VALUING TIME WITH A JOINT MODE CHOICE-ACTIVITY MODEL ...

international journal of transport economics vol. xxxiii · no 2 · june 2006

VALUING TIME WITH A JOINT MODE CHOICE-ACTIVITY MODEL * Marcela A. Munizaga · Rodrigo Correia Sergio R. Jara-Díaz · Juan de Dios Ortúzar ** Abstract : Activity and travel models share a common framework that calls for their joint estimation. This complex problem only recently has received attention in the literature. Explicit equations for travel choice and activity duration have been obtained, allowing disentangling the various components of the subjective value of time (travel time savings, resource value and value of assigning time to travel). Although it is natural to expect some interrelation between equations, only preliminary results assuming independence had been reported. We adapt a discrete/continuous econometric model and postulate a general error structure. We obtain more robust models, and value of time estimates significantly different and more credible. jel Classification : C51, J22, R41.

1. Introduction

A

s travel demand is derived from people’s desires and needs for activity participation it seems natural to try and model jointly the time assigned to activities and the travel choices. Starting from a consistent microeconomic approach originally proposed by De Serpa (1971), Jara-Díaz and Guevara (2003) showed that it is possible to derive a model including equations for both the choice of mode and of the number of hours worked. Although they tested their model using real data and obtained plausible parameters, they acknowledged that by assuming independent error terms in both equations they made the simplest assumption. In this paper we look at the estimation problem of this model in more depth, searching for a method that allows us to make more realistic assumptions and testing it using the

* Final version : November 2005. ** Marcela A. Munizaga and Sergio R. Jara-Díaz are at the Dept. of Civil Engineering, University of Chile ; Rodrigo Correia and Juan de Dios Ortúzar are at the Department of Transport Engineering, Pontificia Universidad Católica de Chile. Acknowledgements : this research was partially funded by FONDECYT Grant 1030694 and the Millennium nucleus in Complex Engineering Systems.

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Marcela A. Munizaga · Rodrigo Correia et al.

same data set of Jara-Díaz and Guevara (2003). As the problem requires the estimation of a joint model of mode choice and number of hours worked, it is by nature a combined discrete/continuous problem with potential endogeneity issues (Greene, 2003). The main objective of the paper is to identify a reliable approach to keep on combining the microeconomic advances in the analysis of travel and activities (e.g. Jara-Díaz and Guerra, 2003) with the most appropriate econometric approaches. The rest of the paper is organized as follows. In section 2 we present the main features of the microeconomic model of Jara-Díaz and Guevara (2003) and show how it can be used to estimate the various components of the subjective value of time. In section 3 we discuss the issue of modeling discrete/continuous choices, while section 4 shows how one particularly useful approach can be adapted to our problem. Section 5 briefly discusses our empirical results and in section 6 we give our main conclusions and directions for research. 2. Problem formulation Following DeSerpa (1971), Jara-Díaz and Guevara (2003) considered the problem faced by an individual that chooses to travel, incorporating the time assigned to all activities and goods consumption in the direct utility function. They assumed that the optimization process is subject to various types of constraints (i.e. time budgets, monetary budgets) including a technological constrain concerning the minimum time required for travel. Assuming a Cobb-Douglas utility function, first order conditions allowed them to derive an explicit model for the amount of time assigned to work (Twq j) by individual q conditional on the choice of mode j : j j Twq = b ax - Tq k + a

cqj

+ wq R V j 2 c cqj S q W j j j + Sb ax - Tq k + a f pW - 72 â + bh - 1A ax - Tq k + h q S wq W wq T X

[1]

where Tqj is its total travel time, cqj stands for its travel cost, x represents the total time available, wq the wage rate for person q, and hqj the random error term associated to work time. By observing that equations depending on the same variables for other activities and for goods consumption can be obtained as well, the authors show that a conditional indirect utility func-

Valuing time with a joint mode choice-activity model

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tion that commands mode choice is obtained by replacing these equations, including [1], in direct utility. The linear approximation to the indirect utility conditional on choosing mode j by person q (Uqj) is : Uqj ≈ c j + ctT qj + ccc qj + fqj

[2]

where c j is a mode specific constant for alternative j, c t and cc are the marginal utilities of travel time and cost respectively, and fij is a randomly distributed error term. Equations [1] and [2] are related, as they are derived from the same model and, therefore, they share explanatory variables. Furthermore, the mode choice ruled by [2] determines the observed values of Tqj and cqj for the chosen mode, which are explanatory variables in [1]. Although a and b are mathematical constructs with no direct interpretation, Jara-Díaz and Guevara (2003) show that these parameters, together with those of the discrete choice model, permit the calculation of all the components of the (subjective) value of travel time savings (VTTS) as defined by De Serpa (1971). In our case the values involved are : • the value of leisure, or value of time as a resource (µ/b) ; • the VTTS (l/m), the willingness to pay to reduce the time assigned to travel ; and • the value of assigning time to a particular activity i, such as work or travel ((∂U/∂Ti)/m).

