basic information about the primary attributes of every mode in order to determine the relevant

attributes in each mode Table 3.1 shows the relevant information.

Table 3.1: current level of attributes in all modes

Ferry Armas Ferry FO Jet Foil Plane

Time (minutes) 210 60 80 30

Average fare 9.3 18 47 40

Frequency 2 per day 4 per day 3 per day Hourly

Accessibility

(from city)In the city

40 by car

60 by busIn the city 20 by car

Modes may be classified into two groups. Modes with car availability, given by Ferries, that are

relatively slow but have the advantage to carry your own car to reach your final destinations. And

Modes without car availability, that are faster and comfortable. Ideal for business travellers who want

a day trip. Jet Foil has the advantage to departure from the port of the city whereas the plane travellers

have to move to the airport situated 20 minutes by car from the capital. However, frequency of the

plane is much higher, almost hourly and the trip last only 30 minutes. In Jet Foil there are two classes

but the plane only has a unique class. Thus, the relevant attributes in the SP experiment could be:

Ferry Armas: fare, time and car availability. Ferry FO: fare, time, car availability and comfort. Jet

Foil: fare, time, comfort (two classes) and frequency. Aeroplane: fare, time and frequency.

Another issue to be taken into account, is the structure of competition. For car travellers competition

takes place between both ferries; although at the same time Ferry Fred Olsen dispute market with

jetfoil and even with aeroplane. On the other hand, business travellers may decide between plane and

jetfoil. As a consequence, we consider that there will be four kind of comparison in the SP exercise: a)

Ferry Armas vs. ferry FO, b) Ferry FO vs. Jetfoil. c) Ferry FO vs. Plane d) Plane vs. Jet foil

The design should be completed determining the type of plan that we are going to use. In order to

simplify we will use a model in differences for costs but not for time because of the particularity of

each mode.


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a) SP Armas-Fred Olsen: There are four attributes: cost difference, time Armas, time Fred Olsen and

Class Fred Olsen. Two attributes with three levels of variation and class with two. According with

Kocur et Al (1982) the suitable master plan is 36 with 9 test required.

b) SPFRED OLSEN-JET FOIL: There are five attributes: cost difference, time jetfoil, time Fred Olsen,

Class Fred Olsen and class jetfoil. Two attributes with three levels of variation and two other with two.

Thus, master plan 45a with 16 test required.

c) SP FRED OLSEN-PLANE: There are five attributes: cost difference, time plane, time Fred Olsen,

Class Fred Olsen and frequency aeroplane. Two attributes with three levels of variation and two other

with two. Thus, master plan 45a with 16 test required.

d) SPJET FOIL-PLANE: There are five attributes: cost difference, time jetfoil, time plane, class jetfoil

and frequency plane. Two attributes with three levels of variation and two others with two. Therefore,

the suitable master plan is 45a with 16 test required.

At the beginning three attributes were used for all designs except jet foil-plane: fare, travel time and

class. For jetfoil and plane travellers, fare, travel time and frequency were tested. Three levels were

chosen for the relevant attributes (fare and travel time) and two for the others. However, after

respondents did not pay attention to class, this attribute was rule out.

Table 5.20 shows all types of SP survey that were tested and their sample size.

Table 3.2: Types of SP

Model Type of SP comparision N

1 FFO-CAR versus FA-CAR 295

2 FFO-CAR versus JF 209

3 FFO versus JF 952

4 FFO versus PLANE 109

5 FFO versus FA 371

6 PLANE versus JF 614

Total SP observations 2,250

4. EMPIRICAL RESULTS

In this section we will explain the stage of modelling. Regarding to MNL models, the entire analysis

has been affected by the low quality of data in terms of waiting time. The majority of models provided

coefficients of waiting time with counterintuitive signs. In addition, some specifications with specific

coefficient in-vehicle-time, did not worked correctly due to the parameter of jet foil. The solution

found was merging waiting and access time in a new variable called acwtime which is shown in the

right side of table 4.1


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N1: nest of

fast modes

Figure 1 shows the design used for HL model: one nest for fast modes and the other three models

hanging separately from the root. This is a rational design, which shows that fast ferry is not sharing

many things with Ferry Armas.

Figure 1: HL structure

To asses among models several test where implemented:

Test for significance of parametert.

Test of to tell between a model restricted and more general models. In this case it is used the

Likelihood Radio Test (Ortuzar y Willunsem, 2001): { })()(2**

ll r

In l*(r) is the final likelihood of the restricted model and l*(r) is the same value in the model with

specific variables.

