ECONOMETRIC ISSUES IN ESTIMATING CONSUMER PREFERENCESFROM STATED PREFERENCE DATA: A CASE STUDY OF THE VALUE

OF AUTOMOBILE TRAVEL TIME

John Calfee, Clifford Winston, and Randolph Stempski*

Abstract—This paper explores a number of methodological issues relatedto the econometric analysis of stated preference data in the context ofestimating the value of automobile travel time. Estimates of parametersand the willingness to pay (WTP) to save time are obtained usingconventional ordered probit and rank-ordered logit models and an inno-vation calledmixed logit.We find that the average WTP is low and doesnot exhibit much variation among motorists. Although our findings usingdata on respondents’ rankings of alternatives are robust, we find thatcaution should be used in estimating stated preferences based on respon-dents’ ratings.

I. Introduction

ECONOMISTS typically analyze consumer preferencesas they are revealed in market settings, but market

transactions are not always available to measure consumerpreferences. Obviously, market data do not exist for prod-ucts that cannot be purchased, and, even if a product is onthe market, it is difficult to use market data to determinehow consumers value product attributes that may not beavailable. For example, what value do consumers place ontelevision sets with 48 in. screens when the largest screenavailable is 36 in.?

Increasingly, researchers have tackled these problems byasking consumers to state their preferences given a hypo-thetical set of alternatives.1 Such stated preference data alsohave statistical advantages over revealed preference data.For example, explanatory variables used to analyze revealedpreferences may have little variation or may be highlycorrelated, so their effects are not identified. Stated prefer-ence analysis overcomes this problem because explanatoryvariables can be varied widely and independently of eachother. Revealed preference data are also limited becausethey capture only individuals’ first choices. Stated prefer-ence data contain complete rankings of several alternatives,thus providing additional information to increase estimationprecision.

But stated preference analysis presents its own problems.It must, for example, employ a survey that generates anaccurate ordering of preferences. It must also use an appro-

priate econometric technique to obtain consistent and effi-cient parameter estimates. We explore these requirements inthe context of estimating the value of automobile traveltime—a critical parameter in determining optimal highwaycongestion tolls and investment levels, but one that cannotbe easily inferred from market transactions.

We sketch the ordered probit and rank-ordered logitmodels commonly used to estimate stated preferences andreport estimates of commuters’ value of automobile traveltime based on them. Conventional ordered models, how-ever, can produce biased or inconsistent parameter estimatesbecause they restrict the “spacing” of consumers’ prefer-ences. Therefore, we reanalyze our data using an innovationcalled mixed logit (Brownstone & Train, 1999), which doesnot incorporate any spacing restrictions. Finally, we inves-tigate whether respondents bias the analysis by payingcloser attention to ordering their top choices than orderingtheir bottom choices.

We find that our value of time estimates are surprisinglyrobust to alternative statistical procedures, which may bodewell for the use of conventional ordered models. But theestimates are sensitive to whether one constructs preferenceorderings from ordinal rankings or cardinal ratings. Inparticular, it appears that estimates based on cardinal ratingscan be very unreliable.

II. A Stated Preference Framework for EstimatingAutomobile Commuters’ Value of Time

A revealed preference analysis estimates urban commut-ers’ willingness to pay (WTP) to save travel time by infer-ring it from commuters’ tradeoff of travel time and travelcost in their choice of transportation mode (such as autoversus bus). This procedure, however, captures the value(disutility) that auto commuters place on time spent on thealternative (bus) mode and the value that bus commutersplace on the alternative (auto) mode. We therefore use astated preference analysis to estimate the WTP for thosecommuters who currently travel by auto and face somecongestion.

We hired a market research firm that draws on surveyrespondents who are members of a well-known nationwidemail panel (National Family Opinion) and who are accus-tomed to preference surveys. Each respondent was pre-sented with thirteen alternatives or packages that describedthe essential elements of a commute, including the con-gested and uncongested travel time, the travel cost (usuallyin the form of a toll), and an indication of whether trucks

Received for publication March 30, 1999. Revision accepted for publi-cation October 19, 2000.