The above parameters (µ, m and l) correspond to the Lagrange multipliers of the maximum utility framework used to derive the model. Although more demanding readers might want to refer to the original source for a more complete explanation, we can briefly state that these quantities represent the variation of the objective function evaluated at the optimum due to a marginal relaxation of the corresponding restriction. In particular, µ is the multiplier associated to the time restriction so we can interpret it as the marginal utility of time. In the same way m, the multiplier for the monetary budget and l, the multiplier for the technological constrain (minimum time assigned to travel in a given mode), are associated to the marginal disutility of travel cost (c c) and the marginal disutility of travel time (c t) respectively. The relation between the VTTS components is established through equation [3], which is derived from the first order conditions of the maximum utility problem ( Jara-Díaz and Guevara, 2003) : l m

=

n m

-

2U/2Tt m

=w+

2U/2TW m

-

2U/2Tt m

[3]

This states that the value of travel time savings (VTTS) is equal to the value

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of leisure (µ/m) minus the value of assigning time to travel (probably negative). On the other hand, the value of leisure is equal to the total value of work, which adds the value of assigning time to work (which can be positive or negative, depending on whether the individual likes or dislikes working) plus the wage rate. Many authors have shown the direct correspondence between the VTTS, l/m, and the ratio of the marginal disutilities of time and cost estimated in discrete choice models such as that represented by equation [1] ( Jara-Diaz, 2000) ; thus, only an expression for the value of leisure (µ/m) is needed to estimate all the components of the VTTS. For this Jara-Díaz and Guevara (2003) derived equation [4] : n m

=d

1 - 2b wTW - ct n$e o 1 - 2a x - TW - Tt

[4]

To obtain estimates of the values of a and b, the model represented by [1] must be calibrated. But, as equations [1] and [2] are related, a joint estimation is required to allow for potential correlation effects. Therefore, we must make certain assumptions about the structure and distribution of their error terms. Note that both equations are subject to disturbances arising from various sources : first, the empirical observation of the variables is subject to data collection errors ; second, the assumptions made imply a certain specification error and, third, there may be differences among individuals due to the randomness inherent to human nature. Once a and b are estimated, typically for two or more income strata, [4] is simply calculated as an average for the members of each stratum. If we look at the mode choice model described by [2] in isolation, the traditional approach is to postulate an additive Gumbel error, leading to the simplest mode choice model form (e.g. the multinomial logit, MNL, see Ortúzar and Willumsen, 2001). This assumption can be modified assuming more general error structures, but we believe 1 it would not change the results dramatically. On the other hand, given the non-linearity of [1] it is harder to suggest a natural distribution for its error term. As the multiple sources of disturbance contributing to the equation have unknown distributions it may be assumed, conservatively, that hqj distributes Normal ( JaraDíaz and Guevara, 2003). Now, as there are variables subject to data collection errors which appear in both equations, it is natural to expect correlation between the error terms 1

In fact we proved it in our empirical work (Correia, 2003).