Statistic is a measure of fit for the whole model, which is the result of)0(

)(1

L

L =

WhereL() represents the likelihood of the model andL(0) is the likelihood considering a model using

only zeros. Although the statistic gives clear assessment when it is close to boundaries 0 and 1, it does

not have an unambiguous interpretation for intermediate values (see Ortzar, 1997).

For this reason it is convenient to use the other statistic)(

)(1

CL

L =

The level of likelihood obtained is another way to test the goodness of a model.

4.1Assessing among RP models

Thus, at first general models will be compared. Then, models using socioeconomic variables will be

shown.

Mode 2:

Jet Foil

Mode 3:

Fast Ferry

Mode 4:

FerryMode 1: Plane


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General models (no socioeconomic variables)

Table 4.1: general models MNL and HLModels without wtime Models with acwtime

simple F/S A/M simple F/S A/M A/M

1 2 3 4 5 6 7 HLFare -.1316E-01

(-2.9)-.1115E-01

(2.4)-.5477E-02

(-1.2)-.2465E-02

(-.5)-.1125E-01

(-2.5)-.978E-02

(-2.1)-.4591E-02

(-1.0)-.3441E-02

(-.7)

Wtime -.3570E-02

(-.5)

Acctime -.9868E-02

(-3.6)

-.1093E-01

(-3.8)-.1548E-01

-.2340E-02

(-3.2)

Acwtime -.1295E-02

(-.6)

-.2468E-02

(-1.2)

-.2444E-02

(-1.2)

-.2517E-02

(-1.2)

-.4011E-02

(-.9)

Ivtime-.1716E-02

-.2580E-02

(-2.5)

-.1162E-02

(-1.1)

IvtimeA -.5249E-01

(-5.9)

-.4008E-01

(-5.1)

-.4369E-01

(-6.0)

IvtimeF -.131E-01

(-2.2)

-.8602E-02

(-1.5)

IvtimeS -.2267E-02(-2.1)

-.1438E-02(-1.4)

IvtimeM -.4206E-02

(-3.6)

-.2591E-02

(-2.4)

-.3184E-02

(-1.4)

Asc2

.5549

(4.4)

1.240

(3.3)

-.1241

(-.7)

.8657E-01

(.6)

.5994

(4.6)

1.060

(2.8)

.1240

(.8)

theta .8359

(1.7)

(0) .0565 .0613 .1041 .0861 .0388 .0410 .0722 .0716 (C) .0266 .0316 .0758 .0572 .0084 .0107 .0428 .0422Final L -378.4521 -376.5178 -359.3287 -366.5547 -385.5406 -384.6440 -372.1390 -372.3872

Table 4.1 shows an overall view of the RP models without socioeconomic variables. The goodness of

fit is certainty poor in all of them. Taking into account this default, the best model seems model 3.

Also, models 4 and 7 offer one of the best statistics. Model 4 has serious problems of significance in

four parameters. HL also presents problems of significance in fare and acwtime. It is useful to split up

these models into two categories: those which divide ivtime between plane and maritime modes and

those with consider fast and slow modes. In the first category, the best model is 3. However, this

model has the shortcoming that it was built without waiting time. Alternatively model 7 may represent

well this group. Into the group of Fast/slow coefficient of ivtime, model 2 performs reasonably better

than 6.

On the other hand, it is useful to test the attribute significance. Models 2 and 3 are extended versions

of the more restricted model 1. In the other group, models 6, 7 and HL are general forms of 5. The test

of LR described above reports the following values:

Table 4.2: LR tests

LR>2

R G LR

Yes 2 3.86

Yes1

3 38.24

No 6 1.79

Yes 7 26.80

Yes

5

HL 26.30


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As we can see in table 4.1 all models pass the test except model 6. One 6 has been rejected, it seems

that model 7 is more appropriate than HL since this model does not have significant coefficients. In

the family of non-waiting time models, 3 seems the strongest. Nevertheless, it could be convenient to

choose 2 because this model has an interesting specification for ivtime. Otherwise, we would not any

model that reports information about fast and slow modes.

Models with socioeconomic variables

Table 4.3: models RP with economic variables: income and work

8a 8b:Subsamples of income 9a 9bMODEL

Income

dummiesLow inc medium High inc Workers W paid trip

fare-.5535E-02

(-1.2)

-.1683E-01

(-2.0)

-.1261E-01

(-2.7)

-.5622E-02

(-.4)

-.5999E-02

(-.6)

-.2369E-01

(-2.4)

acctime -.7070E-02(-1.6) -.8578E-02(-3.2) -.2305E-01(-1.7) -7557E-03(-.2)

acwtime-.2228E-03

(-.1)

-.4073E-02

(-1.0)

ivtime-.4345E-02

(-2.0)

-.1449E-02

(-1.3)

-.1091E-01

(-1.7)

-.1277E-01

(-4.0)

ivtimeA-.4234E-01

(-5.2)

-.6027E-01

(-4.4)

ivtimeM-.2512E-02

(-2.4)-.1599E-01

(-4.7)

faremed.3309E-01

(2.4)

farehigh.5834E-02

(.4)

Asc2.1137

(.7)

-.1750

(-.6)

.3057

(2.1)

1.592

(3.7)

.5764

(2.0)

1.454

(5.5)

(0) .0461 .0389 .0360 .3345 .2329 .2019 (C) .0740 .0203 .0257 -.0128 .0518 -.0048

In order to facilitate the exposition, models have been split up into two groups: those that include

economic variables, like income and work, and those, which include social variables like sex and age.

