Accounting for stochastic variables in discrete choice models

Transportation Research Part B 78 (2015) 222–237

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Transportation Research Part B journal homepage: www.elsevier.com/locate/trb

Accounting for stochastic variables in discrete choice models Federico Díaz a, Víctor Cantillo b, Julian Arellana b,⇑, Juan de Dios Ortúzar c a

Transmetro S.A.S., Calle 45 Carrera 46, Barranquilla, Colombia Department of Civil and Environmental Engineering, Universidad del Norte, Km 5 Antigua Vía a Puerto Colombia, Barranquilla, Colombia Department of Transport Engineering and Logistics, Centre for Urban Sustainable Development (CEDEUS), Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Santiago, Chile b c

a r t i c l e

i n f o

Article history: Received 17 July 2013 Received in revised form 29 April 2015 Accepted 30 April 2015 Available online 22 May 2015 Keywords: Stochastic variables Errors in variables Discrete choice models Mixed logit

a b s t r a c t The estimation of discrete choice models requires measuring the attributes describing the alternatives within each individual’s choice set. Even though some attributes are intrinsically stochastic (e.g. travel times) or are subject to non-negligible measurement errors (e.g. waiting times), they are usually assumed fixed and deterministic. Indeed, even an accurate measurement can be biased as it might differ from the original (experienced) value perceived by the individual. Experimental evidence suggests that discrepancies between the values measured by the modeller and experienced by the individuals can lead to incorrect parameter estimates. On the other hand, there is an important trade-off between data quality and collection costs. This paper explores the inclusion of stochastic variables in discrete choice models through an econometric analysis that allows identifying the most suitable specifications. Various model specifications were experimentally tested using synthetic data; comparisons included tests for unbiased parameter estimation and computation of marginal rates of substitution. Model specifications were also tested using a real case databank featuring two travel time measurements, associated with different levels of accuracy. Results show that in most cases an error components model can effectively deal with stochastic variables. A random coefficients model can only effectively deal with stochastic variables when their randomness is directly proportional to the value of the attribute. Another interesting result is the presence of confounding effects that are very difficult, if not impossible, to isolate when more flexible models are used to capture stochastic variations. Due the presence of confounding effects when estimating flexible models, the estimated parameters should be carefully analysed to avoid misinterpretations. Also, as in previous misspecification tests reported in the literature, the Multinomial Logit model proves to be quite robust for estimating marginal rates of substitution, especially when models are estimated with large samples. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The estimation of discrete choice models requires data such as socioeconomic characteristics of individuals and attributes of the alternatives within their choice sets. These explanatory variables are usually assumed to be inherently deterministic, ⇑ Corresponding author. E-mail addresses: [email protected] (F. Díaz), [email protected] (V. Cantillo), [email protected] (J. Arellana), [email protected] (J.de Dios Ortúzar). http://dx.doi.org/10.1016/j.trb.2015.04.013 0191-2615/Ó 2015 Elsevier Ltd. All rights reserved.

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that is, that they would yield the same values if measured repeatedly. The problem is that some variables are actually intrinsically stochastic (e.g. travel times under congested conditions1) and thus assuming them to be fixed can be fairly heroic. In fact even an accurate measure can be biased if it is different from the value perceived by the decision maker. Furthermore, variables which are intrinsically non-stochastic can still be measured inaccurately producing measurement errors. These errors induce a particular kind of randomness from the modeller’s point of view. For instance, in strategic planning applications it is common practice to use zone-based network models to obtain level of service attributes, such as travel times, instead of measuring this key attribute at an individual level due to the high data collection costs involved. Also, trips with different levels-of-service are usually temporally and spatially aggregated (e.g. the set of trips between two specific zones at a peak hour) and a single level-of-service value (e.g. an ‘‘average’’ value) is assigned to them, which is evidently different from the true values experienced by the users (Train, 1978). Measurement errors also occur when values are directly provided by the individual in a revealed preference (RP) survey (e.g. waiting time to board a bus, income, or preferred departure time). In the latter case the difference between the reported value and the real one can be significant due to cognitive issues or even policy bias (Daly and Ortúzar, 1990). When a discrepancy between the ‘‘true’’ value and the value measured by the modeller exists, an estimation bias arises as discussed by Ortúzar and Willumsen (2011, Section 9.2). Let us consider a simple Multinomial Logit (MNL) model with a typical utility function U = bx + e, where b are parameters to be estimated, x are measured attributes and e is an independent and identically distributed Gumbel error term with mean zero and standard deviation re. Assume there is a difference between the attribute values as perceived by the modeller (x⁄) and the true values (x), such that: x = x⁄ + g, where g distributes with mean zero (i.e. no systematic bias exist) and standard deviation rg. In this case the utility function is transformed to: U = b (x⁄ + g) + e, that is: U = b x⁄ + (e + bg) = bx⁄ + d. The outcome of this is that in the original model, the estimated parameter b0 would be:

p

b0 ¼ pffiffiffi b 6  re

ð1Þ

whilst in the second model the estimated parameter b00 would be2:

p

b00 ¼ pffiffiffi b 6  rd

ð2Þ

and the standard deviation of the distribution function of the new error component d would be:

rd ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2e þ b2  r2g

ð3Þ

Hence b00 < b0 and this estimation bias may affect the model forecasts. There is also experimental evidence about bias estimation and miscalculation of marginal rates of substitution when measurement errors occur. For instance, Train (1978) explores the use of more accurate data in the estimation of mode choice models concluding that it is sometimes advisable to carry out an additional effort to correct for the measurement bias of some attributes, such as transit transfer time, when analysing transport policies. Ortúzar and Ivelic (1987) showed that using very precise real data, measured at the individual level, resulted in better fit and clearly different subjective values of time when estimating mode choice models in comparison with models estimated with aggregate data. More recently, Bhatta and Larsen (2011) show, using synthetic data, how measurement biases may induce biased parameter estimates on a MNL model, besides miscalculation of marginal rates of substitution. Therefore the use of more accurate (but more expensive) data results in better parameter estimates and this clearly establishes a trade-off between data quality and data collection costs (Daly and Ortúzar, 1990). In this paper we deal with the problem of working with incorrigibly biased data due to the stochastic nature (inherent or not) of some variables. After a brief review of relevant literature in Section 2, we carry out an econometric analysis to identify appropriate specifications to account for stochastic variables in discrete choice models (Section 3). Then, in Section 4 the performance of some specifications arising from the econometric analysis will be tested and compared in terms of parameter estimate bias, computation of marginal rates of substitution and forecasting ability using both, synthetic and real datasets. Finally, Section 5presents our main conclusions. 2. The problem of errors in variables (EIV) Much of the effort to specify stochastic variables when estimating econometric models has arisen from the need to solve the EIV problem. In this sense, although there is a vast literature in the case of regression models, research underlying EIV within discrete choice models is scarce, but has shown lately some significant progress. For instance, in the fields of biology and medicine, the EIV problem has been explored in the case of binary models, proposing adaptations of maximum likelihood estimators for specific circumstances (Carroll et al., 1984; Stefansky and Carroll, 1985, 1987, 1990; Kao and Schnell, 1 Related problems arising from the inherent variability of some level-of-service attributes such as travel time are reliability and risk aversion (Jackson and Jucker, 1982). In this research we will only address the difference between the true value and the values measured by the modeller as a result of this variability. 2 Let us assume that d = e + bg also follow an IID Gumbel distribution, just for illustrative purposes.

