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1、Copyright 2000. All rights reserved. Incorporating Observed and Unobserved Heterogeneity in Urban Work Travel Mode Choice Modeling CHANDRA R. BHAT Department of Civil Engineering, University of Texas at Austin, Austin, Texas An individuals intrinsic mode preference and responsiveness to level-of-ser
2、vice variables affects her or his travel mode choice for a trip. The mode preference and responsiveness will, in general, vary across individuals based on observed (to an analyst) and unobserved (to an analyst) individual characteristics. The current paper formulates a multinomial logit-based model
3、of travel mode choice that accommodates variations in mode preferences and responsiveness to level-of-service due to both observed and unobserved individual characteristics. The model parameters are estimated using a maximum simulated log-likelihood approach. The model is applied to examine urban wo
4、rk travel mode choice in a multiday sample of workers from the San Francisco Bay area. Most work travel mode choice models are based on the random utility maximization framework of microeconomic theory. The random utility maximi- zation framework assumes that an individuals choice of mode on any cho
5、ice occasion is a reflection of underlying indirect utilities associated with each of the available modes and that the individual se- lects the alternative that provides her or him the highest utility (or least disutility). The indirect util- ity that an individual associates with each mode is not o
6、bserved to the demand analyst, who then as- sumes that this utility is composed of three compo- nents: a) an intrinsic individual-specific mode bias term that varies across individuals and that repre- sents the bias of the individual toward the mode due to observed and unobserved (to the analyst) in
7、divid- ual factors (such as sex, lifestyle, and culture); b) the utility that the individual derives from observable (to an analyst) level-of-service characteristics of- fered by the mode for the individuals trip; and c) a mean-zero random term that captures the effect of unobserved modal characteri
8、stics or unknown mea- surement error in modal level-of-service attributes (more generally, this final third term represents the effects of all omitted variables that are not individ- ual specific). Ideally, we should obtain individual- specific parameters for the first two utility compo- nents; that
9、 is, for the intrinsic mode biases and for 228 the subjective evaluations of modal level-of-service attributes. However, the data used for mode choice estimation are usually cross-sectional or comprise very few observations on each individual. This pre- cludes estimation at the individual level and
10、con- strains the modeler to pool the data across individ- uals. In such pooled estimations, the analyst should, in some way, accommodate taste differences (i.e., heterogeneity in intrinsic mode biases and hetero- geneity in responsiveness to level-of-service at- tributes) across individuals. In part
11、icular, if the as- sumption of taste homogeneity is imposed when there is taste heterogeneity, the result is inconsis- tent model parameter estimates and even more se- vere inconsistent choice probability estimates (see CHAMBERLAIN, 1980; the reader is also referred to HSIAO, 1986 and DIGGLE, LIANG,
12、 and ZEGER, 1994 for a detailed discussion of heterogeneity bias in discrete-choice models). Taste heterogeneity may be incorporated in travel mode choice models by introducing observed individ- ual socio-economic characteristics as alternative- specific variables and by interacting level-of-service
13、 variables with observed individual characteristics (such as using a travel cost over income specifica- tion or using a market segmentation scheme). How- ever, it is very likely that taste heterogeneity will remain even after accounting for differences in ob- Transportation Science, 2000 INFORMS 004
14、1-1655 / 00 / 3402-0228 $05.00 Vol. 34, No. 2, May 2000 pp. 228 238, 1526-5447 electronic ISSN Copyright 2000. All rights reserved. q q q q qk HETEROGENEITY IN TRAVEL MODE CHOICE / 229 served individual characteristics (see FISCHER and NAGIN, 1981). This taste heterogeneity due to unob- served indiv
15、idual attributes is generally ignored in travel mode choice modeling. In this paper, we formulate a multinomial logit- based model of work travel mode choice that accom- modates taste heterogeneity due to both observed and unobserved individual attributes. The formula- tion ensures the correct sign
16、on the level-of-service parameters (for example, a negative coefficient on the time and cost variables) for all individuals. The model takes the form of a random-coefficients logit (or RCL) structure. The RCL structure has been known for a long time, but there have been few applications of this stru