Table 4.3 shows models of this category. In 8a incomes dummies have the expected sign. However

they are larger than fare and, as a consequence, they cannot be used to obtain segments of value of

time. In addition, it seems thatfarehigh is not significant. Also asc and acwtime posses low tvalues. In

addition, the whole model looks too weak taking into account the low values of tests (0) y(C).

Inside model 8a we have three simple models of sub samples of income. The level of income

increases, the parameter of costs decreases and the opposite in case of acctime. Furthermore the

internal coherent is hold. However, in terms of ivtime, this parameter is slightly smaller in medium

level. The three models show a poor goodness of fit except the model of high income, which in fact, is

the best of this table. For all this reasons, it seems that this system of three sub samples could produce

better results than the dummies of income. Nevertheless it is important to note that these models have

been estimated without waiting time and this is a significant lack of information.


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Table 4.4: Social and other variables: frequency, age, and sex

10 11: age 12:SexMODEL

Freq dummy 11a: dummy 11b: young 12b: male 12b: female

fare -.2620E-03(.0)

-.3569E-01(-2.4)

-.4448E-01(-2.1)

-.4000E-02(-.6)

-.1980E-01(-1.8)

acctime-.1560E-01

(-1.6)-.1280E-01

(-2.2)-.1443E-02

(-.6)

ivtime-.1537E-02

(-.8)

-.1377E-01

(-2.0)

Acwtime-.1959E-01

(-3.6)

-.5671E-04

(.0)

IvtimeA-.5661E-01

(-4.5)

-.3854E-01

(-3.4)

-.4420E-01

(-1.7)

IvtimeM-.2452E-02

(-1.5)

-.779E-02

(-1.9)

-.6401E-02

(-1.9)

Timefreq-.8875E-02

(-2.3)

agefare.3514E-01

(2.4)

Asc2 -.2148(-.9) .2257(1.0) -.8256E-01(-.2) .5839(2.6) .3637(1.2)

(0) .1268 .0804 .1506 .0661 .0702

(C) .1013 .0535 .1333 -.0013 .0847

Models 9a and 9b provide parameters of a subsample of workers and, within this group, a sub sample

of paid workers. It could be interesting tries to compare this model with model 7. Parameters offare,

acwtime and ivtimeA are larger in this model. It has a hard interpretation because most of workers have

their ticket paid. The model looks weak in terms of significance offare and acwtime. Despite of this, it

has one of the highest (0) the statistic (C) shows a low value (as in all of them, in fact). The last

model contains respondents with paid tickets. Surprisingly they show more sensitivity towards costs

than the equivalent model of general table, model 1. It may reflect the lack of real decision in their

choice set.

In table 4.4 the rest of MNL models have been grouped. On the left, there is a model that tries to

reflect the behaviour of frequent travellers. The effect of this variable has been concentrated in the

dummy variable called timefreq. This dummy is 1 when the respondent is a traveller in this route at

least one per week, and 0 otherwise. The effect is an increase in the parameter of time. This outcome

reflects the facts that frequent travellers demand faster trips because this activity is an important

proportion of their available time per week. Unfortunately, the model has an important shortcoming in

the non-significance offare. In contrast, it looks an acceptable goodness of fit within this group of

models.

Models 11a and 11b try to model variable age. Which is better? The goodness of fit is much better in

11b. In addition, 11b does not have problems of significance with important parameters. On the

contrary, 11a has a serious problem of significance in acwtime and offers worse statistics (0) and


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(C). However, the problem of 11b is that, it has not equivalent for the other two subsamples of age.

These models did not report correct signs and were rejected. As a consequence, to study the effect of

age in the entire sample it will be necessary to use 11a.

Variable Sex is modelled in the last two models. These models are the result of two subsamples using

the simplest specification: without waiting time. Female seems more sensitive towards costs and less

sensitive in terms of access time. In contrast, situation is the inverse in ivtime that affects more on

females than males. Males are less concerned about access than women, but females feel less affected

by travel conditions and duration of trip. Goodness of fit is poor in both models and, in addition,

acctime is not significant in 12b.