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of both models. Also, there may be unobserved variables (specification errors) that affect both equations also causing correlation. Finally, there is an analytical reason to postulate the existence of correlation ; the indirect utility function is conditional on the chosen mode and, therefore, it implies calculating the maximum utility that the individual may reach given his/her mode choice. As explained earlier, this implies that the time and cost of the chosen mode are not only mode choice determinants, but also attributes that infl uence the working time assignment decision. For synthesis, the problem can be formulated as two related equations : one continuous and one of discrete choice, both related through underlying factors that should induce correlation between their error terms. 3. Discrete/continuous econometric models The estimation of econometric models for discrete/continuous decisions has been an issue in the literature for some time. Although Train (2003) states that the topic has not reached the same type of maturity as discrete choice modeling, it is worth describing some of their central aspects. However, we agree on his assessment that discussing the different issues and procedures involved could take a book in itself. The first approaches for dealing with the problem were of a sequential type, typically involving two stages. First the discrete choice parameters are obtained using maximum likelihood techniques and in the second stage two different approaches may be used. If the analyst believes that one or more variables of the continuous equation are endogenous to the discrete choice (endogeneity bias), they can be replaced as a function of the probabilities obtained at the first stage. On the other hand, if the analyst believes that the continuous observations depend on the discrete choice (selectivity bias), the regression equation can be augmented by adding an additional term, also constructed as a function of the first stage probabilities. Note that both conditions are present here : the conditional indirect utility is a function of the costs and times of the different alternatives, under the assumption that the individual has endogenously optimised Tw, and Tw depends directly on the cost and time of the chosen alternative in the discrete problem (see equation [2]). Classical references to the sequential method using selectivity correction terms are those by Heckman (1979) and Dubin and McFadden (1984). The first assumes that both disturbances distribute Normal and is very popular due to its simple application. Dubin and McFadden (1984) broaden the approach by treating the discrete choice error as Gumbel distributed (therefore

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a MNL is in hand), whilst the error on the continuous outcome is assumed to distribute Normal also. As the subject expanded and more powerful computational tools became available, full information maximum likelihood models appeared. The idea was, in general, to obtain more efficient, consistent and unbiased estimators, as sequential methods are not as efficient and have problems when constructing an asymptotically correct covariance matrix. Hanemann (1984) pointed out that the building blocks for a discrete/continuous econometric model are the indirect utility function and the joint distribution of the error terms in both the discrete and continuous decisions. The first aspect is related to the microeconomics of the problem and is well treated by Jara-Díaz and Guevara (2003). On the other hand, the joint distribution of the error terms is certainly the key aspect of a stochastic specification. The adequate definition of such a joint distribution allows an unrestricted treatment of correlation between disturbances. The dichotomous case is rather simple. For example, if we assume that both errors are jointly distributed Normal the marginal distributions are also Normal and we obtain a Probit model (Daganzo, 1979). The most common dichotomous case in the transportation literature is the possession and use of a car ; De Jong (1989) models this situation giving a good example of a discrete/continuous model estimated with full information maximum likelihood techniques. However, the plot thickens considerably for discrete polychotomous choice. First, it is well-known that dealing with multiple choices with a Probit model is quite hard and simulation becomes a necessity. Lee (1983) proposed a method for the cases in which there are a priori reasons to assume certain specific marginal distributions. In a way the problem is looked at from a different perspective. Starting from certain specific marginal distributions the attention is not focused on specifying a joint distribution function but on being able to propose useful joint functions. Any such distribution should allow unrestricted correlation between the error terms. The method consists basically in generating an adequate conversion capable of transforming the marginal distributions previously specified into standard Normal disturbances. Hence, the new joint distribution of the error terms (properly transformed) is also standard Normal ; a brief summary of the model derivation can be found in the appendix. Krishnamurthi and Raj (1988) used this technique to model brand choice and consumption. Barnard and Hensher (1992) used it to model shopping location choice and the rate of expenditure. Both papers assumed a Gumbel


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error term in the discrete choice component and a Normal error term in the continuous equation. Recent research by Bhat (1998 ; 2001) has extended the work of Lee (1983) to consider a discrete polychotomous choice and two continuous decisions ; moreover, he also proposed using hierarchical models in the discrete choice analysis. Sanga (1999) estimated a multinomial Probit model through simulation to treat polychotomous discrete choices. However his proposal is hard to apply when the choice set is large. Finally, Bolduc et al. (2002) offered an extension to both Heckman (1979) and Dubin and McFadden (1984), and developed selectivity correction terms with a Mixed Logit model (Train, 2003). Based on our revision we concluded that an approach based on the work of Lee (1983) was appropriate for our case study. The method presents no difficulty in dealing with multiple discrete choices and allows for the presence of correlation between the chosen discrete alternative and the continuous variable. Furthermore, the model remains computationally tractable given its close form structure and a joint simultaneous estimation can be performed. 4. Activity and travel models considering correlation To adapt Lee’s (1983) approach to simultaneously estimate the system of equations [1] and [2], we assumed that the error term fj associated to mode choice distributes Gumbel (0,1) and that the error hj in the work time assignment model distributes Normal (0,v2). As mentioned above, these assumptions are reasonable for our study context and allow keeping the problem within tractable limits. Indeed, thanks to the generality of Lee’s (1983) approach the possibility of incorporating hierarchical specifications that may allow for correlation between the discrete choice alternatives stays open. Let us first consider the discrete choice component. We know that the condition for choosing mode i is that it has the highest utility among the available alternatives : [5] Ui* $ MaxU *j 6j j!i