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1987). More generally, Steinmetz and Brownstone (2005) presented a model that considers EIV using multiple imputations that can be used when there is accurate information for a subsample of observations. More recently, Yamamoto and Komori (2010) estimated a latent class model for handling errors when measuring access distances to public transport. Walker et al. (2010) proposed a hybrid choice model, which includes a latent variable to account for travel time measurement errors given that they were obtained from a network zone based model. In their specification, the true travel time is treated as a latent variable observed through the modeller measured travel time (i.e. the modeller measured travel time is taken as an indicator of the true travel time). Although the theoretical specification of the model is consistent, it is not easy to justify this formulation in practice, especially when high dispersion exists in travel time values, which is the most frequent case. Further, indicators are not available for forecasting. Other examples of hybrid choice models which can be used to deal with the EIV problem can be found in Bolduc and Alvarez-Daziano (2009) and Brey and Walker (2011), where latent variables were used to account for measurement errors in variables such as income in a vehicle choice experiment, or preferred departure time in an airline itinerary choice context. In both cases, the structural equations of the latent variable component are a function of individual characteristics; this has the added benefit of being favourable for forecasting scenarios because the structural equations are used to predict the latent variable’s values. Nevertheless, the specification of a structural equation associated with individual characteristics in the case of an exogenous variable such as travel time is less natural. Wansbeek and Meijer (2000) proposed the use of latent variables for treating EIV simply by adding directly the measurement equations to the utility function, without necessarily specifying an additional structural function. The econometric analysis developed in this paper builds upon the above idea, keeping in mind variables such as travel time. 3. Econometric analysis 3.1. The EIV problem Let us consider the following additive and linear-in-parameters specification for the utility function in a discrete choice context based on random utility theory:

U in ¼

K X bink xink þ ein

ð4Þ

k¼1

where Uin represents the utility of alternative Ai perceived by individual n, xink refers to the value of the kth explanatory attribute of alternative Ai for individual n, bink is an unknown parameter to be estimated (referring to alternative Ai, for individual n and the kth explanatory attribute). ein is an error term that distributes independently and identically (IID) Gumbel. Individual n chooses alternative Ai if and only if the utility of that alternative is the maximum among the utilities of alternatives Aj within her choice set A (i.e. Uin > Ujn for all Aj – Ai). This formulation corresponds to the classical MNL model (Ortúzar and Willumsen, 2011, Chapter 7). The stochastic nature of the kth explanatory variable can be expressed as following:

xink ¼ xink þ gink

ð5Þ

where xink is the mean measured value of the variable (i.e. the value that the modeller would typically use) and gink is the discrepancy between this mean measured value and the true value, perceived by the individual. Eq. (5) can be seen as a measurement equation in the context of a hybrid choice model that includes latent variables in the discrete choice component (Bolduc et al., 2008). The variation or discrepancy gink is a stochastic component that follows a certain distribution. Replacing (5) in (4) we get:

U in ¼

K X bink ðxink þ gink Þ þ ein

ð6Þ

k¼1

Eq. (6) is equivalent to a Mixed Logit (ML) formulation because the random error component is a mix of a Gumbel distribution and some other distribution contained in gink (McFadden and Train, 2000). The above suggests that the EIV problem can be approximated through a particular specification of the ML model, and depends on the definition of gink. Note that the use of MNL models when stochastic variables are present would be a wrong approach because the IID Gumbel error term of the model cannot represent the full error structure given by the data. Below we will discuss two particular ML formulations for dealing with the EIV problem. 3.2. Stochastic variables model The stochastic variations gink can be specified as follows:

gink ¼ rik  uink

ð7Þ

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where uink, the components of a vector u, are independently distributed variables with zero mean and unitary standard deviation; rik is a fixed real number representing the standard deviation of the probability function related with the stochastic variation; this value is alternative specific but constant among individuals. Replacing (7) in (6), and assuming generic tastes in the population, the Stochastic Variables (SV) model can be written as:

U in ¼

K X bik ðxink þ rik  uink Þ þ ein

ð8Þ

k¼1

and the probability Pn that individual n chooses alternative Ai for given values udink can be computed as a class of logit model:

 Pn ðAi b; r; ud Þ ¼

PK d ej k¼1 bik ðxnik þrik unik Þ P K P j k¼1 bjk ðxnjk þrjk udnjk Þ Aj 2C ðnÞ e

ð9Þ

where b and r are vectors with elements bik and rik, j is the scale factor (which is usually normalized by setting it to 1), and C(n) the individual’s choice set. The final choice probability can be obtained by integrating (9) over the range of u values. For example, if u is assumed to distribute standard Normal the probability that individual n chooses alternative Ai can be expressed as follows:

Pin ¼

Z

Pn ðAi jb; r; uÞ  /ðuÞdu

ð10Þ

u

which has a typical Mixed Logit (ML) form and thus can be computed using simulated maximum log-likelihood techniques (Train, 2009) as in (11):

PD Pin ¼

d¼1 P n ðAi

 b; r; ud Þ

D

ð11Þ

for a set of D draws from the distribution of u. It can be shown that in this case, due to identification issues, it is only possible to estimate I  1 covariance matrix parameters (see Appendix A) for I alternatives. As a consequence, the modeller has to choose a specification where all the correspondent covariance matrix parameters must be normalized; this normalization is not simple as it must guarantee that these parameters are fixed at a sufficiently large value in relation to r. In the most general case, these values are unknown, and a trial and error procedure needs to be undertaken. Note that Eq. (8) can be arranged and set also as follows:

U in ¼

K X bik  xink þ min þ ein

ð12Þ

k¼1

where

min ¼

K X bik  rik  uink

ð13Þ

k¼1

Eq. (12) has an error components ML (ECML) structure, frequently used to account for heteroskedasticity among alternatives (Ortúzar and Willumsen, 2011, Section 7.6). Under some circumstances and specially when treating the EIV problem, the SV and ECML structures are mathematically equivalent (i.e. in practice, this equivalence may be interpreted as meaning that the stochastic variables cause heteroskedasticity among alternatives). Furthermore, the effect of the stochastic variables can also be confounded with other sources of heteroskedasticity or heterogeneity in the estimation process. Due to the equivalence between the SV and ECML structures, instead of estimating the rik in the SV model we can estimate an alternative specific parameter Ui for each alternative in the ECML. These parameters correspond to the standard deviations of the error term components:

min ¼ Ui  uin

ð14Þ

where ui distributes Normal with zero mean and unitary standard deviation. The Ui parameters can be expressed as a combination of the attribute variations per alternative:

U2i ¼

K X b2ik  r2ik

ð15Þ

k¼1

Similarly to the SV model, in the ECML structure only I  1 error component variances can be estimated and one of those needs to be normalized. The usual normalization methodology involves estimating a non-identifiable model (i.e. with I variances), and later to estimate a new model but fixing to zero the lowest variance in the preliminary estimation (Walker, 2001). As this normalization is easier in the ECML model than in the SV model, it would appear that estimating the former structure is preferable.

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Moreover, the equivalence between the SV and ECML structures holds when tastes vary across the population. Eq. (8) can be rewritten to account for variations in tastes:

U in ¼

K X ðbik þ sik  wink Þðxink þ rik  uink Þ þ ein

ð16Þ

k¼1

where wink distributes with zero mean and unitary standard deviation and sik is a fixed real number representing the standard deviation of the probability function related with the stochastic variation of parameter bink. In this case (16) can be arranged as follows:

