Representation of Taste Heterogeneity in Willingness to Pay Indicators Using p

7/27/2019 Representation of Taste Heterogeneity in Willingness to Pay Indicators Using p

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Representation of taste heterogeneity in willingness

to pay indicators using parameterization inwillingness to pay space

Stefan Mabit

Sebastian Caussade

Stephane Hess

September 15, 2006

Abstract

One of the most important uses of discrete choice models is as a meanto estimate willingness to pay (WTP). The paper gives a review ofdifferent ways to incorporate taste heterogeneity in WTP indicators.

Furthermore it investigates the difference between the two approachestermed modelling in preference space and WTP space. On one handthe conclusion of the paper is that none of the approaches seems todominate when it comes to model fit, hence they should both be withinthe toolbox of the discrete choice modeller. On the other hand theinvestigations show that not including correlation between coefficientsin mixed logit models is a bigger problem for models in preferencespace. When it comes to the inclusion of explanatory variables theempirical investigation gives evidence to the fact that they should be

incorporate with caution to avoid confounding heterogeneity in WTPwith heterogeniety in scale.

1 Introduction and context

Discrete choice (DC) structures belonging to the family of Random Utility


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across a variety of contexts, ranging from mode choice to destination choice,via the choice of departure time and the choice of route. An ever increasingnumber of different structures are available to modellers, ranging from thebasic Multinomial Logit (MNL) model to more advanced structures, such asMixed Multinomial Logit (MMNL), and other Generalised Extreme Value

(GEV) mixture models.While DC models are also used for purposes such as the forecasting ofmarket shares after the implementation of hypothetical policy changes, themain output of studies using DC models is the computation of willingnessto pay (WTP) indicators. These WTP indicators give an estimate of thereadiness by a respondent to accept an increase in the cost of an alternativein return for an improvement to the alternative along some other dimension,such as travel time. This specific case, namely the trade-off between travel

cost and travel time, gives rise to the most commonly used WTP measure,namely the valuation of travel time savings (VTTS). The VTTS is a measureindicating how much a respondent is willing to pay in additional travel costin return for a reduction of his travel time by a certain measure k, wherethe VTTS is typically calculated for a value of k equal to one hour.

The computation of VTTS measures has been one of the main applica-tions of RUM models, with some recent discussions of the topic includingAlgers et al. (1998), Hensher (2001a,b,c), Lapparent & de Palma (2002), and

Sillano & Ortuzar (2004). The VTTS is used for example for cost benefitanalysis (CBA) in the context of planning new transport systems, or forpricing. In discrete choice models, the computation of VTTS measures isrelatively straightforward, given by the ratio of the partial derivatives of theutility function with respect to travel time and travel cost (i.e. the marginalrate of substitution between travel time and travel cost, at constant utility).Although this is an intuitively plausible approach, it is important to appre-

ciate that the justification for this approach to the valuation of travel timesavings rests not on plausibility but rather on a substantial body of microe-conomic theory that addresses the issue of how individuals allocate timeamongst alternative activities, including travel. Indeed, the topic of timeallocation and valuation has been the subject of intense study from a vari-ety of different perspectives for several decades (see, among others, Becker1965 O t 1969 D S 1971 E 1972 T & H h 1985 B t


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to stress that various other WTP measures are of interest in the context oftransport research. As such, policy planners may be interested in travellerswillingness to pay for increases in frequency, improvements in on-time per-formance or reductions in schedule delay. In fact, the term willingness topay can be rather misleading in this case, as the dimension along which the

respondent accepts a decrease in attractiveness need not necessarily be amonetary one. Indeed, one can similarly be interested in respondents will-ingness to accept increases in travel time in return for improvements alongsome other dimension.

One of the main topics of research in the area of WTP indicators hasbeen the representation of heterogeneity in these indicators, i.e., a variationin their value across respondents. A lot of work has recently been donein this area, and several different ways of modelling heterogeneity in WTP

indicators have emerged. However, there has not been any investigation inthe transportation field of specification in WTP space. Hence, the aim of thispaper is to take another look at the problem of the computation of WTPindicators, with a specific focus on representing the heterogeneity acrossrespondents directly in WTP space as opposed to traditional specifications.

Aside from offering a review of existing work in this area, the maincontribution of the paper comes in several case studies that compare thedifferent ways of accounting for heterogeneity in WTP indicators.

The remainder of this paper is organised as follows: in the next section 2we review the literature on how to represent heterogeneity in discrete choicemodels. In section 3 and section 4 we describe our empirical investigations ofhow to represent taste heterogeneity in mixed logit models on two differentdata sets. Section 5 summarizes the main points of the paper.