We also know that any utility can be written as the sum of an observable component Vi, which is a function of the model parameters, and a random error term fk that in this case distributes Gumbel (0, 1). Considering this we can rewrite [5] as follows :

Vi $ eMaxU *j - f io = ~ i 6j j!i

[6]

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The right hand side of [6] is a random term that we will call ~i. Its probability distribution depends on that of the random variables U and f, which is exactly the same. Also, knowing that the maximum of independent Gumbel terms distributes Gumbel, and that the difference between two identical and independent Gumbel functions distributes Logistic, we get the well-known MNL model (Ortúzar and Willumsen, 2001). Using Lee’s (1983) method the random term ~i can be transformed into a standard Normal error applying the inverse Normal function. An interesting property of such transformation is that, being a strictly increasing function, it has the capacity of maintaining the mode i choice condition, Vi ≥ ~i, in the following way : J(Vi) = Φ-1[F (Vi)] ≥ Φ-1[F (~i)] = ~i*

[7]

Here J (Vi) and ~i* are random variables generated by applying the inverse Normal function Φ-1 to the distribution function of the original variables Vi and Vi. F(·) represents the Logistic distribution function that characterizes the MNL model :

F ^Vih =

exp (Vi) exp ^Vih + !exp _Vji

[8]

j=1 i!j

Thus, we have transformed a polychotomous choice with Gumbel error terms into a dichotomous choice with Normal error terms, and the new random variable ~i* gathers disturbances relative to alternative i in relation to all other alternatives j. We can present now the discrete choice system as follows : R *qi = J (Vi) – ~i* Rqi = 1

[9]

if

R *qi > 0

[10]

Rqi = 0 if

R *qi < 0

[11]

Here Rqi equals one if q chooses alternative i, and zero otherwise. R *qi represents the unobserved propensity of individual q to choose mode i. Note that the new error term ~i* has a negative sign in [9]. This is important in the sign analysis for the correlation between the transformed error term ~i* and the error term of the continuous equation hi. More precisely, if the sign of the estimated correlation coefficient is negative it would indicate that underlying factors increasing the propensity to choose mode i would also increase the amount of time assigned to work.


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As ~ *qi and hqi distribute Normal their joint and conditional distributions are also Normal. Decomposing the joint probability and remembering that Twi is only observed if alternative i is chosen (i.e. equation [7] holds), we can derive an expression for the probability of choosing a given mode i and assigning a certain number of working hours Twi as a function of their error terms : h(~ *qi,hqi) = g1 (~ *qi/~qi)g2(hqi)

[12]

If we define ei as the normalised errors of the continuous choice and bi as the discrete choice errors conditional on the occurrence of the continuous choice, it can be shown that these are given by equations [13] and [14] ; tI is the correlation coefficient between unobserved factors affecting the choice of mode i and the time assigned to work : ei = cj Twi - *b _x - Ti i + a i + wi j

cj cj >b _x - Ti i + a f i pH - 72 â + bh - 1A i _x - Ti ji4 wi wi 2

j

v

bi =

U- 1 7F ^VihA

[13]

- t i ei

[14]

1 - t 2i

Then we can derive a likelihood function using [12] in which ei accounts for error term hi and bi for error term fi conditional on hi. Using standard notation, the full information likelihood function can be written as :

% * % < v1 z êih $ U ^bihF Q

L=

Nq

4

Rqi

q=1 i=1

[15]

Here Nq represents the total number of modes available to individual q, v is the standard deviation of h and z (·) the Normal density function. Although ti must also be estimated within the likelihood maximization process, the idea behind the model is not to find a specific value for the correlation coefficient but to obtain more efficient, consistent and unbiased estimators for the relevant parameters. 5. Numerical application To test our econometric method with real data (an analysis using simulated data can also be found in Correia, 2003) we used the same data bank as Jara-