* American Enterprise Institute, Brookings Institution, and AmericanEnterprise Institute, respectively.

We benefitted from discussions with D. Brownstone, K. Small, and K.Train, and from the comments of an anonymous referee.

1 Contingent valuation asks individuals for an absolute valuation (suchas, how much would they be willing to pay to use an unpollutedrecreational area). This approach has been attacked by many analysts andcan be fatally flawed. The most severe problem is termed “embedding,”where consumers’ valuations are invariant to quantities consumed. Al-though stated preference analysis is not free of methodological concerns,it can overcome most of the problems associated with contingent valua-tion.

The Review of Economics and Statistics,November 2001, 83(4): 699–707© 2001 by the President and Fellows of Harvard College and the Massachusetts Institute of Technology

were allowed on the road.2 The specific packages, presentedin appendix A, were formulated with maximum one-waycommute times of either sixty or forty minutes.3 Respon-dents rated the “acceptability” of each package on a ten-point scale, in which 1 indicated very unacceptable and 10indicated very acceptable; respondents then ranked thepackages, breaking ties among equally rated alternativeswhen necessary.

The stated preference models were estimated from 1,170respondents. Survey respondents were automobile commut-ers in major U.S. metropolitan areas who regularly drive towork and face some congestion.4 Response rates wereroughly 67% approximately three weeks after the mailing,which is slightly better than the usual response rates for theNational Family Opinion panel.

Separate samples of commuters were drawn for each ofthirteen commuting/road pricing scenarios. (See table 1.)Thus, respondents stated their preferences in the context ofa single scenario. The first two scenarios seek to provideperspective on a major concern with stated preferenceanalysis—policy bias. When time savings are the result of apublic intervention, such as tolls, estimates of commuters’

WTP may to some extent capture commuters’ preferencesfor or against tolls rather than their true WTP. Thus, the firsttwo scenarios focused on paying for a “smart car” technol-ogy to reduce travel time.5 Tolls are introduced in the thirdscenario. Scenarios 4 through 7 present alternative ways thatthe toll revenue could be spent.6 Scenarios 8 through 11distinguish between public and private toll roads, and sce-narios 12 and 13 distinguish between flat and rapid trafficgrowth.

III. Econometric Models for Analyzing Ordered Data

Survey respondents who state their preferences among aset of alternatives generate ordered data. Standard probit orlogit models estimate parameters based on the most pre-ferred choice, but more information can be incorporated intoparameter estimation by using an ordered probit or logitmodel that accounts for respondents’ rankings of all alter-natives.

The behavioral foundation of the ordered probit and logitspecifications is a random utility model. For individuali , letthere be a choice setC with J elements, with each elementindexedj 5 1, 2, . . . , J. Let the vector of attributes foreach element in the choice set available be denotedxij , andlet si denote the characteristics of each individual. Theutility of each element in the choice set for each individualis represented asUij 5 V(si, xij ) 1 e ij 5 Vij 1 e ij , whereVij is the deterministic component of utility, ande ij is thestochastic component. Let individuali generate a surveyresponser i 5 { r i1, r i2, . . . , r iJ}, that is, a ranking of thechoice set in descending order of preference. The probabil-ity of a given survey response may then be expressed asPr(r i) 5 Pr[Ui(r i1) . Ui(r i2) . . . . . Ui(r iJ)]. Theprobability of this preference ordering may be decom-posed as

Pr@Ui~ri1! . Ui~ri2! . . . . . Ui~riJ!#

5 Pr@Ui~ri1! . Ui~rij ! for j 5 2, . . . ,J# Pr@Ui~ri2!

. Ui~rij ! for j 5 3, . . . ,J# . . . Pr@Ui~ri ,J21! . Ui~riJ!#.

Hence, theJ-dimensional survey response describing theorder of preferences is equivalent toJ 2 1 binary state-ments of which alternative is preferred, given the censoringof more-preferred elements of the full choice set.