2 Representation of heterogeneity in WTP indi-

cators

In this section, we look at different ways of representing heterogeneity inWTP indicators. While the majority of the discussion focusses on the caseof VTTS, the findings directly extend to other trade-offs, be they monetaryor not


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computed as:

V/T

V/C, (1)

with V giving the observed part of utility, and T and C representing the

travel time and travel cost attributes respectively. In the case of fixed tastecoefficients, and with the commonly used linear-in-attributes utility func-tion, this formula reduces to T/C, where T and C are the time and costcoefficients, giving the marginal utilities of increases by one unit in traveltime and travel cost respectively. Estimates of these marginal utilities areproduced by estimating the model on the choice data. Even with the use ofnon-linear transforms, such as the natural logarithm, the computation re-mains relatively straightforward, although the actual values of the attributesnow enter into the computation of the trade-offs.

2.2 Heterogeneity in WTP indicators

It should be clear that the assumption of a purely homogeneous populationcannot in general be seen to be valid. As such, two individuals will differin their behaviour due to differences in priorities as well as sensitivities. Ina DC context, these differences are exhibited in the form of different values

for the taste coefficients that relate attributes values to the utility of analternative. These differences will clearly have an effect on the calculationof trade-offs, leading to heterogeneity in WTP indicators. Given the impor-tance of accurate estimates of WTP indicators for use in policy analysis, therepresentation of such heterogeneity is of crucial importance. In this sec-tion, we look at the issue of heterogeneous WTP indicators in the context ofthree discussions, namely the representation of deterministic heterogeneity

(Section 2.2.1), the use of continuous or discrete mixtures for the represen-tation of random heterogeneity (Section 2.2.2), and the question of whetherto work in preference space or in WTP space (Section 2.2.3).

2.2.1 Deterministic heterogeneity

Approaches accounting for deterministic heterogeneity aim to relate varia-


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separate population segments. While easy to apply, the main issue with thisapproach is the requirement to specify appropriate segmentations. If too fewsegmentations are used, the majority of heterogeneity is left unexplained.On the other hand, if too many segmentations are used, the number of obser-vations per segment can be so low as to lead to unstable estimation results.

Finally, with the interest being on the computation of WTP indicators, itshould be noted that the use of L segments along one dimension and Msegments along the other dimension leads to LM different WTP measures,where the number of individuals in the combined segment lm might be verylow.

A slightly more advanced method consists of using continuous interac-tions between estimated parameters and socio-demographic attributes. Itis clear that such continuous treatments of interactions have advantages in

terms of flexibility when compared to the more assumption-bound segmen-tation approaches.

As an example, the following approach can be used1:

f(s, x) = x

ss

s,xx, (2)

where s is the observed value for a given socio-demographic variable, suchas income or trip distance, and s is a reference value for this attribute,

such as the mean value across a sample population. In this example, thesensitivity to an alternatives attribute x varies with s. The choice of thereference value s is arbitrary, and has no effect on model fit, or the estimatefor s,x. However, the use of the mean value, s, guarantees that the estimatex gives the sensitivity to x with most precision at the average value of sin the sample population2. The estimate ofs,x gives the elasticity of thesensitivity to x with respect to changes in s; with negative values for s,x,the (absolute) sensitivity decreases with increases in s, with the oppositeapplying in the case of positive values for s,x. Finally, the rate of theinteraction is determined by the absolute value of s,x, where a value of0 indicates a lack of interaction. At this point, it should be said that aproblem with this approach in the present context could be the fact thatincome information is presented in the form of a set of separate income-classes as opposed to absolute income information leading to a requirement


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case where different income classes are grouped together in a segmentationapproach.

2.2.2 Continuous versus discrete mixtures

While it is clearly preferable, from a computational as well as interpreta-

tion point of view, to explain as much of the heterogeneity as possible in adeterministic fashion, there is in general a remaining part of random het-erogeneity, due to issues with data quality as well as inherent randomnessin choice behaviour. To this extent, it is often necessary to explain at leastpart of the heterogeneity with the help of random coefficients models.

Two main approaches are possible, using continuous or discrete mixturemodels, where the former is used far more frequently than the latter.

Continuous mixture models, such as MMNL model, use integration ofthe closed form discrete choice probabilities over the assumed distributionof the random taste coefficients. A detailed presentation of these modelstructures is given by Train (2003).

Aside from heightened computational cost, due to the reliance on simula-tion approaches (cf. Hess, Train & Polak 2006), the biggest issue that needsto be addressed with the use of continuous mixture models is the choice ofmixture distribution. While the discussion here centers on the case of T,

similar issues arise for C, or indeed also when estimating the VTTS directly(cf. Section 2.2.3). These distributional assumptions play a crucial role, andhave a significant effect on model interpretation.