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Díaz and Guevara (2003) and compared results. The sample was built from information about 366 workers gathered at the 1991 Santiago Origin-Destination survey (Ortúzar et al., 1993). The workers were randomly chosen from a set the activity pattern of which was extremely simple : home-tripwork-trip-home ; thus, the data contains information regarding the times assigned to these four different activities during a normal working day. The data bank also includes level-of-service data (i.e. walking, waiting and in-vehicle travel time), albeit measured at a rather aggregate level, and information relative to the modal costs for each individual. The sample was grouped into two strata defined by net family income (i.e. between US$ 240 and US$ 880, 2 and higher). Average travel costs were US$ 0.37 for the medium income stratum and US$ 0.49 for the high income stratum. Table 1 presents information about modal choice and availability by income for the 366 individuals in the sample. From the point of view of an interesting discussion later, the main interest lies in the rather marginal modes Walk and Taxi. Table 1. Mode Choice and Availability by Income Strata. Mode Choice Mode Car Driver Car Pass. Bus Shared Taxi Metro Walk Taxi Bus-Metro Shared Taxi-Metro Total

Medium High Income Income

Availability Total

102 4 138 8 11 17 7 3 4

47 5 12 1 2 0 4 1 0

149 9 150 9 13 17 11 4 4

294

72

366

Medium High Income Income Total 149 5 294 286 72 67 294 145 145

64 5 72 72 15 14 72 45 44

213 10 366 358 87 81 366 190 189

Table 2 presents a model based on our approach and another estimated assuming independent error terms, as in Jara-Díaz and Guevara (2003). 3 In 2

At the time of the survey 1US$ was equal to approximately 460Ch$. Note that the results are very close but not identical to those of the original paper, as a different estimation method was employed ; in our case we used a purposely written routine in Gauss. In particular, we also found that the value of a did not vary across strata. The interested reader can find further model results in Correia (2003). 3

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both cases the bus was considered the base alternative. It is worth mentioning that different NL hierarchical structures were tested, having all public transport modes in a nest, or allowing for nests containing mixed transport modes. Likelihood ratio tests (LR) showed that these were not superior to the MNL ; besides, their structural parameters were not significantly different from one, indicating no correlation between alternatives (Correia, 2003). Table 2. Model Results.

Variables Mode Constants

Car driver Car passenger Shared Taxi Metro Walk Taxi Bus-Metro Shared Taxi-Metro Parameters γ time (medium income) γ cost (medium income) γ time (high income) γ cost (high income) α (1) β medium income β high income σ Correlation ρ Car driver coefficients ρ Bus ρ Walk ρ Taxi

Log Likelihood N° of Observations Likelihood Ratio (LR) χ2 (4, 99.9%)

Independent Errors Estimator t-ratio 0.4299 2.1979 -2.4764 -0.4284 0.5483 -0.7839 -2.3403 -2.4814 -0.0611 -0.0038 -0.0918 -0.0020 -2.2732 0.0961 0.1208 160.44 -2646.3 366 -

1.6 1.1 -7.1 -1.3 1.5 -1.5 -4.4 -4.8 -4.8 -4.2 -4.0 -2.2 -6.3 13.3 13.6 27.1

Considering Correlation Estimator t-ratio 0.5677 2.1391 -2.4399 -0.5534 0.0759 -0.3437 -2.0178 -2.4332 -0.0547 -0.0052 -0.077 -0.0023 -3.2604 0.0948 0.1305 163.69 0.6232 0.3011 -0.4699 0.4905 -2629.6 366 33.4 18.5

2.2 2.0 -7.0 -1.7 0.2 -0.7 -4.0 -4.7 -4.4 -5.2 -3.5 -2.3 -6.7 11.2 13.9 24.7 5.7 1.9 -3.2 4.0

(1) : This parameter was almost identical for both strata so it was re-estimated as a unique value.

Table 2 shows that incorporating correlation among the error terms yields a model that is clearly better than the one assuming independence. The LR

