Discrete Choice Models in Transport: An application to Gran Canaria- Tenerife corridor. José María Grisolía Santos Universidad de Las Palmas de GC

Transcription

1 Discrete Choice Models in Transport: An application to Gran Canaria- Tenerife corridor José María Grisolía Santos Universidad de Las Palmas de GC Departamento de Análisis Económico Aplicado Universidad de Las Palmas de GC Campus Universitario de Tafira Baja Las Palmas de GC Teléfono: Fax: Abstract Discrete choice models analyse individual s 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 the corridor 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 high value 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. 1

2 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 passenger s trips in 2001, traffic between these islands has increased significantly from 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 and ferry Armas use this route 1. 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 of mixed service car/bus-ferry. Thanks to the liberalisation of the market, Ferry Fred Olsen started to offers its service in The shorted 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 passengers 2. 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 per day are transported everyday because it is devoted to freight transport. In order to simplify the exposition we will 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. It must take into account that this work analyses only passengers demand and not freight transport. 2

3 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 of A j alternatives. Theory of Random Utility (see, for instance Ben-Akiva, 1985) postulates that the utility function of an option j for an individual n is determined by U jn = Vjn + ε jn (2.1) In equation (2.1) we can distinguish a deterministic part called V jn and a random component ε 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, V jq is a function of level of attributes of existing options x pondered by coefficients θ. Thus, V jq = K θ χ (2.2) Kj jkq 2 The interested reader about the effect of this liberalisation can read De Rus (1997) 3

4 Give this framework, theory says that individual will select the alternative which maximise his utility. Hence, individual q will select alternative j if: U jn Uin, Ai A( q) (2.3) This leads to: Vjn Vin εiq ε jq (2.4) Depending of distributions of disturbances two types of models arise: if ε in 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 (Ortúzar and Willunsen, 2001) that the probability that individual q chose alternative i equals: P iq exp( βvin) = (2.5) exp( βv ) Aj Aq Where β is a parameter related to the common standard deviation of the Gumbel distribution. In practise, it cannot be estimated separately from parameters θ 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: V I jn = φ EMU + αz (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: EMU = log exp(w j ) (2.7) In (2.7) W j is the utility function of alternative j where all common attributes z of the nest have been taking out. Thus, the second term αz is the vector of common attributes of the nest and parameters α. The estimation process of these models focuses in obtaining estimation of the parameters θ* 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, L ( θ ) = P P (2.8) 12 24P32 j 4

5 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. log L( θ ) g log (2.9) = q j jq P jq Once the set of θ* 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 β n and so, the probability that individual n chooses option j is a conditional probability in β. Assuming that β= b P = P ( β ) P( β b) (2.10) nj nj = Conditional probability P nj is just the simple logit. In the case of fixed parameters, ML collapses into MNL. If β n is a discrete variable, P nj would be the sum of all probabilities conditioned to each β n weighted with every probability β n =b m. This is called the latent classes model: P nj = M m= 1 s m. P ( β ) (2.11) nj n If we consider β n continuous, it is necessary to use an integral where probability is weighted with a density function f(β) which is the most used expression of ML and the one we developed in this article. P nj b' m xni e f ( β ) dβ b' m xni = (2.12) e j 5

6 Now, the modeller has to estimate two sets of parameters: mean b and co-variance matrix W. Bothe can be denominated θ. 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 θ, take draws from the distribution f(β/ θ) (2) Calculate simple logit L ni for each draw (3)After several repetitions average the results. This average is a unbiased estimator of P ni P = 1 R ni L ni R r= 1 ( r β ) (2.13) These simulated probabilities are inserted in the log likelihood function SLL = Maximising (2.14) estimator θ is obtained. N J n= 1 i= 1 d nj L n P nj (2.14) Bayesian estimation does not need to maximise any function. Its results are based in Baye s 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: k ( θ / Y ) L( Y ) = L( Y / θ ) k( θ ) (2.15) Where k(θ) is a prior distribution; k(θ Y) is the posterior distribution; L(Y) is the probability to obtain the observed choices in the sample and L(Y/θ) is the probability of these choices conditional on θ. Then, it is possible to derive: L( Y / θ ) k( θ ) k( θ / Y ) = (2.16) L( Y ) From (2.16) the researcher will have to estimate θ which can be expressed as the mean of posterior distribution θ = θ k ( θ ) dθ (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. 6