The issue of the choice of distribution is discussed at length by Hess et al.(2005), such that only a brief summary is given here. Essentially, the prob-lem lies in making a choice of distribution that is consistent with a prioriexpectations as to the true shape of the distribution, without however mak-ing too strong a shape assumption. The vast majority of applications rely

exclusively on the use of the Normal distribution, potentially leading to se-vere issues in interpretation. As such, the Normal distribution is unbounded,giving positive probability to positive as well as negative coefficient values,while, from a microeconomic perspective, T is required to be non-positive.In the presence of a negative mean value, with a long tail to the left, theestimation of the mean and standard deviation will due to the symmetry


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priori constraint limiting the distribution to the negative space of numbers,as factors such as data issues or model imperfections could indeed mean thatpositive numbers are possible. With the use of the Normal distribution, itis impossible to know whether the presence of positive coefficient estimatesis in fact revealed by the model, or is simply a result of the distributional

assumptions. As such, the use of flexible distributions is most appropriate,allowing for asymmetries as well as the estimation of bounds on the distri-bution. Here, the risk of values with the wrong sign being caused by theshape of the distribution, as with the Normal, largely disappears, althoughproblems may still occur in the case of a significant mass at the endpoints,such as in the presence of individuals with zero VTTS. Finally, it shouldbe noted that another possible approach comes with the use of empirical ornon-parametric distributions. However, such approaches are only beginning

to be exploited in the estimation of MMNL models, and difficult issues ofimplementation and estimation need to be faced (cf. Fosgerau 2005).

An alternative to the use of continuous mixture models comes in the useof discrete mixture approaches. Here, the integration over the distribution(as in the MMNL model) is replaced by a weighted summation over a fi-nite number of different support points for the distribution. For a recentdiscussion of this approach, and a detailed comparison between the resultsobtained with continuous and discrete mixture models, see Hess, Bierlaire &

Polak (2006). Discrete mixture models are not affected by the issue of theshape of the distribution, allowing for symmetrical as well as asymmetricaldistribution, as well as multi-modal distributions. Furthermore, they canpotentially be exploited to allow for a mass at a specific value, such as azero VTTS. Another advantage lies in the absence for a need to use simula-tion, where summation over a finite (and generally low) number of supportpoints is used. However, the estimation of discrete mixture models is oftennot straightforward, especially when using multiple support points, whereissues with clustering can arise.

2.2.3 WTP space versus preference space

The majority of studies looking at the estimation of the VTTS work ina way such that the VTTS estimate is obtained on the basis of separate


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Here, the distribution of the VTTS can often not be obtained directly, andsimulation needs to be used. Further complications arise in the case whereboth coefficients follow a random distribution, where correlation betweenthe coefficients needs to be taken into account. One of the main issues inthe simulation of the ratio arises in the case where the distribution allows

for extreme values (and potentially positive values), leading to very largeor very small values for the ratio, such that special care is required in theanalysis of the results (cf. Hess et al. 2005).

In this context, it may often be preferable to work in (WTP) space asopposed to preference space. Here we use the vocabulary introduced byTrain & Weeks (2005). With this approach, the distribution of the VTTSis estimated directly from the data, as opposed to being based on the ratioof two separately estimated coefficients. Specifically, let:

Ui = + T T+ CC+ . . . .

This can easily be rewritten as:

Ui = + CVT+ CC+ . . . ,

where V =TC

is an estimate of the VTTS that is obtained directly from thedata. This approach is no different from working in preference space when

all coefficients are non-random except for a possible influence on confidenceintervals of the estimates. However, when working with random coefficients,the approach has potential advantages, as issues with the simulation of theratio do no longer arise. However, several questions need to be addressed,notably in terms of the potential impact of the choice of distribution for Con the results for V, where the correlation between the two parameters alsoneeds to be taken into account.

3 Danish data

The empirical investigation of this paper will focus on the questions concern-ing WTP space. We investigate this on two different data sets, one collectedin Denmark and the other collected in Chile. For both data sets estimation


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Table 1: Models

spec/model mnl mix mixv log logv loginvv sbv mixlog mixlogv

basic yes yes yes yes yes yes yes yes yes

corr. no yes yes yes yes yes yes yes yes

ev. yes no no yes yes yes yes yes yes

ev. corr. no no no yes yes yes yes yes yes

ev. II yes no no yes yes yes no yes yes

Individuals in the sample each answered 9 binary SP choices with refer-ence to a recent car trip. Each SP choice had 3 attributes: cost, free flowtime, and congested time. The congested time had a fixed ratio to the to-tal time. Since this was explicit to the respondents the SP can be viewas described by cost, time and congestion percentage, where the congestionpercentage enter as explanatory variable describing the individual.

For each individual 8 of the SP choices came from a balanced design pivotedaround the reference trip values. The 9th choice had a dominated alterna-tive i.e. one of the alternatives was faster and cheaper. The 477 respondents

all chose the fast and cheap alternative which confirms that they understandthe SP task. The dominated SP choice is not included in the present analysishence the sample has 3816 observations. Besides the percentage of conges-tion the data included many background variables. From these we will onlyuse income, xinc and travel time of reference trip, tref. As they are oftenseen to be the two most influential explanatory variables.