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test rejects the null hypothesis of both models being equivalent even at the 99.9% confidence level. The sign of the correlation coefficients indicate that when the chosen mode is Walk, there are common unobserved factors that affect the utility of the chosen mode and the time assigned to work with the same sign ; while, when the chosen mode is Bus, Taxi or Car driver, the effect is of the opposite sign. We can speculate that this sign might be revealing a proximity to the working place ; we will investigate this issue below. Looking further at the models in Table 2 one can see that the two constants most affected in size and significance are those of Walk and Taxi. From Table 1 we know that the first mode is chosen by a significant amount of medium income people (and no one of high income) although it is available to only 22% of that stratum. On the other hand, Taxi is universally available but it is only chosen by 3% of the sample. This, together with the aforementioned sign of its correlation coefficient, led us to consider the case of Walk in more detail. For this we had to go back to the original data source and obtain information regarding the distance between homes and workplaces. We tested the importance of this variable defining a dummy equal to one if the person lived near his/her workplace. This variable was introduced in the indirect utilities of Car Driver, Bus, Walk and Taxi (i.e. those modes presenting significant correlation in the model in Table 2). The model estimated in this way had an enormous increase in log-likelihood (–2603.4) in comparison with those of Table 2. Also, the significance of the correlation coefficient between Car Driver and the time assigned to work increased considerably (whilst the Car Driver constant decreased substantially in size and significance). Finally, and in line with our expectations, the size and significance of the correlation coefficient between Walk and the time assigned to work decreased importantly, confirming the connection between that term and the distance dummy variable (Correia, 2003). As stated earlier, perhaps the main feature of the model by Jara-Díaz and Guevara (2003) is that the various components of the subjective value of time (VTTS) can be calculated using expressions [3] and [4]. These components are the value of leisure (or value of time as a resource), the value of assigning time to work, and the value of assigning time to travel. Estimates for these values are presented in Table 3 ; their t-statistics were calculated using a first-order Taylor approximation of the ratios and the values of the covariance matrix ( Jara-Díaz et al, 1988 ; also in Ortúzar and Willumsen, 2001). Confidence intervals for the VTTS were built using the approach of Armstrong et al. (2001).


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Table 3. Comparison of the VTSS Components [US$/hour]. Results Medium Income Stratum (Wage rate : 3.00) Subjective Value of Travel Value of Value Value Time Savings Leisure of Time of Time Assigned to Assigned to Confidence Model Value t-stat Value t-stat Work Travel Interval Independent 2.10 4.2 1.40 – 3.72 0.117 8.5 -2.883 -1.983 With 1.37 4.2 0.84 – 2.26 0.078 8.7 -2.922 -1.292 Correlation Percentage -34.8 -33.3 1.0 34.8 Variation Results High Income Stratum (Wage Rate : 6.57) Subjective Value of Travel Value of Value Value Time Savings VTTS Leisure of Time of Time Assigned to Assigned to Confidence Model Value t-stat Value t-stat Work Travel Interval Independent 5.99 2.3 4.71 – 45.20 0.310 8.1 -6.260 -5.680 With 4.39 2.2 2.93 – 26.89 0.220 8.0 -6.350 -4.170 Correlation Percentage -26.7 -29.0 1.0 -26.6 Variation

As can be seen, by incorporating correlation effects and other improvements detected through the correlation analysis, both the VTTS and the value of leisure decrease by an important proportion (nearly 30% of the original values) ; in particular, the VTTS decreases from 70% to 46% of the wage rate in the case of medium income people, and from 91% to 67% of the wage rate in the case of high income individuals. This is considered a positive result as VTTS studies in Chile had been criticized for presenting much higher estimates than those reported in the literature (Gaudry et al., 1989). On the other hand, the range associated to each confidence interval is reduced significantly in both strata (almost by half in the high stratum), indicating that there has been a large improvement in the precision of the estimates. So, it appears that the consideration of correlation translates not only into better models, but also into statistically different and more reliable results. 6. Conclusions We have estimated a joint model of travel choice and work time assignment following the framework developed by Jara-Díaz and Guevara (2003), consid-

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ering an error structure within the family of discrete/continuous econometric models using the transformation proposed by Lee (1983). We derived a joint likelihood function for the model including a term accounting for correlation between the errors of the two sub-models and estimated all parameters using full information maximum likelihood. The method is efficient and much simpler than alternative sequential methods. Our results reveal that the theoretically superior treatment of the error structure improves the resulting model from all angles. The joint model allowing for correlation among error terms turned out to be significantly better statistically than the model assuming independent errors and led to more reasonable, consistent and precise estimates of the different components of the subjective value of time. Furthermore, the interpretation of the correlation coefficients proved useful to shed light on improving the specification of the deterministic component of utility, as correlation modeled is between unobserved factors relegated to the error term. In particular we found that by incorporating a dummy representing how distant was home from work, the correlation between Walk and the time assigned to work decreased significantly, and the improved specification presented a substantial increase in likelihood. We have also shown that it is possible and necessary to use an econometric framework allowing for correlation when modeling jointly decisions that come from a common microeconomic model. As the econometric framework is sufficiently flexible, and systems with two continuous equations have already been used in other contexts (Bhat, 1998 ; 2001), we foresee using it to model the more general system derived by Jara-Díaz and Guerra (2003) that includes other activities and other consumptions. This opens many possibilities for further research, such as the estimation of a model system with several continuous and maybe even several discrete choices. Also, the recent literature is generous on discussions about flexible error structures for discrete choice models (see for example Ben-Akiva et al., 2002). In this sense, the solid microeconomic basis of the model allows for such discussions, including a challenge to the hypothesis of additive error terms, or for testing the effect of a non-linear specification of the conditional indirect utility function. References Armstrong, P. M., R. A. Garrido and J. de D. Ortúzar, 2001, “Confidence intervals to bound the value of time”, Transportation Research, 37E, 143-161. Barnard, P. O. and D. A. Hensher, 1992, “Joint estimation of a polychotomous discretecontinuous choice system : an analysis of the spatial distribution of retail expenditures”, Journal of Transport Economics and Policy, 26, 299-312.