A. Ordered Probit7

Assume that individualsi 5 1, 2, . . . , N are asked tospecify an ordered measure of preferencesk 5 0, 1, . . . ,K

2 It is advisable to include an additional attribute such as truck restric-tions because it keeps respondents from focusing exclusively on price-time tradeoffs to the exclusion of other important considerations. (SeeCalfee and Winston (1993).)

3 Respondents whose reported commute times were 51–60 minutes or61–70 minutes were asked to evaluate packages that had a sixty-minutemaximum commute time. Respondents whose reported commute timeswere 21–30 minutes, 31–40 minutes, or 41–50 minutes were asked toevaluate packages that had a forty-minute maximum commute time.

4 The survey was administered in December 1993 by Alison-Fisher, Inc.,a marketing consulting firm in Southfield, Michigan. Demographic char-acteristics, including income, were obtained from the survey respondents.The stated preference studies in MVA Consultancy (1987) and others werebased on surveys that are comparable to the one used here. These studiesobtained credible estimates of modal choice parameters. We also note thatthe problems that frequently arise in contingent valuation studies are notrelevant here. In our analysis, people are asked to rank something that theyare familiar with. The question sequence is not a factor. There are genuinetradeoffs of money and automobile commuting attributes, and respondentsare not given the opportunity to give a single response that they perceiveto be socially appropriate. To be sure, it is possible for respondents torespond strategically to stated preference questions, especially if they havepolicy implications. As explained below, we address this problem bypresenting some choices that do not have public policy implications.

5 The relevant portion of the survey read: “A ‘SMART BOX’ is availablethat can automatically guide you to the best route during your commute.Your commute will take less time, because the box will guide you toshorter or less crowded roads, or guide you around trouble spots.”

6 We did not include a scenario that explicitly asked respondents toassume that the toll revenue would be used to support a general reductionin income or property taxes because this assumption would probably beviewed with considerable skepticism.

7 For a full discussion of this model, see Greene (1993).

TABLE 1.—SCENARIOS

Scenario Description

1 “smart cars”: no tolls, no congestion component2 “smart cars”: no tolls3 toll roads: unspecified use of revenues4 toll roads: revenues to maintenance and construction5 toll roads: revenues to state highway fund6 toll roads: revenues to mass transit7 toll roads: revenues to the poor8 public toll roads: unspecified use of revenues9 private toll roads: unspecified use of revenues

10 public toll roads: revenues to maintenance and construction11 private toll roads: revenues to maintenance and construction12 toll roads: flat traffic growth, unspecified use of revenues13 toll roads: rapid traffic growth, unspecified use of revenues

THE REVIEW OF ECONOMICS AND STATISTICS700

for alternativesj 5 1, 2, . . . , J. Assume the populationshares the utility functionyj 5 b9xj 1 e j, whereyj is theutility associated with alternativej , xj is the vector ofattributes associated with alternativej , b is a vector ofunknown parameters, ande j is a random disturbance. Utilityis unobserved. The preference measure,kj, based on ratingsor rankings assigned to alternatives, is assumed to be arough categorization of the utility associated with an alter-native. Let there be an unobserved categorization ofyj suchthat

kj 5 0 if yj # 0

kj 5 1 if 0 , yj # m1

kj 5 2 if m1 , yj # m2

···

kj 5 J if mJ21 , yj.

The set ofm’s will be estimated along withb resulting inJ 2 2 unobserved threshold parameters to estimate.

Assume thate j is normally distributed. Then

Pr~kj 5 0! 5 F~2b9xj!

Pr~kj 5 1! 5 F~m1 2 b9xj! 2 F~2b9xj!

Pr~kj 5 2! 5 F~m2 2 b9xj! 2 F~m1 2 b9xj!

···

Pr~kj 5 J! 5 1 2 F~mJ21 2 b9xj!,

whereF is a cumulative normal distribution function. Thelikelihood function to be maximized is

ln L~b! 5 Oj51

J Ok5j

ln @F~mj 2 b9xk! 2 F~m1 2 b9xk!#.