3.2 Models

On the sample we investigated different specifications with respect to mixingdistribution, correlation, and explanatory variables in both preference spaceand VTTS space. Table 1 gives an overview of the models tested. The firstrow of models are all basic models with either the specification in preferencespace


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or in inverse VTTS space

Ui = s Ti + s1V T T S Ci + i. (5)

For the basic models the two distributions are assumed independent. Themix models have both coefficients normally distributed. The three log mod-

els have both coefficients follow a log normal distribution. In the Sbvttsmodel the scale is log normal and the VTTS follows an Sb distribution. Thefinal model has a normal VTTS coefficient and a log normal scale. In thecorresponding models in in the second row the correlation between the twodistributions is estimated.

Both of the above model types are enlarged to include explanatory vari-ables income and tref. This is done using the multiplicative specificationfrom equation 2 in the time coefficient for the mnl and preference spacemodels.

Ui = T

xincxinc

incTi + CCi.

In the vtts space models the explanatory variables multiply the VTTS coef-ficient. The models with two normal distributions are not elaborated sincethese models have sign problems that will be discussed in the result section.

The last row of models change the parameterizations by explanatory vari-ables. Here the coefficient not parameterized above is parameterized instead.

3.3 Estimation and results

All the result on models without explanatory variables are reported in Table2a-c. As a base model we estimated an MNL model. It was tested against

an MNL including serial correlation for each individual. The serial correla-tion was insignificant which seems reasonable for an unlabeled route choice.It is seen that the alternative specific constant is significant. The signs ofall coefficients in the MNL are as expected and all coefficients are significant.

First we compare the models without explanatory variables. The first mod-


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sign. For the log normal distributions a shift was tested but in all cases itwas insignificant and left out to keep the VTTS distribution as simple aspossible.

The three models above were next estimated allowing for correlation be-tween the coefficients. For all models the correlation turned out significantand positive. The improvements in log likelihood for the models with oneor two log normals were very large.

Corresponding to these models we estimated 3 specifications in wtp spaceand a fourth with log normal scale and wtp following an Sb distribution.For the specification with normal distributions the signs were as expected.For the log normal distributions and the Sb distribution shifts were found to

be insignificant and hence left out of the final models. The model with Sbdistribution had problems converging and only did so when the parameter0 was fixed based on prior estimation.

The models were then estimated allowing for correlation. For all specifi-cations the correlations were significant. The correlation is found as theproduct oftc and cost. Hence for all models in preference space it is pos-itive. For the models in WTP space the results are mixed with a positive

correlation for the model with two normals while for the remaining it wasnegative.

Last we also estimated a model in inverse wtp space for the specificationwith two log normals. This is reasonable only because the inverse for a lognormal distribution is log normal. This specification actually outperformsthe other specification based on log likelihood. The model was tested witha shift for the inverse VTTS distribution, which turned out insignificantly.When allowing for correlation the correlation turned out insignificant, butthe shift became significant.

Next we discuss models with explanatory variables on income and referencetravel time. The results can be seen in Table 3a-c. In all models the signs areth th t i VTTS i ith i d t l ti F th th


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with the inverse vtts space specification obtaining the highest log likelihood.

The last models test an opposite specification of the explanatory variables.These models highlights the problem of heteroscedasticity mixing with theeffect of explanatory variables.

3.4 VTTS calculation

The parameter estimates are asymptotically normal. Based on this we usethe approach discussed in Hensher & Greene (2003) to simulate the dis-tribution of VTTS. From the simulated distribution it is then possible tocalculate the mean as well as quantiles of interest. For each of the modelswe calculate the mean, the standard deviation and the 95% quantile.

The mean values range from 5 to 19 $ pr. hour if we neglect the strangebehavior of model 16. In 3 cases the mean is the ratio of two normal distri-butions and we see that only in the case of the MNL model this does notcause problems. Two immediate observations are that log normal specifica-tion gives rise to higher VTTS and that estimation in WTP space lowersthe VTTS. In preference space correlation is seen to lower VTTS and inWTP space the VTTS rises in the models allowing for correlation. Since

the true VTTS is unknown it is difficult to assess which distribution/spacegives rise to the best mean VTTS but it seems reasonable to expect thatthe true mean has the preference space estimate as upper bound and theWTP space estimates as lower bound. One useful reference could be thatthe mean net hourly wage in the full sample is around 18 $/hr.

4 Chilean data

4.1 Data

For the second application we use the SP data collected by Caussade et al.(2005) on car route choices. Here we use 2166 observations out of the 8020in the full data set. These are the observations where individuals make routechoices based on five attributes: Free flow time (FFT), Moderate congestion


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in Chile.