U in ¼

K X bik  xink þ min þ ein

ð17Þ

k¼1

where

min ¼

K X

ðxink  sik  wink þ bik  rik  uink þ sik  rik  wink  uink Þ

ð18Þ

k¼1

The distribution of min is fairly complex depending on the structure of wink and uink ; so, in this case, especial considerations must be taken into account while specifying and estimating an ECML model. On the other hand note that, as a result, the effect of the stochastic variables might also be confounded with variations across tastes in the population. 3.3. Random coefficients model Stochastic variations in the SV model have been assumed to be independent of the corresponding attribute values. Furthermore, the standard deviations rik are equal for all individuals implying homoskedasticity across observations. However, under some circumstances it is reasonable to expect that the higher an attribute value the larger should be its level of randomness. If such is the case, a proportional direct relationship may be specified:

gink ¼ kink  xink

ð19Þ

where kink follows a probability distribution function (pdf) with zero mean and unknown standard deviation hik:

kink ¼ hik  uink

ð20Þ

In this case then, heteroskedasticity across both respondents and alternatives would be found in the model. Replacing (19) in (6), and assuming taste homogeneity, we can write: K X

aink  xink þ ein

ð21Þ

aink ¼ bik ð1 þ hik  uink Þ

ð22Þ

U in ¼

k¼1

where

Even though the taste variations across respondents are fixed, Eq. (21) represents a random coefficients (RC) model. Due to formulation equivalence, it is possible that RC model estimates could be confounded with (apparent) random taste heterogeneity in the population if stochastic variables are included in the formulation, and the randomness is directly proportional to the variable size. However, confounded effects in ML estimates are possible (see Cherchi and Ortúzar, 2008; Swait and Bernardino, 2000), when the stochastic variables in the model are interpreted as taste heterogeneity, even when stochastic variations are independent from the size of the variables as shown in Section 3.2 4. Experimental analysis We adopted the classic procedure of Williams and Ortúzar (1982) to generate synthetic datasets from known parameters. Then we estimated different discrete choice models using this data. Specifically, we estimated MNL, ECML and RC models in several scenarios of randomness on taste parameters and explanatory variables. In order to evaluate their performance, WTP (willingness to pay) measures were computed and compared with the real values. We also tested the proposed model specifications using a real databank where travel times measured with different levels of accuracy were available for each individual observation. 4.1. Synthetic population generation A collection of datasets was generated in which pseudo-observed individuals behaved according to a known choice rule, with a defined error structure, and had to choose among three options (Ai) labelled: Taxi (i = 1), Bus (i = 2) and Metro (i = 3).

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Attributes included in the choice set description were: cost (x1), travel time (x2) and access time (x3). The simulated (i.e. pseudo observed) choices for each individual j, represented the alternative with higher associated utility Uij, which was computed as:

U ij ¼ bij1  xij1 þ bij2  xij2 þ bij3  xij3 þ eij

ð23Þ

where b1, b2 and b3 are cost, travel time and access time parameters respectively. The random error terms, eij, were generated from an iid standard Gumbel pdf. In turn, attribute and taste parameter randomness was included as follows:

xink ¼ xink þ gink

ð24Þ

bijk ¼ bik þ dijk

ð25Þ

where the first terms represent base values (i.e. modeller measured values in the case of attributes), and the second ones correspond to variation terms. The base values of the attributes were, in turn, generated from a truncated Normal distributed pdf to avoid negative attribute values. Table 1 shows the base attribute values and base taste parameters used in the synthetic sample generation process. Three main scenarios were considered: 1. Independently distributed stochastic variations in the Bus and Metro travel times with fixed taste parameters 2. Independently distributed stochastic variations in the Bus and Metro travel times, and the travel time taste parameter allowed to vary across the population with mean 0.12 and standard deviation 0.03 3. Correlation between the stochastic variations of the Bus and Metro travel times with fixed taste parameters; correlation was induced by adding a common error term / which distributes Normal with zero mean and unitary standard deviation. For each main scenario, synthetic samples with 500 and 5000 observations were generated for each of the two levels of randomness in the attributes. Two out of the nine attributes were of a stochastic nature in the first level of randomness (I) and only one was stochastic in the second level (II). Stochastic variations were obtained from Normal and Log-Normal pdf with zero mean and a given standard deviation. The standard deviations used are shown in Table 2. 4.2. Specification of the estimated models The following discrete choice model structures were estimated using the synthetic databank generated by the procedure described in the previous section: MNL model

U i ¼ b1  xi1 þ b2  xi2 þ b3  xi3 þ ei

ð26Þ

RC (Random coefficients model with generic coefficients)

U i ¼ ðb1 þ r1  ui1 Þ  xi1 þ ðb2 þ r2  ui2 Þ  xi2 þ ðb3 þ r3  ui3 Þ  xi3 þ ei

ð27Þ

ECML (Error Components Mixed Logit model)

U i ¼ b1  xi1 þ b2  xi2 þ b3  xi3 þ ni  ui þ ei

ð28Þ

All random terms u in the utility functions above were specified as standard Normal, whilst the error terms e were assumed to distribute iid Gumbel. For the ECML model, all the error component terms ni were estimated, but due to identification issues the Metro error components were fixed at zero in all cases. 4.3. Parameter estimation Estimation results for the levels of randomness I and II are given in Tables 3 and 4. The following null hypothesis was statistically tested through the estimation analysis: Hypothesis I. bestimated = 0, in order to evaluate parameter significance; the corresponding t ratios are shown in parentheses within the tables.

The log-likelihood (LL) values are shown as model fit measures and the likelihood ratio (LR) test was used for model comparisons (Ortúzar and Willumsen, 2011, page 279), considering the MNL as the restricted model; the computed LR value was contrasted with the critical v2 value at the 5% significance level and degrees of freedom defined by the difference in the number of parameters between both models (critical v2 values are shown with an asterisk within the tables). Parameter values with two asterisks were not estimated (i.e. they were fixed a priori).

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Table 1 Attribute base values and parameters used in synthetic sample generation. Attribute

Taxi

Bus

Metro

Parameter

Cost Mean Standard deviation Minimum value

50 5 25

20 4 15

22 3 12

0.08

Travel time Mean Standard deviation Minimum value

15 4 7

30 10 10

16 3 4

0.12

10 3 2

18 4 2

0.16

Access time Mean Standard deviation Minimum value

5 2 0.5

Table 2 Standard deviations for attribute stochastic variations. Attribute

a

Mode

Log-Normala

Normal I

II

I

II

Cost

Taxi Bus Metro

– – –

– – –

– – –

– – –

Travel time

Taxi Bus Metro

– 9 3.2

– 9 –

– 5 2

– 5 –

Access time

Taxi Bus Metro

– – –

– – –

– – –

– – –

Log-Normal distributions were generated by converting and scaling Normal distributions with zero mean and standard deviation of 0.5.

For higher sample sizes (5000) and Normal distributed variations, the ECML estimation results show a better fit than the MNL specification. This is not the case for the RC model except in scenario 3. However, we found significant taste differences for the travel time attribute, suggesting that confounding effects could be an issue when attribute variability is considered in the estimation of random coefficients models (the travel time parameter absorbs part of that randomness). For lower sample sizes (500) both ML specifications performed poorly, suggesting that more flexible models may encounter problems when estimated with low sample sizes (Cherchi and Ortúzar, 2008). In general, the RC and ECML models failed to recover any randomness collapsing to the MNL model. The population taste parameters were found to be non-significant due to their high standard errors in estimation. Furthermore, most of the covariance matrix error component parameters were found to have low significance. In scenarios 1 and 3 (level I), in the case of Log-Normal distributed variations, the ECML model failed to recover significant taste parameters except for access time. Finally, for Log-Normal distributed variations, the RC model did not recover any randomness in the cases where taste variations do not exist across the population even for large datasets; this suggests that confounding issues are largely dependent on error structure. 4.4. Willingness to pay computations Willingness to pay (WTP) values for travel time (WTPTT) and access time (WTPAT) are shown in Tables 5 and 6 for levels of randomness I and II respectively. The ratio between the estimated and true WTP values is defined by r = WTPestimated/WTPtrue. The estimated ratios are presented in square brackets. In the case of the RC models, the reported values represent the mean of the parameter distribution. Additionally, the standard errors of the WTP estimates were computed using the Delta method (Bliemer and Rose, 2013) and used to evaluate the following hypothesis: Hypothesis II. WTPestimated  WTPtrue) = 0; the corresponding t ratios are shown in round brackets. The critical value for comparison is 1.96 for a 5% level. The WTP estimations with all model structures perform fairly well. There is no strong difference in terms of WTP recovery capacity when using advanced or simpler models with large sample sizes, except for complex scenarios (i.e. when correlations and taste variation exist).