Conceptually, this ordered model appears to fit well withthe way ratings or rankings data are generated. Ratings data,however, could be seriously flawed if respondents start withdifferent “anchors” in the ratings exercise.8 For example, arespondent who generated ratings of 2, 4, 7, and 7 for fourpackages could have a utility function (up to a lineartransformation) that is identical to that of a respondent whogave ratings of 5, 7, 10, and 10. When the data are aggre-gated across subjects, a common ordering is estimated, andwith it a common anchoring strategy. If respondents hadsimilar tradeoffs but different anchoring strategies, the es-timation process would be unable to discriminate betweendiverse tradeoffs and diverse anchoring strategies. Esti-

mated tradeoffs could therefore be highly unreliable even ifrespondents’ underlying preferences were very similar.

Alternatively, one could analyze the rankings data, butthis raises a different potential problem because the behav-ioral assumptions underlying the ordered probit model maybe inconsistent with the nature of the choice process. If arespondent rank-orders ten packages from 1 to 10, the probitmodel assumes that the utility of each package falls into aspecific utility interval. But, when the data are aggregated,the model estimates only a single set of boundaries for theten utility intervals, and uses those boundaries for all re-spondents. Thus, the model assumes that all respondentsperceive approximately the same utility differences betweenalternatives. In all likelihood, however, respondents willtend to have preferences that “bunch” some packages to-gether and space others apart. On the other hand, unevenutility spacings tend to be “smoothed” as respondents’orderings are aggregated; thus, the spacing assumptionimplied by the ordered probit model may not have a largeeffect on parameter estimates.

B. Rank-Ordered Logit9

Assume again the random utility specification,yj 5b9xj 1 e j. Rank-ordered logit assumes that the disturbanceparameter in the random utility function takes on the ex-treme value distribution. The model makes full use of allranking information by repeatedly applying the multinomiallogit model to an “exploded” data set. Each choice setconsists of a ranked choice and the lower-ranked alterna-tives.

The probability that a given rank ordering will be ob-served has the closed-form solution

Pr@U~r1! . U~r2! . . . . . U~rJ!# 5 Ph51

J21 eb9x~rh!

¥m5hJ eb9x~rm! ,

where x(r h) is the vector of attributes of the alternativerankedh in the ordering. Given an independent sample ofNindividuals facing independent and identically distributede j, the log-likelihood function to be maximized is

L~b! 5 Oi51

N

ln FPh51

J21 eb9x~r ih!

¥m5hJ eb9x~r im!G

5 Oi51

N Oh51

J21

b9x~rih! 2 Oi51

N Oh51

J21 Fln Om5h

J

eb9x~r im!G,wherex(r ih) represents the attributes of the alternative thatindividual i assigned in rankingh.

8 It may be possible to mitigate the anchoring problem by standardizingratings, but this is likely to be unsuccessful when a survey allows for awide range of ratings.

9 For a full discussion of this model, see Beggs, Cardell, and Hausman(1981).

ECONOMETRIC ISSUES IN ESTIMATING STATED PREFERENCE DATA 701

Rank-ordered logit may appear not to suffer from orderedprobit’s “spacings” problem because it is a purely ordinalmodel that makes no assumptions about utility intervals. Fora given choice set, all the unchosen (lower-ranked) alterna-tives simply provide lower utility than the chosen (highest-ranked) alternative. But, as in other econometric contexts,the difference between logit and probit models is not great.Logit models assume that the ratio of choice probabilitiesdepends on the ratio of the choices’ utilities but not on anyother alternative’s utility (the well-known independencefrom irrelevant alternatives property). Thus, the logit modelimposes its own spacings restriction, which, for example,could make it appear that a particular alternative has muchgreater relative utility than it actually has simply because it isconsistently chosen over a slightly less attractive alternative.