Even though it will not be of importance in this study it is worthwhile men-tioning that the sample is far from being a random sample of the Chileanpopulation. The reason why this is not important is that we are not inter-ested in the level of WTP but in the way it changes depending on estimationin preference and WTP space.

4.2 Estimation and results

To have a base model we first estimated an MNL model. The preferredmodel was estimated taking serial correlation into account. The parametershave the expected signs and are all significant. The derived WTPs are rea-sonable in the sense that WTP rises with congestion.

The MMNL models are all estimated with normal distributions only or lognormals only. The first type of model has uncorrelated coefficients in prefer-ence space. These models are what is most common in research and appli-cation. For both distributions we see that they outperform the MNL modeland that all the mixing parameters are significant except for the mixing ofTTV in the log normal case. The signs in the normal case are negative as

expected. In the log normal specification we have entered shifts to test thesign constraint implied by a log normal coefficient. Here we see that the shiftis only significant for FFT. This should not be seen as an implication of theexistence of negative WTP merely as an indication that the log normal isan approximation to the true distribution and that it approximates it betterover the whole range of WTP by being shifted to the left.

Both of the above specifications were then estimated allowing for correlation.

For both distributions the model with correlation outperform the modelwithout. For the normal specification the signs and magnitudes remain thesame, but all variances become less significant which seem reasonable. Thecorrelations are all significant except for FFT-MCT and MCT-TC. All thenegative correlations contain HCT. The fact that most correlations are pos-itive contrary to the intuition that individuals with lower marginal utility


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4.3 WTP calculations

Here we calculate the mean, standard deviation and 95% quantile as de-scribed in section 3.4. Again we observe the problems that arise when theVTTS is the ratio between two normal distributions. The values for lognormal models are very high. This could be due to poor identification of

the mixing distribution. For both distributions the values make more sensefor the WTP models. One the other hand it is a problem that MCT has alower value than FFT for the models in WTP space.

5 Summary and conclusions

This paper gives a review of how to incorporate taste heterogeneity in will-

ingness to pay indicators. Furthermore it investigates on real data sets thedifference between the two approaches termed modelling in preference spaceand WTP space. The conclusions of the paper is that there is non of theapproaches that seems to dominate when it comes to model fit but that notincluding correlation between variables seem like a bigger problem for mod-els in preference space. This underlines that it is very probable that mostof the correlation comes from a random scale when individuals evaluate SPchoices. When it comes to the inclusion of explanatory variables the empir-

ical investigation of the paper gives evidence to the fact that they should beincorporate as in WTP space models to avoid mixing effects of scale.

Based on the review and the empirical investigation of the paper three futuredirections of research that deserve attention are: non parametric methodsto aid in the guessing of appropriate distributions, semi parametric methodsand diagnostic tests to assess whether the chosen distributions are correct,and finally the question of which explanatory variables to use on one hand to

avoid the problem of omitted variables and on the other to avoid endogenousexplantory variables.

6 Acknowledgment


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TABLE 2a 1 3 4 5 6

Model: MNL ML - norm ML - norm ML - norm ML - norm

(w. corr.) (vtts sp.) (vtts sp. w. corr )

N. Of draws 500 500 500 500

N. of par.: 4 6 7 6 7

N. of obs.: 3816 3816 3816 3816 3816

N.of ind.: 3816 477 477 477 477

Null LL: -2645.05 -2645.05 -2645.05 -2645.05 -2645.05

Final LL: -2494.47 -2189.04 -2185.52 -2182.27 -2180.72

Ll ratio test: 301.151 912.013 919.069 925.569 928.669

Rho^2: 0.056927 0.1724 0.173734 0.174962 0.175549

Adj rho^2: 0.055415 0.170132 0.171087 0.172694 0.172902Final grad.: 0.015469 0.369839 0.371625 0.659231 4.84738

Var-covar: hessian hessian hessian hessian hessian

Coefficients Value t-test Value t-test Value t-test Value t-test Value t-test

ASC_1 0.104 3.113 0.172 3.962 0.176 3.981 0.174 3.945 0.175 3.965

BETA_cng -0.265 -5.218 -0.816 -3.892 -0.855 -6.736 2.312 4.466 2.375 4.764

BETA_cost -0.103 -8.834 -0.420 -9.956 -0.442 -9.569 -0.448 -9.681 -0.449 -9.957

BETA_time -0.086 -9.148 -0.201 -8.654 -0.219 -8.112 0.556 8.353 0.706 10.514

SIGMA_cost 0.302 9.979 0.341 9.282 0.314 8.688 0.306 9.183SIGMA_time -0.278 -10.428 -0.314 -11.810 -0.956 -12.083 -0.888 -13.054

rho_time_cost 0.107 4.674 0.345 6.384

BETA_0

BETA_1

At zero

congestion

E(VTTS) $/hr 8.32 11.01 1.80 5.46 6.94std. dev. 1.35 7019.90 1116.50 9.49 9.4595% fractile 10.71 44.82 32.15 21.10 22.53std. dev. of

mean 0.07 405.29 64.30 0.54 0.56


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TABLE 2b 7 8 9 10 11 12

Model: ML - log ML - log ML - log ML - log ML - log ML - log

(w. corr.) (vtts sp.) (vtts sp. w. corr ) (inv. vtts sp.) (inv. vtts sp. w. corr )