7 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) Average fare Frequency 2 per day 4 per day 3 per day Hourly Accessibility 40 by car In the city (from city) 60 by bus In 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. 7

8 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 36ª with 9 test required. b) SP FRED 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) SP JET 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 FFO-CAR versus JF FFO versus JF FFO versus PLANE FFO versus FA PLANE versus JF 614 Total SP observations 2, 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 8

9 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. N 1 : nest of fast modes Ø Mode 1: Plane Mode 2: Jet Foil Mode 3: Fast Ferry Mode 4: Ferry Figure 1: HL structure To asses among models several test where implemented: Test for significance of parameter t. 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{ l ( θ r ) l ( θ )} In l*(θ r ) is the final likelihood of the restricted model and l*(θ r ) is the same value in the model with specific variables. L( θ ) Statistic ρ is a measure of fit for the whole model, which is the result of ρ = 1 L(0) Where L(θ) represents the likelihood of the model and L(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 Ortúzar, 1997). L( θ ) For this reason it is convenient to use the other statistic ρ = 1 L( C) The level of likelihood obtained is another way to test the goodness of a model. 4.1 Assessing among RP models Thus, at first general models will be compared. Then, models using socioeconomic variables will be shown. 9

10 General models (no socioeconomic variables) Fare Wtime Acctime Acwtime Ivtime IvtimeA IvtimeF IvtimeS IvtimeM Asc2 theta Table 4.1: general models MNL and HL Models without wtime Models with acwtime simple F/S A/M simple F/S A/M A/M HL E E E E E E E E-02 (-2.9) (2.4) (-1.2) (-.5) (-2.5) (-2.1) (-1.0) (-.7) E-02 (-.5) E E E E-01 (-3.6) (-3.8) (-3.2) E E E E E-02 (-.6) (-1.2) (-1.2) (-1.2) (-.9) E E E-02 (-2.5) (-1.1).5549 (4.4) -.131E-01 (-2.2) E-02 (-2.1) (3.3) E-01 (-5.9) E-02 (-3.6) (-.7).8657E-01 (.6).5994 (4.6) E-02 (-1.5) E-02 (-1.4) (2.8) E-01 (-5.1) E-02 (-2.4).1240 (.8) E-01 (-6.0) E-02 (-1.4) ρ (0) ρ (C) Final L (1.7) 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>χ 2 R G LR Yes Yes No Yes Yes HL

11 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 MODEL fare acctime acwtime ivtime ivtimea ivtimem faremed farehigh Asc2 Table 4.3: models RP with economic variables: income and work 8a 8b:Subsamples of income 9a 9b Income dummies E-02 (-1.2) E-03 (-.1) E-01 (-5.2) E-02 (-2.4).3309E-01 (2.4).5834E-02 (.4).1137 (.7) Low inc medium High inc Workers W paid trip E-01 (-2.0) E-02 (-1.6) E-02 (-2.0) (-.6) E-01 (-2.7) E-02 (-3.2) E-02 (-1.3).3057 (2.1) E-02 (-.4) E-01 (-1.7) E-01 (-1.7) (3.7) E-02 (-.6) E-02 (-1.0) E-01 (-4.4) E-01 (-4.7).5764 (2.0) E-01 (-2.4) -7557E-03 (-.2) E-01 (-4.0) ρ (0) ρ (C) (5.5) 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 that farehigh is not significant. Also asc and acwtime posses low t values. In addition, the whole model looks too weak taking into account the low values of tests ρ (0) y ρ (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. 11