In table 4.5 can be see general models of SP: one model represent each SP exercise. In terms of

significance of parameters, model 1 posses the highest t ratios. In contrast, model 4 shows weak

parameters oftime JFand the intercept. The same problem is found in asc of model 5. According with

ttest this variable should be eliminated. In terms of goodness of fit, model 3 has the best performance

with a 0.3542 of (0) and the next in this ranking would be model 6 with .3142 of this statistic.

However if we consider the most rigorous test of(C) the best model is model 5.

It is worthy to aware that model 3 represents one of the hybrid cases (combination of car-ferry against

jet foil) and has provided satisfactory results. In addition, Fred Olsen versus plane, which at firstseems unrealistic, appears robust as well. On the other hand, model 6 is the result of the most

important exercise of SP and seems robust in terms of goodness of fit and significance of parameters.

Table 4.5: general models of SPFFO FA

(cars)FFO-FA

FFO-JF

(car in FFO)FFO-JF FFO-P P-JF

MODEL

1 2 3 4 5 6

Fare-.5647E-01

(-6.2)

-.1250

(-6.6)

-.1745E-01

(-1.4)

-.5649E-01

(-5.4)

-.8286E-01

(-4.7)

-.5745E-01

(-6.5)

Time-.1576E-01

(-6.7)

-.5197E-02

(-1.1)

-.3748E-01

(-1.9)

-.1598E-01

(-1.3)

-.1614E-01

(-2.4)

Time JF-.2067E-01

(-2.6)

Time FFO-.1245E-01

(-.6)

Head-.2912E-02

(-1.7)

Asc1.727

(2.6)

1.146

(4.3)

-.4036

(-.2)

-.2698

(-.4)

(0) .1614 .2397 .3542 .3091 .2979 .3142

(C) .1285 .2318 .0335 .0591 .2977 .1620


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14

Table 4.6: Sp models with socio-economic variables and subsamplesType of

SPFFO-FA (car) FFO-JF JF-Plane

Model1b

FA users

1c

FFO users

4b

JF users

6b

P users

6c

JF users

6d

income d

6e

paid

6f

Non paid

fare-.1028

(-2.6)

-.2981E-01

(-3.1)

-.6016E-01

(-3.4)

-.4885E-01

(-3.1)

-.5840E-01

(-9.3)

-.5821E-01

(-5.8)

-.4363E-01

(-3.6)

-.6786E-01

(-5.8)

time-.1838E-01

(-1.8)

-.9552E-02

(-3.4)

-.2365E-01

(-4.1)

-.3539E-02

(-.6)

-.2466E-01

(-8.4)

-.1174E-01

(-1.4)

-.1773E-01

(-3.2)

-.2720E-01

(-5.0)

fareinc.7701E-02

(.4)

asc .4497E-01

(.1)

.5911

(1.8)

(0) .1614 .0669 .3186 .1637 .3019 .3153 .1884 .3584 (C) .1285 .0449 .0590 .1637 .1591 .1634 .1106 .2019

In table 4.6 it is shown the rest of models produced in SP. It is difficult, may be impossible, make

comparisons among models, which came from different SP exercise, because they will have different

type of errors. It may be guess that the most cost preference travellers are the FA users. Results seem

confirm this idea. In fact, its parameter of fare is really large, reflecting this special sensitivity towards

fare. In the other extreme, inside the same SP, are situated FFO users with a parameter of fare 34 times

smaller. However, in terms of time parameter results are the opposite that expected because time

parameter in FA users is slightly bigger than FFO users.

It is possible to compare time coefficient of 4b with the general model 4 in table above. It seems that,

inside this SP, jet foil users shows more sensitivity on time than FFO users. This is a logic outcome.

Nevertheless, inside the SP6, plane-jet foil, JF users posses the higher time parameter. Regarding with

paid and non-paid users it seems that, as we expect, non-paid users are more cost sensitive.In terms of

coefficients, model 6b seems to be too weak: time is not significant and there are only two parameters

in the model. Also asc in model 4b is not significant at all; moreover, this parameter has problems of

correlation with time. The ranking of goodness of fit is head by 6f, model that shows extraordinary

robustness. It may confirm the hypothesis of consider paid users as a captive.

4.2 Mixed Logit results

Table 4.7:ML for 4 normal distributions

Parameters Estimates Standard Errors

Fare -0.00998501 0.007096580.01315751 0.01570268

Access time 0.00910062 0.003926410.00021131 0.00086644

Waiting time -0.01906961 0.007577720.01920147 0.01771682

In-vehicle time -0.00229251 0.00129467

0.00020438 0.00008634Function value: -455.61034710


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15

ML yields two statistics for each parameter: mean and standard deviation. On of the most important

issues of modelling here is the correct choice of parameter distribution. In this work, we faced a sign

problem with waiting time which has been solved merging waiting and access time in case of MNL

models.