Table 3 Parameter estimates for level of randomness I. Parameter

Scenario 1

Scenario 2

500 Observations

Normal Cost (0.08) Travel time (0.12) Access time (0.16)

500 Observations

5000 Observations

MNL

RC

ECML

MNL

RC

ECML

MNL

RC

ECML

MNL

RC

ECML

MNL

RC

ECML

0.06 (9.24) 0.09 (9.83) 0.14 (9.36)

0.06 (8.83) 0.10 (6.92) 0.14 (9) 0.00 (0.00) 0.03 (0.86) 0.00 (0.01)

0.10 (2.00) 0.17 (2.03) 0.20 (2.31)

0.07 (30.93) 0.10 (32.00) 0.14 (29.41)

0.07 (29.04) 0.10 (22.09) 0.14 (28.02) 0.00 (0.00) 0.03 (3.00) 0.00 (0.17)

0.07 (22.65) 0.11 (16.64) 0.15 (23.08)

0.06 (9.78) 0.09 (9.23) 0.13 (9.21)

0.06 (8.83) 0.10 (7.16) 0.13 (8.89) 0.00 (0.00) 0.02 (0.66) 0.00 (0.02)

0.09 (2.21) 0.16 (2.23) 0.19 (2.58)

0.07 (30.66) 0.09 (31.04) 0.13 (29.05)

0.07 (28.44) 0.10 (21) 0.14 (27.31) 0.00 (0.00) 0.03 (3.21) 0.01 (0.24)

0.07 (24.17) 0.11 (16.84) 0.15 (23.7)

0.06 (9.32) 0.10 (10.04) 0.13 (9.18)

0.07 (8.88) 0.11 (6.67) 0.14 (8.81) 0.00 (0.00) 0.04 (1.26) 0.00 (0.01)

0.08 (1.99) 0.15 (2.11) 0.17 (2.36)

0.07 (30.83) 0.09 (31.1) 0.13 (29.02)

0.07 (28.93) 0.10 (21.3) 0.14 (27.71) 0.00 (0.00) 0.04 (5.08) 0.00 (0.05)

0.08 (12.77) 0.11 (11.66) 0.15 (15.28)

n1: Error component

1.58 (0.93) 2.44 (1.48) 0.00⁄⁄

n2: Error component n3: Error component Log-likelihood 478 Log-likelihood ratio test 2 Critical v at 5% 0.07 (10.1) 0.11 (10.8) 0.16 (10.38)

rcost: RCg rtime: RCg raccess: RCg

0.01 (0.03) 1.13 (5.07) 0.00⁄⁄

0.31 (0.54) 1.33 (4.84) 0.00⁄⁄

0.97 (0.50) 2.20 (1.5) 0.00⁄⁄

475 4.85 5.99

4731

4729 3.07 7.81

4727 8.42 5.99

479

479 0.20 7.81

477 4.28 5.99

4773

4771 3.58 7.81

4768 10.23 5.99

476

476 0.61 7.81

474 4.81 5.99

4770

4765 10.13 7.81

4762 15.54 5.99

0.07 (9.61) 0.11 (8.18) 0.16 (9.84) 0.00 (0.00) 0.03 (0.91) 0.01 (0.13)

0.08 (8.27) 0.12 (6.34) 0.17 (8.66)

0.08 (32.83) 0.11 (34.64) 0.15 (31.14)

0.08 (30.12) 0.11 (23.45) 0.15 (28.57) 0.00 (0.00) 0.02 (1.54) 0.01 (0.28)

0.08 (14.17) 0.12 (13.11) 0.16 (16.95)

0.07 (9.87) 0.10 (10.24) 0.15 (9.78)

0.07 (9.22) 0.12 (7.14) 0.15 (9.15) 0.00 (0.00) 0.05 (2.26) 0.00 (0.12)

0.08 (7.47) 0.13 (5.58) 0.16 (7.32)

0.07 (32.69) 0.11 (33.92) 0.15 (31.05)

0.08 (30.17) 0.12 (22.93) 0.15 (28.9) 0.00 (0.00) 0.04 (4.05) 0.01 (0.28)

0.08 (12.01) 0.13 (11.3) 0.17 (14.41)

0.07 (10.2) 0.11 (10.77) 0.16 (10.42)

0.07 (9.72) 0.11 (8.15) 0.16 (9.98) 0.00 (0.00) 0.02 (0.84) 0.00 (0.04)

0.08 (8.36) 0.121 (6.35) 0.17 (8.69)

0.08 (34.63) 0.11 (34.63) 0.15 (31.05)

0.08 (30.74) 0.11 (23.81) 0.15 (29.34) 0.00 (0.00) 0.02 (1.14) 0.00 (0.18)

0.08 (12.89) 0.12 (12.16) 0.16 (15.56)

0.00⁄⁄

n2: Error component

0.99 (1.58) 0.01 (0.01)

n3: Error component 457 0.31 7.81⁄

457 3.71 5.99⁄

0.00⁄⁄

0.41 (0.97) 0.77 (2.60) 0.00⁄⁄ 4575

4574 0.73 7.81⁄

4573 2.67 5.99⁄

1.47 (2.19) 0.16 (0.19) 470

469 0.00 7.81⁄

469 0.00 5.99⁄

0.00⁄⁄

0.60 (1.59) 1.09 (3.73) 0.00⁄⁄ 4611

4608 5.58 7.81⁄

4607 7.72 5.99⁄

0.52 (1.33) 0.84 (2.81) 0.00⁄⁄

0.90 (1.39) 0.00 (0.00) 457

457 0.23 7.81⁄

455 4.17 5.99⁄

4576

4576 0.42 7.81⁄

4575 3.30 5.99⁄ 229

Values in Bold represent estimates.

1.50 (0.99) 2.27 (1.54) 0.00⁄⁄

478 0.29 7.81

n1: Error component

Log-likelihood 457 Log-likelihood ratio test Critical v2 at 5%

0.11 (0.29) 1.05 (4.62) 0.00⁄⁄

F. Díaz et al. / Transportation Research Part B 78 (2015) 222–237

ECML

raccess: RCg

Access time (0.16)

5000 Observations

RC

rtime: RCg

Travel time (0.12)

Scenario 3

500 Observations

MNL

rcost: RCg

Log-Normal Cost (0.08)

5000 Observations

Parameter

230

Table 4 Parameter estimates for level of randomness II. Scenario 1

Scenario 2

500 Observations

Normal Cost (0.08) Travel time (0.12) Access time (0.16)

5000 Observations

5000 Observations

MNL

RC

ECML

MNL

RC

ECML

MNL

RC

ECML

MNL

RC

ECML

MNL

RC

ECML

0.06 (9) 0.09 (9.79) 0.13 (8.82)

0.06 (8.61) 0.11 (6.83) 0.13 (8.45) 0.00 (0.00) 0.05 (1.98) 0.00 (0.04)

0.07 (7.35) 0.13 (5.69) 0.14 (7.22)

0.07 (31) 0.10 (32.2) 0.14 (29.44)

0.07 (29.15) 0.11 (22.13) 0.11 (22.13) 0.00 (0.00) 0.04 (4.29) 0.00 (0.08)

0.07 (22) 0.12 (16.39) 0.15 (22.55)

0.06 (8.89) 0.09 (9.53) 0.13 (0.0144)

0.07 (8.51) 0.11 (6.79) 0.14 (8.65) 0.00 (0.00) 0.07 (2.87) 0.00 (0.12)

0.08 (3.22) 0.15 (2.77) 0.17 (2.74)

0.07 (30.68) 0.09 (31.21) 0.14 (29.24)

0.07 (28.66) 0.10 (21.28) 0.14 (27.69) 0.00 (0.00) 0.04 (5.45) 0.00 (0.16)

0.08 (22.52) 0.12 (16.6) 0.15 (22.86)

0.06 (9.27) 0.10 (9.97) 0.14 (9.35)

0.06 (8.79) 0.10 (6.66) 0.14 (8.91) 0.00 (0.00) 0.04 (1.27) 0.00 (0.02)

0.09 (2.2) 0.16 (2.21) 0.19 (2.56)

0.07 (31.18) 0.09 (31.35) 0.14 (29.64)