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Ben-Akiva, M., D. McFadden, K. Train, J. Walker, C. Bhat, M. Bierlaire, D. Bolduc, A. Boersch-Supan, D. Brownstone, D. Bunch, A. Daly, A. De Palma, D. Gopinath, A. Karlstrom and M. A. Munizaga, 2002, “Hybrid choice models : progress and challenges”, Marketing Letters, 13, 165-173. Bhat, C. R., 1998, “A model of post-home arrival activity participation behaviour”, Transportation Research, 32B, 387-400. Bhat, C. R., 2001, “Modelling the commute activity-travel pattern of workers : formulation and empirical analysis”, Transportation Science, 35, 61-79. Bolduc, D., L. Khalaf and E. Moyneur, 2002, “Joint discrete/continuous models with possibly weak identification” (Working Paper, Department of Economics, Université Laval, Québec). Correia, R., 2003, “Especificación de Errores en Modelos Conjuntos de Asignación de Tiempo a Actividades y Viajes” (MSc Thesis, Pontificia Universidad Católica de Chile. in Spanish). Daganzo, C. F., 1979, Multinomial Probit : The Theory and its Application to Demand Forecasting, Academic Press, New York. De Jong, G. C., 1989, “Simulating car costs changes using an indirect utility model of car ownership and annual mileage” (Proceedings 17th PTRC Summer Annual Meeting, Brighton). De Serpa, A., 1971, “A theory of the economics of time”, The Economic Journal, 81, 828846. Dubin, J. and D. McFadden, 1984, “An econometric analysis of residential electric appliance holdings and consumption”, Econometrica, 52, 345-362. Gaudry, M. J. I., S. R. Jara-Díaz and J. de D. Ortúzar, 1989, “Value of time sensitivity to model specification”, Transportation Research, 23B, 151-158. Greene, W. H., 2003, Econometric Analysis. Fifth Edition, Prentice Hall, New Jersey. Hanemann, W. M., 1984, “Discrete/continuous models of consumer demand”, Econometrica, 52, 541-561. Heckman, J. J., 1979, “Sample selection bias as a specification error”, Econometrica, 47, 153161. Jara-Díaz, S. R., 2000, “Allocation and valuation of travel time savings”, in : D. Hensher and K. Button, eds., Handbooks in Transport, Vol. 1 : Transport Modelling (Pergamon Press, Oxford) 303-319. Jara-Díaz, S. R. and C. A. Guevara, 2003, “Behind the subjective value of travel time savings : the perception of work, leisure and travel from a joint mode choice-activity model”, Journal of Transport Economics and Policy, 37, 29-46. Jara-Díaz, S. R. and R. Guerra, 2003, “Modeling activity duration and travel choice from a common microeconomic framework” (10th International Conference on Travel Behaviour Research, Lucerna, Suiza). Jara-Díaz, S. R., J. de D. Ortúzar and R. Parra, 1988, “Valor subjetivo del tiempo considerando efecto ingreso en la partición modal” (Actas del V Congreso Panamericano de Ingeniería de Tránsito y Transporte, Puerto Rico, in Spanish). Krishnamurthi, L. and S. P. Raj, 1988, “A model of brand choice and purchase quantity price sensitivities”, Marketing Science, 7, 1-20. Lee, L. F., 1983, “Generalized econometric models with selectivity”, Econometrica, 51, 507512. Ortúzar, J. de D., A.M. Ivelic, H. Malbrán and A. Thomas, 1993, “The 1991 Great San-

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tiago origin-destination survey : methodological design and main results”, Traffic Engineering and Control, 34, 362-368. Ortúzar, J. de D. and L. G. Willumsen, 2001, Modelling Transport, Third Edition, John Wiley and Sons, Chichester. Sanga, D., 1999, “Estimation des Modèles Econométriques de Choix Discrets/Continus avec Choix Polytomiques Interdépendants : Une Approche par Simulation » (PhD Thesis, Université Laval, Québec, in French). Train, K., 2003, Discrete Choice Methods with Simulation, Cambridge University Press, Cambridge. Train, K. and D. McFadden, 1978, “The goods/leisure trade off and disaggregate work trip mode choice models”, Transportation Research, 12, 349-353.