Using ML, we have allowed all parameters vary with many combination of distribution and the sign

problem was reported in most of them. Log normal distribution is an option for this cases but, as it has

been reported it leads to flat log likelihood function where is difficult to achieve the maximum.

Finally, the only option to obtain correct signs for all parameters was fixed waiting time and allow the

others coefficients to vary according to a normal distribution; despite of the fact that log likelihood

function offers an slightly higher value, this seem the best model. Results are shown in table 4.8

Table 4.8: ML for 4 normal and one fixed variable

Parameters Estimates Standard Errors

fare -0.01259882 0.00275809-0.00464131 0.00457605

Access time -0.00398426 0.00504412

0.01073593 0.00408301

Waiting time -0.01754933 0.00000000

In-vehicle time -0.00108957 0.000915540.00007157 0.00003850

Function value: -457.33207865

Individual parameters

ML is completed estimating individual parameters. Using the results of model ML1 individualparameters were estimated, considering waiting time a fixed coefficient and furthermore, it will be the

same for all costumers. It is useful show individual parameters in an histogram shape, which allows

for all interpretations. Thus figures 2, 3 and 4 shows histogram for access time, fare and in-vehicle

time, respectively.

Figure 2: Histogram for access time

,50,38

,25,13

0,00-,13

-,25-,38

-,50-,63

-,75-,88

400

300

200

100

0

Desv. tp. = ,06

Media = -,05

N = 420,00


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16

Figure 2: Histogram of fare

,200,175

,150,125

,100,075

,050,025

,000-,025

-,050

-,075

-,100

-,125

120

100

80

60

40

20

0

Desv. tp. = ,04

Media = -,040

N = 420,00

Figure 3: histogram of in-vehicle time

,75,63

,50,38

,25,13

0,00

-,13

-,25

-,38

-,50

-,63

-,75

-,88

-1,00

500

400

300

200

100

0

Desv. tp. = ,09

Media = -,01

N = 420,00

First of all, these histograms do not show the expected normal shape which could be the result of the

limited sample. However, it seems too concentrated around the average, especially in the case of in-

vehicle time. One important issue related to individual parameters is the question of the number of

individuals who has the correct sign in their coefficients. The next table 4.9 summarises this problem

It seems that Fare is the variable which contains more individuals who report counterintuitive signs.

This level of estimation has the advantage that we can detect and remove those individuals with

problematic estimation from the sample.


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17

Table 4.9: individuals with wrong sign

individuals

with wrong sign (per cent)

Access time 45 (10%)

Fare 75 (17%)

In-vehicle time 57 (13%)

5. RESULTS

5.1 Value of time in RP models

Within these models we have split up between models with and without socioeconomic variables.

Models without SE variables

Table 5.1: value of time of all models

Models without wtime Models with acwtime

simple F/S A/M simple F/S A/M A/M

1 2 3 4 5 6 7 HL

Wtime1.45E+00

(0.52)

Acctime7.50E-01

(2.64)

9.80E-01

(2.06)

2.83E+00

(3.17)

9.49E-01

(4.98)

Acwtime5.25E-01

(0.2)

2.19E-01

(0.23)

2.50E-01

(1.03)

5.48E-01

(0.41)

1.17E+00

(0.69)

Ivtime1.30E-01

(2.63)1.05E+00

(0.6)1.03E-01

(0.14)

IvtimeA9.58E+00

(10.5)8.73E+00

(8.27)1.27E+01

(8.84)

IvtimeF1.17E+00

(1.82)8.80E-01

(0.23)

IvtimeS2.03E-01

(0.4)

1.47E-01

(0.25)

5.64E-01

(0.63)

IvtimeM7.68E-01

(0.97)

9.25E-01

(0.68)

Table 5.1 shows values of different kinds of time in euros per minute. Eventually, it has been

calculated values of time for all modes of table 5.35. Our purpose was to use only these models that

reported the best goodness of fit according with the discussion in previous section. However, due to

the lack of reasonable results, it was necessary to extend calculations to all models. In fact, table 5.1

shows 7 unacceptable values that we reject totally. The rest of values have been grouped in table 5.2.

Values of access and regress time (VAT) are situated between 45 and 58.8 per hour. The aggregate

of this time plus waiting time (VAWT) is valued in a range between 13.1 and 32.9 per hour. Value

of time in vehicle (VIT) is found between 7.8 and 6.18 per hour. However, if we split up this VIT

into VIT of fast and slow modes, VIT change completely.