0.07 (29.14) 0.11 (21.34) 0.14 (28.17) 0.00 (0.00) 0.05 (5.73) 0.00 (0.01)

0.08 (13.09) 0.12 (11.88) 0.16 (15.39)

n1: Error component

0.00⁄⁄

n2: Error component

1.82 (2.72) 0.09 (0.08)

n3: Error component Log-likelihood 482 Log-likelihood ratio test 2 Critical v at 5% 0.07 (10.33) 0.11 (10.87) 0.16 (10.43)

rcost: RCg rtime: RCg raccess: RCg

2.55 (1.83) 0.66 (0.41)

0.35 (0.72) 1.53 (5.62) 0.00⁄⁄

1.37 (0.88) 2.17 (1.49) 0.00⁄⁄

479 5.92 5.99⁄

4722

4719 6.59 7.81⁄

4717 10.76 5.99⁄

483

480 5.37 7.81⁄

479 7.00 5.99⁄

4764

4758 12.18 7.81⁄

4754 18.84 5.99⁄

476

476 0.64 7.81⁄

474 3.67 5.99⁄

4748

4741 13.80 7.81⁄

4737 23.14 5.99⁄

0.07 (9.83) 0.11 (8.26) 0.16 (9.95) 0.00 (0.00) 0.02 (0.48) 0.01 (0.11)

0.12 (2.26) 0.18 (2.19) 0.24 (2.55)

0.08 (33.25) 0.11 (34.95) 0.15 (31.59)

0.08 (31.31) 0.12 (24.34) 0.15 (30.09) 0.00 (0.00) 0.02 (0.99) 0.00 (0.17)

0.08 (12.9) 0.13 (12.27) 0.17 (15.48)

0.07 (9.94) 0.10 (10.18) 0.15 (9.95)

0.07 (9.25) 0.11 (7.15) 0.16 (9.25) 0.00 (0.00) 0.05 (2.12) 0.01 (0.21)

0.08 (4.74) 0.14 (3.95) 0.18 (4.06)

0.08 (32.85) 0.11 (34.05) 0.15 (31.21)

0.08 (30.47) 0.12 (22.98) 0.15 (29.31) 0.00 (0.00) 0.03 (3.9) 0.01 (0.28)

0.08 (12.37) 0.13 (11.73) 0.17 (14.79)

0.07 (10.33) 0.11 (10.86) 0.16 (10.45)

0.07 (9.81) 0.11 (8.11) 0.16 (10.01) 0.00 (0.00) 0.02 (0.5) 0.00 (0.08)

0.12 (2.16) 0.18 (2.09) 0.24 (2.42)

0.08 (33.09) 0.11 (34.86) 0.15 (31.47)

0.08 (31.55) 0.11 (25.23) 0.15 (30.39) 0.00 (0.00) 0.01 (0.55) 0.00 (0.09)

0.08 (12.36) 0.13 (11.76) 0.17 (14.85)

1.89 (1.26) 1.89 (1.47) 0.00⁄⁄

n2: Error component n3: Error component

Values in Bold represent estimates.

0.09 (0.21) 1.41 (6.43) 0.00⁄⁄

481 1.91 7.81⁄

n1: Error component

Log-likelihood 456 Log-likelihood ratio test 2 Critical v at 5%

0.00⁄⁄

0.12 (0.34) 1.16 (5.11) 0.00⁄⁄

456 0.09 7.81⁄

455 2.69 5.99⁄

0.00⁄⁄

0.55 (1.47) 0.87 (3.04) 0.00⁄⁄ 4549

4548 0.34 7.81⁄

4546 4.24 5.99⁄

1.80 (2.04) 0.56 (1.16) 469

468 2.28 7.81⁄

467 4.53 5.99⁄

2.02 (1.26) 1.97 (1.26) 0.00⁄⁄

0.49 (1.16) 1.17 (4.24) 0.00⁄⁄ 4601

4599 5.12 7.81⁄

4596 10.87 5.99⁄

456

456 0.07 7.81⁄

454 3.07 5.99⁄

0.62 (1.71) 0.90 (3.05) 0.00⁄⁄ 4556

4556 0.13 7.81⁄

4554 4.09 5.99⁄

F. Díaz et al. / Transportation Research Part B 78 (2015) 222–237

ECML

raccess: RCg

Access time (0.16)

500 Observations

RC

rtime: RCg

Travel time (0.12)

5000 Observations

MNL

rcost: RCg

Log-Normal Cost (0.08)

Scenario 3

500 Observations

Table 5 Willingness to pay for travel time (WTPTT) and access time (WTPAT) computations for level of randomness I. Scenario 1

Scenario 2

500 Observations

Normal WTPTT (1.5)

WTPAT (2.0)

Log-Normal WTPTT (1.5)

WTPAT (2.0)

5000 Observations

Scenario 3

500 Observations

5000 Observations

500 Observations

5000 Observations

MNL

RC

ECML

MNL

RC

ECML

MNL

RC

ECML

MNL

RC

ECML

MNL

RC

ECML

MNL

RC

ECML

1.49 [0.99] (0.05)

1.54 [1.02] (0.11)

1.71 [1.14] (0.18)

1.41 [0.94] (1.2)

1.47 [0.98] (0.3)

1.52 [1.01] (0.13)

1.48 [0.98] (0.1)

1.51 [1.01] (0.03)

1.68 [1.12] (0.17)

1.36 [0.91] (1.86)

1.42 [0.95] (0.82)

1.49 [0.99] (0.08)

1.52 [1.01] (0.07)

1.62 [1.08] (0.3)

1.83 [1.22] (0.25)

1.35 [0.9] (1.97)

1.46 [0.97] (0.45)

1.50 [1] (0.02)

2.18 [1.09] (1.03)

2.17 [1.09] (0.45)

2.04 [1.02] (0.04)

2.01 [1.01] (0.3)

2.00 [1] (0.03)

2.01 [1] (0.06)

2.13 [1.07] (0.81)

2.13 [1.06] (0.37)

2.00 [1] (0)

2.00 [1] (0.08)

2.00 [1] (0)

2.00 [1] (0)

2.10 [1.05] (0.61)

2.09 [1.05] (0.22)

2.02 [1.01] (0.02)

1.99 [0.99] (0.33)

1.97 [0.98] (0.3)

1.97 [0.98] (0.19)

1.52 [1.01] (0.08)

1.56 [1.04] (0.21)

1.62 [1.08] (0.30)

1.48 [0.98] (0.33)

1.50 [1] (0.02)

1.52 [1.02] (0.15)

1.44 [0.96] (0.25)

1.58 [1.06] (0.24)

1.63 [1.08] (0.28)

1.42 [0.95] (1.05)

1.50 [1] (0.03)

1.51 [1.01] (0.08)

1.49 [0.99] (0.06)

1.53 [1.02] (0.1)

1.58 [1.05] (0.20)

1.47 [0.98] (0.36)

1.49 [0.99] (0.1)

1.52 [1.01] (0.1)

2.23 [1.12] (1.35)

2.22 [1.11] (0.7)

2.022 [1.01] (0.45)

1.99 [1] (0.13)

1.99 [0.99] (0.11)

1.97 [0.99] (0.17)

2.11 [1.06] (0.76)

2.10 [1.05] (0.27)

2.10 [1.05] (0.21)

2.00 [1] (0.07)

2.00 [1] (0)

1.97 [0.98] (0.18)

2.21 [1.11] (1.29)

2.21 [1.1] (0.67)

2.20 [1.10] (0.46)

1.98 [0.99] (0.54)

1.98 [0.99] (0.22)

1.96 [0.98] (0.26)

In square brackets [. . .] we present the ratio between the estimated and true WTP values. In round brackets (. . .) we present the corresponding t ratios regarding the statistically comparison between estimated and real WTP values. The critical value for comparison is 1.96 for a 5% level. Values in Bold represent estimates.