Because we do not have theoretical grounds for choosingbetween rankings or ratings data, or between rank-orderedlogit or ordered probit, we explored these alternative ap-proaches in estimating our stated preference models.Whether one uses ratings or rankings data, ordered probitestimates a utility function in which the coefficient for eachinfluence (price, congested time, uncongested time, andwhether trucks are on the road) indicates the effect on utilityof a change in the attribute. When respondents are asked torate packages on a 1-to-10 scale (with 10 indicating thegreatest satisfaction), we expect the sign for price, traveltime, and the dummy variable indicating trucks on the roadto be negative because higher prices and times and trucks onthe road should bring lower numerical and thus worseratings. When respondents are asked to rank packages, withthe most-favored package ranked number 1, and so on, theexpected signs are positive because higher values of thevariables bring higher numerical and thus worse rankings.Finally, rank-ordered logit also estimates a utility functionin which the coefficient for each variable indicates the effecton utility of a change in the variable. The expected signs ofthe coefficients in this model are negative because highervalues of the variables reduce the likelihood of a better(numerical) ranking.

IV. Estimation Results

The final survey responses were first checked to see thatrespondents gave rational orderings. In particular, we foundthat respondents always rated and ranked a “dominated”alternative below a “dominating” alternative.10

We begin with maximum likelihood parameter estimatesof the ordered probit model based on the ratings data.11 (Seetable 2.) Although most of the parameter estimates arestatistically reliable, the coefficients for travel time andtrucks on the road always have counterintuitive positivesigns, and the price coefficients have counterintuitive posi-tive signs in many scenarios. These unsatisfactory estimatessuggest that the “anchoring” problem described above—inwhich different subjects with similar preference orderingsgive very different ratings—can prevent one from obtainingcredible estimates of stated preferences based on ratingsdata.

We obtained much more encouraging results by estimat-ing the rank-ordered logit model.12 (See table 3.) Thecoefficients for price and congested travel time have theexpected negative sign and are statistically reliable. Thecoefficients for uncongested travel time also have the ex-pected negative sign, and are statistically reliable in themajority of scenarios. But the magnitude of these coeffi-cients is roughly one-third the magnitude of the coefficientsfor congested travel time, which suggests that uncongested

10 For example, package 2 in the forty-minute maximum scenarios($0.35 price, thirty-minute travel time, and trucks on the road) was alwaysrated and ranked below package 5 ($0.35 price, twenty-minute travel time,and trucks on the road). Dominated alternatives are included to check forrational orderings and to increase estimation precision (Calfee & Winston,1998).

11 We estimated models that explicitly included income by interacting itwith other variables, but this did not lead to changes in our findings. Wealso specified separate travel-time coefficients for discrete intervals oftravel time, but this did not lead to improvements in the model.

12 The ordered probit parameter estimates based on rankings data arevery similar to the rank-ordered logit estimates; thus, they are not reportedhere (but are available upon request).

TABLE 2.—ORDERED PROBIT PARAMETER ESTIMATES BASED ON RATINGS

Scenario

Price (cents) Congested Time (min.) Uncongested Time (min.) Trucks on the RoadNumber of

Observations**Coefficient T-statistic Coefficient T-statistic Coefficient T-statistic Coefficient T-statistic

1* 20.00014 20.79769 0.02332 15.59593 0.39858 5.49473 7282 20.00020 20.93515 0.00257 1.04682 0.04778 8.91076 0.29730 3.52064 7153 20.00028 21.40107 0.00373 1.55602 0.04915 9.41988 0.29438 3.57416 7804 0.00070 3.41868 0.01576 6.50115 0.07971 15.25957 0.35713 4.62509 8325 0.00016 0.67520 0.01242 4.52609 0.06290 10.68649 0.45001 5.53061 7156 0.00087 4.06561 0.02571 10.12432 0.04809 8.86867 0.59595 8.08880 8717 0.00057 2.43798 0.01847 6.67712 0.05325 9.01764 0.58688 7.37693 7418 0.00065 2.92814 0.01750 6.61883 0.05936 10.49238 0.53008 6.82138 7809 20.00129 25.40192 0.00243 0.90374 0.03829 6.66306 0.42253 5.38669 806

10 0.00083 3.55627 0.01454 5.24214 0.05783 9.74965 0.53599 6.56403 71511 20.00115 24.97608 0.00584 2.19702 0.04130 7.25558 0.29933 3.70753 76712 0.00055 2.34376 0.01982 7.13605 0.05889 9.91407 0.59202 7.20590 70213 20.00206 27.62923 0.00568 1.90053 0.04261 1.90053 0.26777 3.07566 676

* Travel time in scenario 1 was not subdivided into congested and uncongested components. For this scenario, the travel-time coefficient is the effectof total travel time.** Number of observations equals the number of respondents times the number of packages placed in order (513).