12 MODEL fare acctime ivtime Acwtime IvtimeA IvtimeM Timefreq agefare Asc2 Table 4.4: Social and other variables: frequency, age, and sex 10 11: age 12:Sex Freq dummy 11a: dummy 11b: young 12b: male 12b: female E-03 (.0) E-01 (-3.6) E-01 (-4.5) E-02 (-1.5) E-02 (-2.3) (-.9) E-01 (-2.4) E-04 (.0) E-01 (-3.4) -.779E-02 (-1.9).3514E-01 (2.4).2257 (1.0) E-01 (-2.1) E-01 (-1.6) E-01 (-1.7) E-02 (-1.9) E-01 (-.2) E-02 (-.6) E-01 (-2.2) E-02 (-.8).5839 (2.6) E-01 (-1.8) E-02 (-.6) E-01 (-2.0) ρ (0) ρ (C) (1.2) 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 of fare, 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 of fare and acwtime. Despite of this, it has one of the highest ρ (0) the statistic ρ (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 of fare. 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 ρ (0) and ρ 12

13 (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 of time JF and the intercept. The same problem is found in asc of model 5. According with t test this variable should be eliminated. In terms of goodness of fit, model 3 has the best performance with a of ρ (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 ρ (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 first seems 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 SP MODEL Fare Time Time JF Time FFO Head Asc FFO FA FFO-JF FFO-FA FFO-JF FFO-P P-JF (cars) (car in FFO) E-01 (-6.2) E-01 (-6.7) (-6.6) E-02 (-1.1) (2.6) E-01 (-1.4) E-01 (-1.9) (4.3) E-01 (-5.4) E-01 (-2.6) E-01 (-.6) E-01 (-4.7) E-01 (-1.3) E-01 (-6.5) E-01 (-2.4) E-02 (-1.7) ρ (0) ρ (C) (-.2) (-.4) 13

14 Type of SP Table 4.6: Sp models with socio-economic variables and subsamples FFO-FA (car) FFO-JF JF-Plane Model 1b FA users 1c FFO users 4b JF users 6b P users 6c JF users 6d income d 6e paid 6f Non paid fare E E E E E E E-01 (-2.6) (-3.1) (-3.4) (-3.1) (-9.3) (-5.8) (-3.6) (-5.8) time E E E E E E E E-01 (-1.8) (-3.4) (-4.1) (-.6) (-8.4) (-1.4) (-3.2) (-5.0) fareinc.7701e-02 (.4) asc.4497e-01 (.1).5911 (1.8) ρ (0) ρ (C) 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 Access time Waiting time In-vehicle time Function value:

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 Access time Waiting time In-vehicle time Function value: Individual parameters ML is completed estimating individual parameters. Using the results of model ML1 individual parameters 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 Desv. típ. =,06 Media = -,05 N = 420,00 15

16 Figure 2: Histogram of fare ,200,175,150,125,100,075,050,025,000 -,025 -,050 -,075 -,100 -,125 Desv. típ. =,04 Media = -,040 N = 420,00 Figure 3: histogram of in-vehicle time Desv. típ. =,09 Media = -,01 N = 420,00,75,63,50,38,25,13 0,00 -,13 -,25 -,38 -,50 -,63 -,75 -,88-1,00 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 invehicle 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. 16

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 Wtime Acctime Acwtime Ivtime IvtimeA IvtimeF IvtimeS IvtimeM simple F/S A/M simple F/S A/M A/M HL 7.50E-01 (2.64) 1.30E-01 (2.63) 9.80E-01 (2.06) 1.17E+00 (1.82) 2.03E-01 (0.4) 2.83E+00 (3.17) 9.58E+00 (10.5) 7.68E-01 (0.97) 1.45E+00 (0.52) 9.49E-01 (4.98) 5.25E-01 (0.2) 1.05E+00 (0.6) 2.19E-01 (0.23) 1.03E-01 (0.14) 2.50E-01 (1.03) 8.80E-01 (0.23) 1.47E-01 (0.25) 5.48E-01 (0.41) 8.73E+00 (8.27) 5.64E-01 (0.63) 1.17E+00 (0.69) 1.27E+01 (8.84) 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 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. 17