F. Díaz et al. / Transportation Research Part B 78 (2015) 222–237

Parameter

231

232

Table 6 Willingness to pay for travel time (WTPTT) and access time (WTPAT) computations for level of randomness II. Scenario 1

Scenario 2

500 Observations

Normal WTPTT (1.5)

WTPAT (2.0)

Log-Normal WTPTT (1.5)

WTPAT (2.0)

5000 Observations

Scenario 3

500 Observations

5000 Observations

500 Observations

5000 Observations

MNL

RC

ECML

MNL

RC

ECML

MNL

RC

ECML

MNL

RC

ECML

MNL

RC

ECML

MNL

RC

ECML

1.53 [1.02] (0.11)

1.67 [1.12] (0.43)

1.86 [1.24] (0.61)

1.42 [0.95] (1.04)

1.51 [1] (0.06)

1.55 [1.04] (0.4)

1.50 [1] (0.01)

1.75 [1.17] (0.57)

1.94 [1.3] (0.34)

1.37 [0.91] (1.72)

1.49 [0.99] (0.14)

1.55 [1.04] (0.39)

1.51 [1.01] (0.05)

1.61 [1.07] (0.28)

1.72 [1.15] (0.2)

1.35 [0.9] (2.09)

1.47 [0.98] (0.25)

1.53 [1.02] (0.21)

2.08 [1.04] (0.45)

2.07 [1.03] (0.17)

2.07 [1.04] (0.12)

2.01 [1] (0.15)

2.00 [1] (0.03)

2.00 [1] (0.02)

2.21 [1.11] (1.15)

2.18 [1.09] (0.39)

2.20 [1.1] (0.17)

2.01 [1.01] (0.31)

2.00 [1] (0)

2.01 [1] (0.04)

2.17 [1.08] (0.95)

2.17 [1.08] (0.39)

2.04 [1.02] (0.04)

2.01 [1.01] (0.32)

2.00 [1] (0.03)

2.00 [1] (0)

1.49 [0.99] (0.06)

1.50 [1] (0)

1.53 [1.02] (0.03)

1.47 [0.98] (0.4)

1.49 [0.99] (0.11)

1.52 [1.01] (0.13)

1.41 [0.94] (0.37)

1.55 [1.03] (0.15)

1.64 [1.09] (0.20)

1.43 [0.95] (1.03)

1.49 [1] (0.07)

1.53 [1.02] (0.19)

1.48 [0.99] (0.07)

1.50 [1] (0)

1.53 [1.02] (0.03)

1.47 [0.98] (0.42)

1.48 [0.99] (0.23)

1.51 [1.01] (0.08)

2.18 [1.09] (1.13)

2.17 [1.09] (0.59)

2.01 [1] (0.01)

1.99 [1] (0.2)

1.99 [1] (0.06)

1.97 [0.99] (0.17)

2.15 [1.08] (1.01)

2.13 [1.07] (0.37)

2.15 [1.08] (0.25)

2.00 [1] (0)

2.00 [1] (0)

1.98 [0.99] (0.09)

2.19 [1.09] (1.19)

2.18 [1.09] (0.6)

2.02 [1.01] (0.02)

2.01 [1] (0.19)

2.00 [1] (0.03)

1.96 [0.98] (0.22)

In square brackets [. . .] we present the ratio between the estimated and true WTP values. In round brackets (. . .) we present the corresponding t ratios regarding the statistically comparison between estimated and real WTP values. The critical value for comparison is 1.96 for a 5% level Values in Bold represent estimates.

F. Díaz et al. / Transportation Research Part B 78 (2015) 222–237

Parameter

233

F. Díaz et al. / Transportation Research Part B 78 (2015) 222–237 Table 7 Parameter estimates in the real data application. MNL1

MNL2

RC

ECML

Estimate

Rob. t-test

Estimate

Rob. t-test

Estimate

Rob. t-test

Estimate

Rob. t-test

Parameter TIME-A TIME-B WALK WAIT COMP COST/w

0.09363 – 0.1118 0.3739 1.992 0.04002

6.10 – 9.30 6.87 5.77 4.74

– 0.08668 0.11123 0.34071 2.03520 0.03906

– 7.36 9.17 6.28 5.86 4.61

0.09363 – 0.11181 0.37392 1.9920 0.04002

6.10 – 9.30 6.87 5.77 4.74

0.10230 – 0.16999 0.41792 2.29270 0.04399

5.57 – 7.54 6.57 5.66 4.26

Alternative specific constant Car Driver (1) Car Passenger (2) Shared Taxi (3) Metro (4) Bus (5) Park ‘n’ Ride (6) Kiss ‘n’ ride (7) Shared Taxi – Metro (8) Bus-Metro (9)

1.9807 2.4713 1.7444 2.3327 – 1.7519 1.267 1.0117 0.64063

6.20 13.70 9.01 11.34 – 5.70 7.44 5.00 3.56

2.0399 2.4815 1.7692 2.2142 – 1.9669 1.4889 1.2023 0.8435

6.36 14.04 8.63 10.68 – 6.36 8.12 5.68 4.19

1.9807 2.4713 1.7444 2.3327 – 1.7519 1.2670 1.0117 0.6406

6.16 13.30 8.13 11.12 – 5.75 7.24 4.84 3.20

2.8356 3.5855 2.1772 3.8052 – 2.0812 1.3389 0.9285 0.5341

6.10 9.31 2.72 5.82 – 5.73 7.31 3.90 2.43

0.00024

0.49228 2.3769 2.5506 1.0854 0.0088 0.0104

4.55 3.01 1.10 0.92 0.77

Standard deviation – attribute TIME-B Standard deviation – random component Car driver and Car passenger (1,2) Shared Taxi (3) Metro (4) Bus (5) Combinations (6,7,8,9) Number of observations Draws Log-likelihood

1374 – 1578.38

Value of in-vehicle travel time

2.34

0.01

1374 – 1570.83

1374 1000 1578.38

2.22

2.34

1374 1000 1565.09 0.01

2.33

0.01

When assuming Normal distributed measurement errors in the case of small samples, the MNL performs slightly better than the other model structures in the terms of the capacity to recover the WTP for travel time. This is not the case for the WTP of access time, where the advanced models do a better job. For large samples and with the same measurement errors distribution, the ECML shows the best performance. In particular, null hypothesis II is not accepted at the 5% level when estimating MNL models with large sample sizes in scenarios 2 and 3. On the other hand, when assuming Log-Normal distributed measurement errors, the MNL performs well in all cases (i.e. hypothesis II is always accepted at the 5% level). But again, when working with large samples, the more advanced models perform slightly better. To conclude, these results let us to infer that when working with small sample sizes it might be convenient to estimate simpler model structures such as the MNL. However, when working with large samples the estimation of more flexible models seems warranted. In particular, when suspecting the presence of high randomness in variables, the ECML model could be preferred. Of course, more experimentation is needed to give more support and generalize this conclusion. 4.5. Real data application In this section we test the ability of different model formulations to account for stochastic variations due to variable measurement errors. We have available datasets on personal mode choice for morning peak journey-to-work trips in Santiago, Chile. Workplace interviews were collected for 697 individuals living in the Las Condes-CBD corridor (to the richest East End of the city and following Line 1 of the underground), and 677 individuals living in the San Miguel-CBD corridor (to the middle/low income South of the city and following Line 2 of the metro). Nine mode alternatives were considered: Car Driver (1), Car Passenger (2), Shared Taxi (3), Metro (4), Bus (5), Park ‘n’ Ride (6), Kiss ‘n’ ride (7), Shared Taxi-Metro (8) and Bus-Metro (9). More details about these datasets can be found in Ortúzar and Ivelic (1987). Explanatory variables used in the modelling are TIME (In-vehicle Travel Time, in min), WALK (Walking Time, in min), WAIT (Waiting Time, in min), COMP (Car competition, defined as the number of cars divided by the number of driving licences in the household with a ceiling of one), and COST/w (Travel cost divided by the wage rate, in Chilean pesos -CH$- per min).