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18

Taking averages, we see that VAT is the highest, followed by IVT of fast modes. Next is the IVT of

maritime modes with an average of 50.79 per hour, then VAWT with 23.13, IVT of slow modes

with 18.28 and the generic IVT with 6.99 per hour.

Table 5.2: value of time for RP models (t ratio)Value of time per minute Value of time in per hour Average

Acctime 0.98(2.06) 0.949(4.98) 0.75(2.64) 58.8 56.9 45 53.58

Acwtime 0.54(0.41) 0.52(0.2) 0.25(1.03) 0.21(0.23) 32.9 31.5 15 13.1 23.13

Ivtime 0.13(2.63) 0.10(0.14) 7.8 6.18 6.99

IvtimeF 0.88(0.23) 52.8 52.8

IvtimeS 0.56(0.63) 0.20(0.4) 0.14(0.25) 33.8 12.2 8.82 18.28

IvtimeM 0.82(0.68) 0.76(0.97) 55.5 46.1 50.79

Are these values reasonable? First, it is useful to wonder about the internal coherence of these values.

The intuition would allow us to establish a ranking like: VAWT>VAT>VIT. This coherence is hold.

Nevertheless, VAT, which does not contain waiting time, is smaller than VAWT. On the other hand, it

is reasonable to expect that faster modes had larger VIT than slower modes as we have obtained in this

work. On the other hand, in terms of t ratio, these results seem poor. Only 4 pass the test and three of

them are the VAT.

Models with SE variables

Table 5.3 illustrates values of time according to income groups, and susample of workers. Again, the

problem here is the lack of realism: these figures represent euros per minute and, at least four of them

(underlined) are too large. It seems that the whole sample has a bias towards large values of time or

reduced parameters of fare. Only two of these VT pass clearly the ttest.

Table 5.3: values of time according with economic variables

8b:Subsamples of income 9a 9bMODEL

Low inc Medium High inc Workers W paid trip

acctime4.20E-01

(0.51)

6.80E-01

(1.5)

4.10E+00

(1.48)

3.19E-02

(.0)

acwtime6.79E-01

(0.34)

ivtime2.58E-01

(0.36)

1.15E-01

(0.19)

1.94E+00

(0.79)

5.39E-01

(1.57)

ivtimeA1.00E+01

(6.04)

ivtimeM2.67E+00

(1.73)

Tables 5.4 and 5.5 show values of times and compare them with the average of the whole sample.

Some values are extremely large as VAT of high-income segment. In addition, all VT for workers

must be rejected. Throughout the income segments, VT reveals an internal logic in VAT. However,


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19

IVT decreases up to 5.9 and, even in the highest group is smaller than the lowest. Despite of this lack

of coherence, these values are probably the most reasonable of the table. Actually, the problem is the

VIT of low-income segment: it looks too high, but the other two figures seem rational.

Table 5.4: value of time for segments of income

Value of time in per hour

low Medium high Workers paid Whole sample

Acctime 25.2 40.8 246 1.91 53.58

Ivtime 15.5 6.9 11.6 32.3 6.99

VAT-average -28.38 -12.78 192.42 -51.67

VIT-average 8.51 0 4.61 25.31

Table 5.5: value of time of workers

Value of time in per hourWorkers Whole sample VT-average

acwtime 40.7 23.13 16.87

Ivtime A 600 -

IvtimeM 160 50.79 109.21

Finally, we would need to calculate value of time according the rest of RP models. Table 5.6 illustrates

the results of these models. Model 10, which tried to represent the effect of frequency, is weighed

down by its lack of significance in cost parameter. Consequently, results are enormous and they must

be rejected. Table 5.6 shows only segmentation of sex and age. Figures show VT in euros per minute

and per hour.

Table 5.6: value of time for different types of travelers

Value of time in per minute (t ratio) Vot in per hourMODEL

11: age 12:Sex age sex

Under 30 Over 30 12b: male 12b: female < 30 >30 male Fem

acctime 0.0159(0) 3.20 (1.55) 0.072(0.9) 0.954 192 4.32

ivtime 0.1(0.6) 0.38(0.26) 0.69(0.05) 22.8 41.4

IvtimeA 1.08(2) 70(0.02) 64.8 4,200

IvtimeM 0.21(0.14) 14.2(5) 12.6 252

First, it is clear that age is an incremental factor of willingness to pay. It is rational expect this result;

however, except figures remarked in bold, these VT are extremely big. The only possible conclusion is

that, in fact, there is a substantial difference between these kinds of travellers and that age is an

important explanatory variable in the model. On the other hand, only two VT pass the t test.