17、cture. The primary reason is that the choice probabilities in the RCL structure do not have a closed-form expression and generally in- volve high dimensional integration. However, in the past few years, the advent of simulation techniques to approximate integrals has facilitated the applica- tion of
18、 the RCL structure (see BHAT, 1998; BROWN- STONE and TRAIN, 1997; and TRAIN, 1998). The mode choice model in this paper is estimated from repeated work travel mode choices of workers The utility Uqit that an individual q associates with an alternative i on choice occasion t may be written in the fol
19、lowing form: Uqit qxqit qit (1) where xqit is a vector of observed variables (includ- ing alternative specific constants), q is a corre- sponding coefficient vector that may vary over indi- viduals but does not vary across alternatives or time, and qit is an unobserved extreme value ran- dom term th
20、at captures the idiosyncratic effect of all omitted variables that are not individual specific. qit is assumed to be identically and independently distributed across all choice occasions and indepen- dent of q and xqit. A number of different specifications may be used for the coefficient vector q in
21、 Eq. 1. To facilitate the following discussion, we partition the coefficient vec- tor q: q asc , ls , (2) q q where asc is the coefficient sub-vector on the alter- obtained from a multi-day travel survey conducted in the San Francisco Bay area. It is important to note that repeated mode choice data
22、from workers is needed to accommodate unobserved variations in native specific constants and ls is the coefficient sub-vector on the level-of-service variables. One pos- sible specification is then to write each element of the asc as a deterministic function of an observed asc asc intrinsic mode bia
23、ses across individuals. In conven- tional cross-sectional work mode choice models that use a single observation for each individual, it is impossible to separate the effect of unobserved het- erogeneity in intrinsic bias from the effect of omitted variables that are generic to all choice occasions (
24、see BHAT, 1998 for an application that allows variation in level-of-service responsiveness, but is unable to accommodate unobserved heterogeneity in intrinsic mode preferences because it uses cross-sectional data). The rest of this paper is organized as follows. The vector zq of individual character
25、istics ( qi i izq), and to maintain a fixed value (across individ- uals) on the level-of-service coefficients ( ls ls). This specification corresponds to the standard multinomial logit (or MNL) model and is the one generally adopted in mode choice modeling. A sec- ond specification is similar to the
26、 first, except that it relaxes the assumption of homogeneity (across indi- viduals) in response to level-of-service changes by specifying the level-of-service coefficient ls associ- ated with the kth level-of-service variable (k 1, 2, . . . , K) as a function of an observed vector wqk of ls exp( ls
27、w ). next section discusses the formulation and estima- individual attributes, qk k k qk tion of the RCL model used in the paper. Section 2 presents the empirical results obtained from apply- ing the model to an urban mode choice context. The final section provides a summary of the research findings
28、. 1. MODEL FORMULATION WE DEVELOP THE model formulation assuming that all alternatives are available on all choice occasions. The sign is applied for a non-negative response coefficient (such as the coefficient on frequency of service) and the sign is applied for a non-positive response coefficient
29、(such as the coefficient on travel time or travel cost). This second specification corre- sponds to a MNL model with parameters entering the utility non-linearly. We will refer to this speci- fication as the deterministic coefficients logit (DCL) model. A third specification superimposes random (uno
30、bserved) heterogeneity over the deterministic (observed) heterogeneity of the second specification, asc asc ls ls Extension of the formulation to the case where only qi i izq qi and qk exp( k a subset of alternatives are available on some choice occasions is straightforward. kwqk vqk), where qi and
31、vqk are assumed to be normally distributed across individuals. We will re- Copyright 2000. All rights reserved. 230 / C. R. BHAT fer to this specification as the random coefficients logit (RCL) model. Our RCL model specification dif- fers from (and is more general than) the RCL model specification u
32、sed by REVELT and TRAIN (1997), JAIN, VILCASSIM, and CHINTAGUNTA, (1994), MEHN- the probabilities, Pqit exp xqit j exp xqit . (3) DIRATTA (1996), and Train (1998). Specifically, these other studies do not allow the distribution of the random taste coefficients to vary based on observed individual ch