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F. Díaz et al. / Transportation Research Part B 78 (2015) 222–237

In-vehicle travel times, TIME-A and TIME-B, were measured using two different methods. TIME-A refers to travel times obtained by performing several measurements during the peak period (two and a half hours). On the other hand, TIME-B refers to more accurate measurements done at 15 min intervals during the peak, more closely conforming to the actual times experienced by each traveller. Thus, we can assume that the TIME-A values may contain measurement errors (in relation to the TIME-B values). Four models were estimated combining the two data sources: two MNL models using TIME-A (MNL1) and TIME-B (MNL2) values, respectively; a random coefficients ML model (RC), which uses TIME-A values and considers a randomly distributed in-vehicle travel time parameter; and an error components model (ECML), using TIME-A values and adding an error component to each alternative to allow for measurement errors in the travel time attribute. We also estimated the value of in-vehicle time for all models and the corresponding t-values regarding a comparison with the estimated value for the MNL2 model. Estimation results of the above models are presented in Table 7. Even though both MNL models have correct signs and significant variables at the 5% level, model MNL2 (i.e. considering a more accurate measurement of travel time) has a better goodness of fit than MNL1, as expected. A brief perusal of model RC allows us to conclude that the random parameters approach is not able to capture stochastic variations in the travel time attribute. In the case that randomness in travel times exists, it is not directly proportional to the variable value because the standard deviation parameter estimated in the RC model is not significant. Moreover, the log-likelihood values for the RC and MNL models are almost equal. The ECML model appears to be superior to both the MNL1 and RC models, capturing somehow the measurement errors in the TIME-A variable, even though there is no significant gain regarding willingness to pay measures. The ECML model also presents a better goodness of fit than MNL2, but required five additional parameters. From the results we inferred that using aggregate measures and simultaneously considering an ECML structure could yield to similar results as those obtained when using more accurate measures. However the added error terms can also capture other effects, in addition to the stochastic variations in travel time, but as these are confounded it is not possible to isolate them.

5. Conclusions We examined the inclusion of stochastic variables in the estimation of discrete choice random utility models. We conducted an econometric analysis of the problem and designed an experimental study using simulated data with variables subject to stochastic variations. MNL and different ML model specifications were estimated and their results statistically compared on the basis of willingness-to-pay measures which are free from scaling problems. The econometric analysis shows that the problem can be approximated using ML formulations with a flexible covariance matrix that allows including heteroskedasticity among alternatives and/or observations, due to the presence of the stochastic variables. The most appropriate specification would depend on the data variation structure. The simple MNL model cannot solve the problem properly. If the variable randomness does not depend on the attribute magnitude and its variance is constant across observations, the problem can be approximated through a stochastic variable (SV) model, which is a specific kind of ML model and can be shown to be equivalent to an error components (ECML) model. The ECML structure has certain estimation benefits in term of parameter identification in comparison with the SV model; furthermore, the structure also allows treating the stochastic variable effect as a particular kind of heteroskedasticity among alternatives. On the other hand, when stochastic variations are related with the magnitude of the explanatory variables, randomness can be captured more adequately through a random coefficients (RC) model. Nevertheless, the effect of the stochastic variables might also be confounded with variation across tastes in the population even when the stochastic variations are independent from the variable size. However, the level of confounding seems to be dependent on the error structure. In any case, special care must be taken when dealing with and interpreting estimations results from ML specifications as it is possible to confuse unobserved taste heterogeneity with stochastic variability. The proposed model specifications were also tested using a real databank where the travel time attribute was measured with different levels of precision. Results show that the ECML structure captures partially the stochastic variations in the attribute, but confounded effects could exist and it is not possible to isolate them. Experiments with our simulated dataset suggest that the MNL model is fairly robust for computing marginal rates of substitution, particularly when the model is estimated using large datasets. Notwithstanding, when dealing with the stochastic variables problem the best results were obtained using the ECML structure, but still care should be taken if the model is estimated with small sample size data. In sum, some benefits are obtained from using more advanced models to account for stochastic variables in discrete choice models when having large datasets. Benefits are related with a better representation of randomness and a better capacity to recover parameters and marginal substitution rates. The issue to highlight is the need for large samples to estimate more advanced models. When having a small sample size, it would be difficult to estimate reasonably advanced models and then, maybe, the conservative approach would be to consider conventional and simpler model structures, even if they may produce biased results. Our findings could be useful to formulate more robust specifications to better explain behaviour and when significant randomness in the explanatory variables is suspected. Our findings also expose the possibilities of confounding effects

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between unobserved taste heterogeneity and stochastic variability. Confounding effects are very difficult, if not impossible, to isolate from unobserved taste heterogeneity or heteroskedasticity. In consequence, when models with complex covariance matrices are estimated, results should be looked carefully because of the presence of such confounding effects. Further research should focus on generalizing our findings using other real databanks and multiple simulated datasets varying the values and stochastic levels of the attributes and the sample sizes. Another important issue confirmed here, relates to the appropriate estimation of models using low sample sizes; following Sillano and Ortúzar (2005), maybe this should be performed using Bayesian estimation techniques. Acknowledgements We are grateful for the very useful comments of four unknown referees. We also want to acknowledge Findeter and the General Royalties System, for having partially financed this work through the project Diamante Caribe y Santanderes (Cod. 2014000100012). We also wish to thank the Administrative Department of Science, Technology and Innovation (COLCIENCIAS) through Contract RC No 0791-2013 (Cod. 121562238496), the Atlantic Department Government and the General Royalties System (Project LOGPORT BPIN 20120001001911050672, Agreement No. 0103-2013-000016), the Millennium Institute in Complex Engineering Systems (ICM: P05-004F; FONDECYT: FB016), the ALC-BRT Centre of Excellence, funded by the Volvo Research and Educational Foundations, and CEDEUS, the Centre for Urban Sustainable Development (Conicyt/Fondap/15110020) for their financial support. Appendix A. Identification and normalization for SVM models In this section identification and normalization issues in SV logit models are discussed using a three alternatives case, I = 3, which are available for all individuals in the dataset. Order and rank conditions are applied using the covariance matrix of utility differences of the specified model. A.1. Covariance matrix The covariance matrix (X) for a logit model with three alternatives, I = 3, and three explanatory variables is the same for all individuals can be written as:

0

B1 X¼B @0 0

0 B2 0

1 0 C 0A B3

ðA:1Þ

where

" # 3 X b2ik r2ik þ n2

Bi ¼

ðA:2Þ

k¼1

n2 is the variance of a Gumbel distribution, which in turn can be written as a function of the scale parameter l as follows:

n2 ¼

g

ðA:3Þ

l2

where

g¼

p2

ðA:4Þ

6

From the above definitions, the covariance matrix of utility differences (XD) is as follows:



XD ¼

B1 þ B3

B3

B3

B2 þ B3



ðA:5Þ

A.2. Order condition The order condition states that the maximum number (Morder) of alternative-specific covariance terms from X that can be estimated is given by:

Morder ¼

IðI  1Þ 1¼2 2

Morder is in addition equal to the number of distinct cells in XD minus one to set the scale parameter l.