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Appendix : the general method In this appendix we summarize the central ideas of the transformation proposed by Lee (1983). The method derives a likelihood expression for a discrete/continuous model in terms of a Normal distribution and permits an adequate econometric calibration including certain correlation terms. In particular, assume a situation in which person i has M possible discrete choices (in our case M travel modes) with a latent utility (not observed) denoted as Usi* . In the same way suppose a continuous assignment, conditional on the choice of the discrete alternative s, Ysi. With these definitions we can formulate the following equation system : Usi* = zsi cs + hsi

(s = 1, ....., M)

Ysi = xsi bs + vsnsi

[A.1] [A.2]

In [A.1] zsi is a vector of exogenous variables for person i such as, for example, the time for mode s. Consequently, cs is a vector of parameters to be estimated and hsi a random error term with a given marginal distribution (which has been previously defined). In equation [A.2], xsi is a vector of exogenous variables such as the level of education of person i, bs is a vector of parameters to be estimated, µsi is a random variable the marginal distribution of which has been already defined and vs represents its standard deviation. Furthermore it is assumed that the expectations E(µsi / x1,...,xM, z1,...,zM) and E(hsi / x1,...,xM, z1,...,zM) are both equal to zero. Expression [A.3] considers the necessary condition for choosing alternative s. To save notation we omit the individual sub index.

U s*> Max U *j j = 1, f, M j!s

[A.3]

This expression is equivalent to say that alternative s is chosen when its utility is greater than that of the second best alternative. Us* may be decomposed, as usual, in an observed portion plus a random term. Therefore we can isolate the random terms of [A.3] and rewrite it as [A.4] :

Vs> Max U *j - h s j = 1, f, M j!s

[A.4]

Let us define I as a variable that can take values from 1 to M ; I=s if alternative s is chosen. Using equations [A.1] and [A.4] the choice condition for alternative s can be written as :

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Marcela A. Munizaga · Rodrigo Correia et al. zscs > hs*

I = s if

where

h *s / Max U *j - h s j = 1, f, M j!s

[A.5]

Let G(µ) and F (hs*) be the previously specified marginal distributions for µ and hs*. Lee (1983) shows that error terms µ and hs*, which distribute according to functions G and F, may be transformed in standard Normal random variables using the inverse Normal function. If Φ represents the standard Normal distribution, it is possible to create the following new random terms : = J1(hs*) ≡ Φ–1(F (hs*)) n* = J2(n) ≡ Φ–1(G (n))

hs**

[A.6] [A.7]

A desirable property characterizing this transformation is that the new error terms hs** and µ* are now jointly Normal distributed, with zero mean and correlation coefficient t. This allows unrestricted correlation between them. In particular, in order to specify a likelihood function the central idea is to define the probability of observing both errors jointly. Using classic probability theory we can write : fp (hs**, n* = g1(n*) · g2 (hs**/n* ) [A.8] Taking gs ( · ) as the density function for Gs ( · ) and the dummy variables Ds =1 if and only if I=s, the log likelihood of the multiple choice model for a random sample of size N, is given by : ln L (b, c, t, v) =

! ! )Dsi $ ln gs d Ysi -vxs si b s n + Dsi $ M

N

s = 1i = 1

[A.9] R V t $ J2s _^Ysi - xsi b sh /v si W S $ fln U S J1s ^zsi c sh WWp4 S 1 - t 2s T X The first term corresponds to the continuous realization and the second term describes the discrete choice conditional on the continuous outcome. t and v are parameters to be estimated within the model. The correlation coefficient t represents the correlation between the chosen alternative of the modified discrete problem (the error of which appears with a negative sign) and the continuous variable. Therefore, a negative value of t indicates the presence of positive correlation between the error of a certain discrete alternative, if it is chosen, and the continuous variable.