N. Of draws 500 500 500 500 500 500

N. of par.: 6 7 6 7 6 8

N. of obs.: 3816 3816 3816 3816 3816 3816

N.of ind.: 477 477 477 477 477 477

Null LL: -2645.05 -2645.05 -2645.05 -2645.05 -2645.05 -2645.05

Final LL: -2147.16 -2093.61 -2112.11 -2107.25 -2094.05 -2093.53

Ll ratio test: 995.785 1102.88 1065.88 1075.61 1102 1103.04

Rho^2: 0.188236 0.20848 0.201485 0.203325 0.208314 0.20851

Adj rho^2: 0.185967 0.205833 0.199217 0.200678 0.206045 0.205485Final grad.: 0.014883 1.41786 1.60251 2.36007 0.914216 1.79354

Var-covar: hessian hessian hessian hessian hessian hessian

Coefficients Value t-test Value t-test Value t-test Value t-test Value t-test Value t-test

ASC_1 0.177 4.013 0.199 4.181 0.197 4.159 0.198 4.169 0.198 4.146 0.198 4.143

BETA_cng 2.193 5.169 1.914 26.686 2.897 318.386 2.600 31.998 2.414 32.491 2.283 56.227

BETA_cost -0.981 -9.011 -0.704 -5.975 -0.677 -4.645 -0.701 -4.968 0.595 31.026 0.448 10.398

BETA_time -1.629 -13.046 -1.166 -11.027 -0.920 -148.379 -0.732 -24.852 -1.274 -11.611 -1.225 -10.986SIGMA_cost -1.258 -13.006 -2.102 -12.865 2.513 11.055 2.482 10.199 1.205 62.323 1.284 26.106

SIGMA_time -0.884 -9.750 -0.846 -42.762 -1.250 -139.866 -1.169 -249.803 -1.769 -10.306 -1.721 -9.038

rho_time_co

st -1.302 -8.125 -0.183 -26.847 0.130 0.717

BETA_0 0.109 2.366

BETA_1

At zerocongestion

E(VTTS) $/hr 16.99 12.90 8.60 9.60 11.28 14.66

std. dev. 54.61 25.99 17.25 17.33 21.56 33.30

95% fractile 64.67 44.50 31.03 33.62 39.93 53.22

std. dev. of

mean 3.16 1.52 0.99 1.05 1.23 1.95


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TABLE 2c 13 14 15 16 17 18

Model: ML - Sb ML - Sb ML - mixlog ML - mixlog ML - mixlog ML - mixlog

(vtts sp.) (vtts sp. w. corr ) (corr ) (vtts sp. ) (vtts sp.w. corr )

N. Of draws 500 500 500 500 500 500

N. of par.: 6 7 6 7 6 7

N. of obs.: 3816 3816 3816 3816 3816 3816

N.of ind.: 477 477 477 477 477 477

Null LL: -2645.05 -2645.05 -2645.05 -2645.05 -2645.05 -2645.05

Final LL: -2111.61 -2107.83 -2157.56 -2115.66 -2158.87 -2156.37

Ll ratio test: 1066.88 1074.43 974.984 1058.78 972.361 977.363

Rho^2: 0.201675 0.203102 0.184304 0.200143 0.183808 0.184753Adj rho^2: 0.199406 0.200456 0.182035 0.197497 0.181539 0.182107

Final grad.: 3.66402 6.25139 0.0177412 0.933118 0.266678 5.37607

Var-covar: hessian hessian hessian hessian hessian hessian


ASC_1 0.198 4.171 0.193 4.067 0.177 4.051 0.196 4.239 0.185 4.024 0.185 4.019

BETA_cng 3.623 89.761 3.585 16.522 0.600 5.075 0.583 4.142 2.590 14.784 2.598 7.110

BETA_cost -0.640 -4.128 -0.687 -4.979 -1.033 -9.712 -0.901 -8.567 -0.959 -8.282 -0.953 -6.952

BETA_time -2.923 -1760.553 -2.658 -10.941 0.254 9.638 0.433 11.486 0.580 24.515 0.680 20.692SIGMA_cost -2.477 -9.620 -2.430 -10.292 -1.248 -15.012 -1.837 -21.832 -1.767 -9.193 -1.757 -9.103