With reference to sex, it seems that male are more concerned about travel to access and regress and

female are more aware about the length of trip in vehicle. It is possible that this outcome reflect the

fact that females are more worried about safety and also, it may be possible that they feel more


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20

affected by travel conditions. Figures of female seem reasonable, but, once again, it is necessary to

reject VAT of male.

5.2 Value of time of SP models

Table 5.7: value of time in SP models (t ratio)

Value of time in per minute

MODELFFO FA

(cars)FFO-FA

FFO-JF

(car in FFO)FFO-JF FFO-P P-JF

Time 0.27 (3.32) 0.41 (0.04) 2.15 (2.47) 0.19 (0.23) 0.28(0.65)

Time JF 0.36 (-0.8)

TimeFFO 0.22 (0.12)

Head 0.05(0)

Value of time in per hour

Time 16.7 2.49 129 11.6 16.9

Time JF 22

Time FO 13.2Head 3.04

Table 5.7 shows VT calculated from SP models. Unlike the RP, these figures seem realistic, except

VT in FFO-JF (with car), which is too high. Taking out this case, VT is situated in a range from 2.49

and 16.9 per hour. Value of time from P-JF is the highest as we could expect and, VT from ferries is

the lowest which is a rational result. Car market is a different case because is affected by the massive

presence of transport workers but its VT remains reasonable. Moreover, table 5.7 shows a VT in JF

almost two times value of time in FFO. This result is fairly balanced. In addition, value of head is five

times VT. This seems a rational relation. However, t ratio only is acceptable in two VT.

Tables 5.8 and 5.9 show VT in SP for different types of travellers. All results seem rational. In FFO-

FA it is logic to find a higher VT in FFO users. In addition, VT of JF users is higher than the other two

and fairly close to figure in table 6.9. Results show a sort of coherence inside the whole set of SP

exercise. In JF-Plane SP we find that JF users have much higher VT than plane travelers. This latter

relation does not seem realistic. On the other hand, there is not too much difference between low and

high income and VT of paid and non-paid are practically the same. In terms of significance, except JFusers in table 6.9 there are not VT with enough tratio.

Table 5.8: VT of SP for different types of users (t ratio)

Type of SP FFO-FA (car) FFO-JF

Model FA users FFO users JF users

VT per minute 0.17(0.74) 0.32(1.7) 0.39(1.05)

VT per hour 13.32 19.2 23.4


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21

Table 6.9: VT in SP for different types of users within JF-Plane (t ratio)

JF-Plane

P users JF users Low inc H inc paid Non paid

VT per minute 0.072(0.4) 0.42(2.83) 0.202(0.27) 0.232(0) 0.40(0.9) 0.40(1.78)

VT per hour 4.2 25.2 12.2 13.92 24.36 24.06

5.3 Comparison of results

It may useful to evaluate these results with the values of time reported by Ortzar and Gonzalez

(2001). It will be indispensable to update those results since they are based on data gathered in 1992.

Therefore figures will be converted from pesetas to euros3. Table 5.10 shows this transformation. It

was consider an average of annual inflation rate of 3% for these 10 years.

Table 5.10: Updating VT from Ortzar and Gonzalez.

VT 1992 Updated results

Low 630 (1.99) 888.67 5.34Medium 794 (3.64) 1120.01 6.73

Income

High 1,809 (1.45) 2551.77 15.33

Aeroplane 1,360 (9.45) 1918.41 11.52JF 1,466 (9.03) 2067.93 12.42

mode

Ferry 256 (2.57) 361.11 2.17

Table 5.11: comparisons of VT

SPVT Ortuzar FFO-FA

(car) FFO-JF JF-P FFO-JF

RP

Low income 5.34 13.92 15.5

Medium income 6.73 6.9

High income 15.33 24.36 11.6

Airplane 11.52 4.2

Jet foil 12.42 23.4 25.2 22

Ferry A 2.17 13.32

Ferry FO4 - 19.2 13.2

In terms of significance of parameters, it is obvious that our results are inferior; it is more interesting

to concentrate in the level of VT estimated. IVT of RP has been taken for this comparison.

Surprisingly many figures seem to find the same pattern. Medium and high income of Ortzar survey

are close to those equivalent values calculated in this work. In fact, VT in medium income is almost

exactly the same figure. VT from this work seems higher in all types except the strange VT in

airplane. VT in JF is situated in a narrow range of 22-25.2 in SP; nevertheless, the same figure in

Ortuzars work is a half.

3

1 =166.386 pesetas4 It must take into account, as we have already said, that this mode did not exist at the time that Ortuzar andGonzalezs survey.


22/25

22

5.4 Value of time in individual level

Since we have estimated individual parameters is it possible to obtain value of time for each

individual. We will follow the same procedure that we used when we showed individuals parameters,

displaying these value of time using histograms. Thus, figures 4 and 5 represents histograms of value

of time for access and in-vehicle time respectively.