33、aracteristics. In the RCL specification of this paper, we assume that the elements in each of the random vectors q ( q1, q2, . . . , q1 ) and vq( vq1, vq2, . . . , vqK ) are independent of the elements in the other vector, and that each element in a vector is indepen- dent from other elements in tha
34、t vector. The normal distribution assumption for the ele- ments in the vq vector in the RCL model implies a log-normal distribution for the level-of-service coef- ficients. Specifically, the kth level-of-service coeffi- cient is log-normally distributed with the following properties (see JOHNSON and
35、 KOTZ, 1970): a) Me- dian exp( qk), b) mode exp( qk)/ k, c) mean 1/2 The unconditional probability of choosing alterna- tive i on choice occasion t for a randomly selected individual q can now be obtained by integrating the conditional multinomial choice probabilities in Eq. 3 over all possible valu
36、es of , Pqit * Pqit f * d . (4) The disaggregate-level self- and cross-elasticities are cumbersome though straightforward to compute from the choice probability expression in Eq. 4. The aggregate-level elasticities may be computed from the disaggregate-level elasticities in the usual way (see BEN-AK
37、IVA and LERMAN, 1985, page 113). To develop the likelihood function for parameter estimation, we need the probability of each sample individuals sequence of observed travel mode exp( qk) k , and d) variance exp(2 qk) k( k choices. Let Tq denote the number of choice occa- 1), where qk k kwqk, k exp k
38、), and k sions observed for individual q. Conditional on , ls 2 2 the likelihood function for individual qs observed is the variance of the kth element of the vector vq. A useful property of the log-normal distribution is that the ratio of two independent log-normally distrib- sequence of choices is
39、 Tq I uted variables is also log-normally distributed. Therefore, a log-normal distribution assumption for Lq P qit yqit, the level-of-service coefficients implies a log-normal distribution for the money value of time, which is obtained as the ratio of the travel time and travel where t 1 i 1 1 if t
40、he qth individual chooses cost coefficients (BEN-AKIVA, BOLDUC, and BRADLEY, 1993, in contrast, specify a log-normal distribution for the money value of time by imposing the a priori assumption that the cost coefficient is fixed, while yqit alternative i on choice occasion t 0 otherwise. (5) allowin
41、g the travel time coefficient to be lognormally distributed; the specification in this paper is more general than the one used by those authors. The coefficient vector q in the RCL model de- pends on both observed and unobserved individual attributes, as indicated earlier. The assumptions about the
42、functional form of this dependence, and the distributional assumptions regarding the unob- served attributes, imply that q varies in the popu- lation with density f( *), where * is a vector of the true parameters (mean and variance) character- izing the distribution (to be precise in notation, we sh
43、ould subscript the distribution function f by an index for the elements of , because different ele- ments may follow different distributions; however, for convenience, we forego this notational formality). Conditional on , we get the familiar MNL form for The unconditional likelihood function of the
44、 choice sequence is Lq * Lq f * d . (6) The goal of the maximum likelihood procedure is to estimate *. The log-likelihood function is ( ) q ln Lq ( ). The log-likelihood function involves the evalua- tion of a multi-dimensional integral. Conventional quadrature techniques cannot compute the integral
45、s with sufficient precision and speed for estimation via maximum likelihood when the dimensionality of the integration is greater than 2 (see Revelt and Train, 1997 and HAJIVASSILIOU and RUUD, 1994). We apply Monte Carlo simulation techniques to approximate the integrals in the log-likelihood func-
46、Copyright 2000. All rights reserved. HETEROGENEITY IN TRAVEL MODE CHOICE / 231 tion and maximize the resulting simulated log-like- lihood function. The simulation procedure is similar to the one used by Revelt and Train (1997). For a given value of the parameter vector , we draw a particular realiza
47、tion of from its distribution, and, subsequently compute the individual likelihood function Lq( ) (Eq. 5). We then repeat this process M times for each individual for the given value of the parameter vector . The individual likelihood func- tion is then approximated by averaging over the different L
48、q ( ) values. rate simulations of the individual log-likelihood functions and to reduce simulation variance of the MSL estimator. All estimations and computations were carried out using the GAUSS programming language on a personal computer. Gradients of the simulated log- likelihood function with respect to the parameters were coded. 2. EMPIRICAL ANALYSIS OF URBAN MODE CHOICE M 2.1 Data and