SIGMA_time -1.545 -333.655 -1.457 -14.502 0.195 8.706 0.171 7.197 -0.972 -20.126 -0.924 -8.151

rho_time_cost 0.152 2.191 -0.309 -11.509 0.108 8.017

BETA_0

BETA_1 7.000 7.000

congestion

E(VTTS) $/hr 7.26 8.53 15.54 -17.49 5.70 6.69

std. dev. 9.52 10.52 39.04 384.03 9.61 9.2595% fractile 28.24 31.25 62.93 39.05 21.56 21.90std. dev. of

mean 0.54 0.63 2.24 22.51 0.55 0.56


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TABLE 3a 19 20 21 22 23

Model: MNL ML - log ML - log ML - log ML - log

(w. corr.) (vtts sp.) (vtts sp. w. corr )

N. Of draws 500 500 500 500N. of est. Par.: 7 9 11 9 10

N. of obs.: 3816 3816 3816 3816 3816

N.of ind.: 3816 477 477 477 477

Null LL: -2645.05 -2645.05 -2645.05 -2645.05 -2645.05

Final LL: -2399.95 -2108.02 -2071.01 -2094.92 -2094.85

Ll ratio test: 490.201 1074.06 1148.08 1100.26 1100.4

Rho^2: 0.092664 0.203032 0.217025 0.207985 0.208012


Var-covar: hessian hessian hessian BHHH hessian


ASC_1 0.114 3.338 0.179 4.039 0.198 4.181 0.197 4.415 0.197 4.150

BETA_cng -0.298 -6.364 1.834 5.275 1.504 8.024 2.257 5.460 2.258 27.043

BETA_cost -0.131 -13.023 -1.134 -11.553 -0.846 -7.340 -0.667 -4.396 -0.663 -4.627

BETA_inc 0.290 2.578 0.263 1.972 0.327 3.024 0.647 5.591 0.647 27.165

BETA_nlinc 0.373 2.038 0.700 3.988 0.189 2.414 0.011 0.038 0.015 0.322BETA_time -0.084 -8.591 -1.609 -13.710 -1.048 -9.432 -0.736 -9.332 -0.727 -9.747

BETA_tt 0.586 10.018 0.723 9.490 0.482 18.149 0.378 5.562 0.377 49.049

SIGMA_cost -0.990 -14.389 -1.871 -10.767 2.507 10.670 2.508 9.856

SIGMA_time -0.976 -16.401 -0.765 -14.999 -1.196 -18.448 -1.195 -91.882

rho_time_cost -1.073 -6.918 -0.005 -0.156

BETA_0 -0.031 -2.186

BETA_1At zero cong,

log(inc), log(tt)

E(VTTS) $/hr 6.35 16.13 11.45 9.73 9.82

std 0.9 38.13 32.73 17.74 18.21

95% fractile 7.88 60.13 42.2 34.28 34.44std. dev. of

mean 0.048 2.25 1.91 1.03 1.09


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TABLE 3b 24 25 26 27 28

Model: ML - log ML - log ML - sb ML - sb ML - mixlog

(inv vtts sp.) (inv vtts sp. w. corr ) (vtts sp. ) (vtts sp.w corr )

N. Of draws 500 500 400 500 500N. of est. Par.: 9 11 10 10 9

N. of obs.: 3816 3816 3816 3816 3816

N.of ind.: 477 477 477 477 477

Null LL: -2645.05 -2645.05 -2645.05 -2645.05 -2645.05

Final LL: -2068.5 -2066.89 -2100.47 -2091.63 -2154.75

Ll ratio test: 1153.1 1156.31 1089.17 1106.84 980.596

Rho^2: 0.217973 0.218581 0.205888 0.209228 0.185364


Var-covar: hessian hessian hessian hessian hessian


ASC_1 0.199 4.172 0.198 4.174 0.199 4.190 0.200 4.196 0.177 4.046

BETA_cng 1.805 18.803 1.745 12.809 2.706 68.057 2.752 75.404 0.650 4.402

BETA_cost 0.375 27.846 0.276 4.286 -0.604 -4.152 -0.612 -4.251 -1.028 -9.587

BETA_inc 0.268 20.415 0.452 10.381 0.160 22.469 0.404 55.838 0.218 1.723

BETA_nlinc 0.342 17.711 0.556 7.042 0.068 0.575 0.070 6.446 -0.215 -1.336BETA_time -1.203 -10.751 -1.262 -11.065 -2.683 -152.580 -2.516 -69.494 0.251 8.669

BETA_tt 0.522 23.697 0.639 11.433 0.254 37.938 0.346 60.048 -0.102 -1.541

SIGMA_cost 1.078 41.411 1.206 15.391 -2.478 -10.236 -2.412 -11.035 -1.291 -13.943

SIGMA_time -1.649 -9.531 -1.638 -9.874 -1.422 -125.125 -1.349 -220.180 0.179 8.211

rho_time_cost -0.267 -1.923 0.082 6.077

BETA_0 0.156 2.381 0.023 5.109

BETA_1 6.930 7.000At zero cong,

log(inc), log(tt)