Value of access time is clearly concentrated around 35 euros per hour. Numbers of counterintuitive

cases raise calculating value of time, since this is the ratio between time and fare coefficient. However

most of cases are under the positive part of the distribution. Value of in-vehicle time reaches an

average of 77,95 euros per hour, also very concentrated around the average. It might be convenient

compare these results with the average wage paid in the Canary Economy in 20025

which was 11.8

euros per hour. Thus, Value of access time represents almost three times the medium wage and value

of in-vehicle time is up to seven time this figure.

Figure 4: value of access time

1750,0

1250,0

750,0

250,0

-250,0

-750,0

-1250,0

-1750,0

-2250,0

-2750,0

-3250,0

-3750,0

-4250,0

400

300

200

100

0

Desv. tp. = 413,19

Media = 35,0

N = 420,00

5 Source: Canary Institute of Statistics. This figure is the wage in service sector which is the most important inthis economy.


23/25

23

Figure 5: the value of in-vehicle time

550,0

450,0

350,0

250,0

150,0

50,0-50,0

-150,0

-250,0

-350,0

-450,0

-550,0

400

300

200

100

0

Desv. tp. = 77,95

Media = 6,6

N = 421,00

It is known that hour-wage is usually considered a proxy of value of time and it is used in most of

public investment analysis. The findings of this article could suggest that this mean could not be

appropriate. Also, it could show the highest cost of travelling between islands considering the fact of

these high fares compared with the actual length of trips.

In addition, it is interesting to see that in-vehicle value of time represents seven times the value of

access time. Therefore, costumers will be willing to accept transferences from in-vehicle time to

access time and this is exactly what has happened in this market with the strongest competitor FFO.

This ferry relocated the port to a closer point to Tenerife, transferring part of trip costs to travellers

who prefer face this longer access if it means a shorter trip.

6. CONCLUSIONS

We have analysed an inter-island corridor served by three modes and two routes. Using a survey with

RP and SP we have developed several MNL, HL and ML models. Several values of time have been

reported using these models. The main conclusions of this work could be the following:

Among the simplest models the best specification seems to be an MNL model with specific

coefficients for plane and maritime modes. Also the HL model was able to describe the natural

connection between jetfoil and aeroplane. On the other hand, SP exercises provided robust models

although applied in pairs of competition. ML has shown powerful features, especially in the ability to

avoid estimation problems with counterintuitive signs. These problems were controlled using a


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24

combination of normal and fixed variables. Also, individual estimation could be used to detect and, in

case, remove from the sample those individuals who report wrong parameters. This deep level of

estimation has an enormous potential.

The work provided a wide variety of values of time. From RP we found a value of access, regress and

waiting time of 23.13 per hour, a value of in-vehicle-time of 6.99 per hour. Also value of in-

vehicle time reported for fast and slow modes were 52.8 and 18.28 respectively. SP models

generated more reasonable values, although not better in statistical terms. Value in vehicle time for JF

was 25 and 13.2 for ferry Fred Olsen. Also, values reported for medium and high income were

fairly plausible: 6.9 and 11.6 . Other specifications proved positive relationship between age and

willingness to pay, highest value of time for females in vehicle time and highest value of time for

frequent travellers.

Results were compared with updated empirical evidence in the same market and some coincidences

were found. The closest values of time were VT per medium and high-income segment. For modes,

our values of in-vehicle-time were larger; nevertheless we coincided in founding jetfoil with the

highest value of in-vehicle-time.

Broadly speaking, values of time obtained are higher than averaged wage paid in this economy.

Taking into account that this statistic is used in investment projects, it might suggest a re-estimation of

this procedure. In addition these high value of time could be consider an estimation of high travelling

costs between two islands with large population density: to certain extend they could express an

unsatisfied demand.

On the other hand, comparing access value of time and in-vehicle time, could be interpreted the recent

evolution of this market. In effect, in-vehicle VT is seven times larger than access VT which could

suggest a potential improvement transferring time from in-vehicle time to access time, which is

exactly what has happened in this market with FFO: this company relocated the departure place to acloser point to Tenerife, reducing in-vehicle time and enlarging access time for users. The massive

answer from travellers, willing to accept this exchange, is coherent with the results showed in this

work.

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(in Spanish)

Train K. (2003),Discrete Choice Models with Simulation. Cambridge University Press.

Wardmand, M. (1988), A comparison of revealed and stated preference models of travel behaviour.Journal of Transport Economics and Policy 22(1): 71-92

Documents

Discrete Choice Models in Transport. - An Application to Gran Canaria- Tenerife Corridor