ðA:6Þ

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A.3. Rank condition The rank condition states that the maximum number of estimable parameters is given by:

MRank ¼ rank½Jacobian½v ecuðXD Þ  1

ðA:7Þ

where vecu(XD) is the function to vectorize the unique elements of XD into a column vector. For our case, vecu(XD) is:

0

1 B1 þ B3 C v ecuðXD Þ ¼ B @ B2 þ B3 A

ðA:8Þ

B3 The Jacobian matrix of Eq. (A.8) is given by:

0

b21

B Jacobian½v ecuðXD Þ ¼ @ 0 0

1

b22

b23

0

0

0

b21

b22

b23

0

0

b21

b22

b23

b21

b22

b23

C 2A

0

b21

b22

b23

1

0

0

0

0

2

ðA:9Þ

The rank of the Jacobian matrix of Eq. (A.9) is three. Then, according to Eq. (A.7), only two parameters can be estimated. A.4. Equality condition It is necessary to know which normalization could be appropriate (i.e. which parameters should be fixed and which value they should take) to guarantee that the original model structure remains unaltered. The above rationale can be written as:

XND ¼ XD

ðA:10Þ

where XND is the normalized matrix. Eq. (A.10) implies:

BN1 ¼ B1

ðA:11Þ

BN2 ¼ B2

ðA:12Þ

BN3 ¼ B3

ðA:13Þ

Defining kik as the estimated value or the fixed parameter rik, and being t the fixed or estimated value for the scale parameter l of the model; then Eqs. (A.11)–(A.13) can be written as:

b211 k211 þ b212 k212 þ b213 k213 þ b221 k221 þ b222 k222 þ b223 k223 þ b231 k231 þ b232 k232 þ b233 k233 þ

g

t2 g

t2 g

t2

g

¼ b211 r211 þ b212 r212 þ b213 r213 þ

ðA:14Þ

l2 g

¼ b221 r221 þ b222 r222 þ b223 r223 þ

ðA:15Þ

l2 g

¼ b231 r231 þ b232 r232 þ b233 r233 þ

ðA:16Þ

l2

From Eqs. (A.14) and (A.15):

b211 k211 þ b212 k212 þ b213 k213 ¼ b211 r211 þ b212 r212 þ b213 r213 þ b221 k221 þ b222 k222 þ b223 k223 ¼ b221 r221 þ b222 r222 þ b223 r223 þ





g

g

l

t2



 2



g

g

l

t2

 2

ðA:17Þ

ðA:18Þ

And from Eq. (A.16):

b231 k231 þ b232 k232 þ b233 k233  ðb231 r231 þ b232 r232 þ b233 r233 Þ ¼



g

g

l

t2

 2

 ðA:19Þ

Substituting Eq. (A.19) in Eqs. (A.17) and (A.18) and sorting:

b211 k211 þ b212 k212 þ b213 k213 ¼ b211 ðk231  r231 þ r211 Þ þ b212 ðk232  r232 þ r212 Þ þ b213 ðk233  r233 þ r213 Þ

ðA:20Þ

b221 k221 þ b222 k222 þ b223 k223 ¼ b221 ðk231  r231 þ r221 Þ þ b222 ðk232  r232 þ r222 Þ þ b223 ðk233  r233 þ r223 Þ

ðA:21Þ

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No rik can be estimated at least for one alternative. Without loss of generality, it can be assumed to be the third alternative. There are two possibilities: to estimate two parameters on one of the two remaining alternatives or to estimate one parameter of each alternative. In the first case, all parameters in one of the two remaining alternatives must be fixed. For instance, to estimate r11 and r12 (i.e. k11 and k12); r21, r22 and r23 must be fixed to k21, k22 and k23 respectively. In consequence, correct values for k21, k22, k23, k31, k32 and k33 must be found to satisfy Eq. (A.21), which in turn could be an impossible task. On the other hand, fixing r31, r32 and r33 to k31, k32 and k33 respectively should be sufficient when only one parameter is estimated per alternative. Parameters must be fixed at a sufficiently large number to ensure obtaining a positive value on the right hand side of equations (A.20) and (A.21).3 Remaining parameters can be fixed freely. Estimated parameters should not correspond to the real values, because they are confounded according to equations (A.20) and (A.21); this is a direct result from the identification problem. For I > 3, a similar result would be reached. It can be noted that identification and normalization issues are not simple for SV models. References Bhatta, B., Larsen, O., 2011. Errors in variables in multinomial choice modelling: a simulation study applied to a multinomial model of travel mode choice. Transport Policy 18, 326–335. Bliemer, M., Rose, J., 2013. Confidence intervals of willingness-to-pay for random coefficient logit models. Transportation Research Part B: Methodological 58, 199–214. Bolduc, D., Alvarez-Daziano, R., 2009. On estimation of hybrid choice models. In: International Choice Modelling Conference, Harrogate. Bolduc, D., Boucher, N., Alvarez-Daziano, R., 2008. Hybrid choice modelling of new technologies for car choice in Canada. Transportation Research Record 2082, 63–71. Brey, R., Walker, J.L., 2011. Estimating time of day demand with errors in reported preferred times: an application to airline travel. Procedia Social and Behavioural Sciences 17, 150–168. Carroll, R.J., Spiegelman, C.H., Gordon Lan, K.K., Bailey, K.T., Abbott, R.D., 1984. On errors-in-variables for binary regression models. Biometrika 71, 19–25. Cherchi, E., Ortúzar, J. de D., 2008. Predicting best with mixed logit models: understanding some confounding effects. In: Inweldi, P.O. (Ed.), Transportation Research Trends. Nova Science Publishers, Inc., New York, pp. 215–235. Daly, A.J., Ortúzar, J. de D., 1990. Forecasting and data aggregation: theory and practice. Traffic Engineering and Control 31, 632–643. Jackson, W.B., Jucker, J.V., 1982. An empirical study of travel time variability and travel choice behaviour. Transportation Science 16, 460–475. Kao, C., Schnell, J., 1987. Errors in variables in the multinomial response model. Economics Letters 25, 249–254. McFadden, D., Train, K.E., 2000. Mixed MNL models for discrete response. Journal of Applied Econometrics 15, 447–470. Ortúzar, J. de D., Ivelic, A.M., 1987. Effects of using more accurately measured level-of-service variables on the specification and stability of mode choice models. In: Proceedings 15th PTRC Summer Annual Meeting, University of Bath, September 1987, England. Ortúzar, J. de D., Willumsen, L.G., 2011. Modelling Transport, fourth ed. John Wiley and Sons, Chichester. Sillano, M., Ortúzar, J. de D., 2005. Willingness-to-pay estimation with mixed logit models: some new evidence. Environment and Planning 37A, 525–550. Stefansky, L.A., Carroll, R.J., 1985. Covariate measurement error in logistic regression. The Annals of Statistics 13, 1335–1351. Stefansky, L.A., Carroll, R.J., 1987. Trust conditional scores and optimal scores for generalized linear measurement error models. Biometrika 74, 703–716. Stefansky, L.A., Carroll, R.J., 1990. Structural logistic regression measurement model. In: Proceedings Summer Research Conference on Statistical Analysis of Measurement Error Models and Applications. Providence, Rhode Island. Steinmetz, S.S.C., Brownstone, D., 2005. Estimating commuters ‘value of time’ with noisy data: a multiple imputation approach. Transportation Research Part B 36, 865–889. Swait, J.D., Bernardino, A., 2000. Distinguishing taste variation from error structure in discrete choice data. Transportation Research Part B 34, 1–15. Train, K.E., 1978. The sensitivity of parameter estimates to data specification in mode choice models. Transportation 7, 301–309. Train, K.E., 2009. Discrete Choice Methods with Simulation, third ed. Cambridge University Press, Cambridge. Walker, J.L., Jieping, L., Sirinivasan, S., Bolduc, D., 2010. Travel demand models in the developing world: correcting for measurement errors. Transportation Letters: The International Journal of Transportation Research 4, 231–243. Walker J.L., 2001. Extended Discrete Choice Models: Integrated Framework, Flexible Error Structures and Latent Variables. Ph.D Thesis. MIT. Wansbeek, T., Meijer, E., 2000. Measurement Error and Latent Variables in Econometrics. Elsevier, Amsterdam. Williams, H.C.W.L., Ortúzar, J. de D., 1982. Behavioural theories of dispersion and the mis-specification of travel demand models. Transportation Research Part B 16, 167–219. Yamamoto, T., Komori, R., 2010. Mode choice analysis with imprecise location information. Transportation 37, 491–503.

3 The covariance matrix for Probit errors must be positive semi-definite. In this case, diagonal elements must be non-negative, which is trivial as they are squared numbers. In consequence, the right hand part of equations (A.20) and (A.21) must be a positive number.