E(VTTS) $/hr 12.15 10.12 8.15 8.7 16.15

std 18.77 13.76 9.43 9.7 41.96

95% fractile 40.32 30.64 28.78 29.66 64.79

std. dev. of 1.07 0.84 0.57 0.59 2.42


23/25

TABLE 3c 29 30 31

Model: ML - mixlog ML - mixlog ML - mixlogcorr. ) vtts sp) vtts sp corr. )

N. Of draws (MLHS) 500 500 500N. of est. Par.: 10 9 10

N. of obs.: 3816 3816 3816

N.of ind.: 477 477 477

Null LL: -2645.05 -2645.05 -2645.05

Final LL: -2110.18 -2138.4 -2138.17

Ll ratio test: 1069.74 1013.29 1013.76

Rho^2: 0.202215 0.191546 0.191633Adj rho^2: 0.198435 0.188143 0.187852

Final grad.: 1.62239 1.02916 12.1373

Var-covar: hessian hessian hessian

Coefficients Value t-test Value t-test Value t-test

ASC_1 0.196 4.231 0.186 4.056 0.187 4.050

BETA_cng 0.531 4.013 1.967 28.950 1.967 22.749

BETA_cost -0.918 -8.932 -0.931 -7.653 -0.931 -7.706

BETA_inc 0.243 4.001 0.241 2.488 0.238 2.938

BETA_nlinc -0.050 -0.526 0.107 2.493 0.109 0.998

BETA_time 0.447 11.251 0.693 18.100 0.712 12.102

BETA_tt 0.013 0.338 0.426 6.962 0.427 11.476

SIGMA_cost -1.881 -18.716 -1.724 -7.616 -1.725 -8.128

SIGMA_time 0.163 6.806 -0.887 -27.011 -0.890 -63.159

rho_time_cost -0.343 -8.218 0.010 0.466BETA_0

_At zero congestion,

log(inc), log(tt)

E(VTTS) $/hr -28.98 6.81 7

std 544.3 8.77 8.8

95% fractile 37.06 21.28 21.51

std. dev. of mean 31.79 0.5 0.53


24/25

TABLE 4 32 33 34 35 36 37

Model: MNL log logvtts loginvvtts mixlog mixlogvtts

N. Of draws 500 500 500 500 500

N. of est. Par.: 7 9 9 9 9 9

N. of obs.: 3816 3816 3816 3816 3816 3816

N.of ind.: 3816 477 477 477 477 477

Null LL: -2645.05 -2645.05 -2645.05 -2645.05 -2645.05 -2645.05

Final LL: -2458.89 -2144.07 -2087.55 -2076.85 -2124.24 -2135.14

Ll ratio test: 372.315 1001.95 1115 1136.39 1041.61 1019.82

Rho^2: 0.0703795 0.189401 0.210771 0.214815 0.196898 0.192779Adj rho^2: 0.0677331 0.185998 0.207368 0.211412 0.193496 0.189376

Final grad.: 0.0161071 0.0594964 1.82554 0.28903 0.011302 2.22981

Var-covar: hessian hessian BHHH hessian hessian BHHH


ASC_1 0.107 3.197 0.177 3.989 0.190 4.270 0.194 4.106 0.180 4.100 0.181 4.166

BETA_cng -0.277 -5.631 2.157 3.814 2.521 5.082 2.375 7.941 0.484 3.744 2.481 6.702

BETA_cost -0.109 -9.684 -0.985 -9.133 -0.904 -6.689 0.573 6.734 -1.134 -11.169 -1.126 -10.749

BETA_inc 0.346 2.405 0.356 2.187 0.161 0.470 0.394 1.478 -0.345 -1.812 0.178 0.663

BETA_nlinc 0.133 0.863 -0.296 -0.485 -1.780 -3.465 -1.402 -2.951 -0.672 -2.607 -1.186 -3.133

BETA_time -0.069 -6.826 -1.603 -12.457 -0.935 -10.550 -1.371 -11.109 0.276 9.065 0.575 8.965

BETA_tt 0.356 4.903 -0.077 -0.849 -1.062 -6.487 -0.646 -5.153 -0.642 -6.964 -0.818 -6.543

SIGMA_cost -1.287 -12.641 1.918 10.047 1.180 21.572 -1.068 -13.806 -1.345 -9.288

SIGMA_time -0.854 -10.135 -1.232 -15.831 -1.420 -7.372 0.196 6.884 -0.932 -19.478

At zero cong,

log(inc), log(tt)

E(VTTS) $/hr 6.30 16.13 9.73 12.15 16.15 6.81

std 1.15 38.13 17.74 18.77 41.96 8.77

95% fractile 8.29 60.13 34.28 40.32 64.79 21.28

std. dev. of mean 0.06 2.25 1.03 1.07 2.42 0.50


25/25

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