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 per hour, then VAWT with 23.13, IVT of slow modes with 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) Acwtime 0.54(0.41) 0.52(0.2) 0.25(1.03) 0.21(0.23) Ivtime 0.13(2.63) 0.10(0.14) IvtimeF 0.88(0.23) IvtimeS 0.56(0.63) 0.20(0.4) 0.14(0.25) IvtimeM 0.82(0.68) 0.76(0.97) 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 t test. Table 5.3: values of time according with economic variables MODEL acctime acwtime ivtime ivtimea ivtimem 8b:Subsamples of income 9a 9b Low inc Medium High inc Workers W paid trip 4.20E-01 (0.51) 2.58E-01 (0.36) 6.80E-01 (1.5) 1.15E-01 (0.19) 4.10E+00 (1.48) 1.94E+00 (0.79) 6.79E-01 (0.34) 1.00E+01 (6.04) 2.67E+00 (1.73) 3.19E-02 (.0) 5.39E-01 (1.57) 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, 18

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 Ivtime VAT-average VIT-average Table 5.5: value of time of workers Value of time in per hour Workers Whole sample VT-average acwtime Ivtime A IvtimeM 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 MODEL Value of time in per minute (t ratio) Vot in per hour 11: age 12:Sex age sex Under 30 Over 30 12b: male 12b: female < 30 >30 male Fem acctime (0) 3.20 (1.55) 0.072(0.9) ivtime 0.1(0.6) 0.38(0.26) 0.69(0.05) IvtimeA 1.08(2) 70(0.02) ,200 IvtimeM 0.21(0.14) 14.2(5) 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 19

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 MODEL FFO FA (cars) Table 5.7: value of time in SP models (t ratio) Value of time in per minute FFO-JF FFO-FA FFO-JF FFO-P P-JF (car in FFO) 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 Time JF 22 Time FO 13.2 Head 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 JF users in table 6.9 there are not VT with enough t ratio. 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

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 Comparison of results It may useful to evaluate these results with the values of time reported by Ortúzar and Gonzalez (2001). It will be indispensable to update those results since they are based on data gathered in Therefore figures will be converted from pesetas to euros 3. 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 Ortúzar and Gonzalez. Income mode VT 1992 Updated results Low 630 (1.99) Medium 794 (3.64) High 1,809 (1.45) Aeroplane 1,360 (9.45) JF 1,466 (9.03) Ferry 256 (2.57) Table 5.11: comparisons of VT SP VT Ortuzar FFO-FA RP FFO-JF JF-P FFO-JF (car) Low income Medium income High income Airplane Jet foil Ferry A Ferry FO 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 Ortúzar 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 in SP; nevertheless, the same figure in Ortuzar s work is a half. 3 1 = pesetas 4 It must take into account, as we have already said, that this mode did not exist at the time that Ortuzar and Gonzalez s survey. 21

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 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 ,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 Desv. típ. = 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 in this economy. 22

23 Figure 5: the value of in-vehicle time ,0 450,0 350,0 250,0 150,0 50,0-50,0-150,0-250,0-350,0-450,0-550,0 Desv. típ. = 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 23

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 per hour, a value of in-vehicle-time of 6.99 per hour. Also value of invehicle time reported for fast and slow modes were 52.8 and 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 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 a closer 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. REFERENCES Ben-Akiva, M. and Steven R. Lerman (1985), Discrete Choice Analysis. The MIT press. Ben-Akiva, M; D. Bolduc, and Bradley (1993), Estimation of travel model choice models with randomly distributed values of time. Transportation Research Record 1413,