THE REVIEW OF ECONOMICS AND STATISTICS702

time is not perceived to be as onerous as congested time.13

Finally, the presence of trucks on the road has a statisticallyinsignificant effect in most scenarios.

Differences and similarities among the parameter esti-mates are more apparent when we use them to calculatecommuters’ WTP to reduce travel time. WTP estimates areobtained as the ratio of the (congested) travel time coeffi-cient to the price coefficient, multiplied by 60 to yield anhourly value. The calculations, presented in table 4, showthat commuters’ WTP as a fraction of their wage based onthe ordered probit model using ratings is very unstable,and even takes on implausible negative values for somescenarios.

In contrast, the WTP/wage estimates based on the rank-ordered logit model and the ordered probit model usingrankings are stable and nearly identical. In addition, they areprecisely estimated as indicated by the standard errors of theWTP estimates. Of substantive importance is that, accordingto these estimates, the average commuter does not appearwilling to pay much to reduce congested travel time underany scenario. Average WTP per hour ranges from 14% to27% of the average gross hourly wage, with an average overthe entire sample of ordered probit and rank-ordered logitestimates of 18% and 19% respectively—much lower thanmost of those based on transportation mode choice androute choice models that cluster around 50% (Small, 1992;Miller, 1989). In all likelihood, we have found that medium-to long-distance automobile commuters have a low value oftravel time because those commuters who have a high valueof time are not in our sample by virtue of their residentialand workplace location decisions that result in a shortcommute and/or one with little congestion. (For furtherdiscussion, see Calfee and Winston (1998).)

V. Mixed Logit Analysis of Ordered Data

Although ordered probit and rank-ordered logit modelsare easy to estimate, they impose restrictions on the orderingof preferences that arise from the implicit assumption thatthe stochastic component of a respondent’s utility functionis uncorrelated across alternatives. If errors are correlated,then parameter and WTP estimates are inconsistent. Wetherefore relax the assumption of uncorrelated errors byestimating our utility functions with an econometric model,mixed logit (Brownstone & Train, 1999), that allows errorsto be correlated.

Such correlation could arise because of preference heter-ogeneity along the price and travel-time dimensions. Mixedlogit captures this correlation by introducing stochasticterms caused by deviations from mean tastes and allowingthese terms to be correlated across an individual’s alterna-tives (packages). Although the choice process assumes util-ity-maximizing behavior, we do not know how this processis made; thus, we cannot model other unobserved sources ofcorrelation (unrelated to price and time) that would affect anindividual’s rankings.

Mixed logit assumes that estimates of utility functioncoefficients are randomly distributed across the population.The output of the model consists of parameters that char-acterize the probability distribution of the coefficients in-cluding a population mean and a standard deviation frommean tastes; the latter coefficient is part of an error term thatis correlated across alternatives.

The random utility function isyij 5 b9ixij 1 e ij , whereyij

is the utility that individuali associates with alternativej ,andbi is unobserved for each member of the population andis randomly distributed with densityf(bi uu*), where u*represents the true parameters describing the distribution.The other source of variation,e ij , is unobserved and distrib-uted i.i.d. extreme value (as in a simple logit model), anduncorrelated withbi andxij .

13 Based on a route choice model, MVA Consultancy (1987) and othersfound that the value of travel time in congested traffic was as much as 50%higher than in free flow.

TABLE 3.—RANK-ORDERED LOGIT PARAMETER ESTIMATES BASED ON RANKINGS

Scenario

Price (cents) Congested Time (min.) Uncongested Time (min.) Trucks on the RoadNumber of

Observations**Coefficient T-statistic Coefficient T-statistic Coefficient T-statistic Coefficient T-statistic

1* 20.00946 236.730 20.05697 233.032 20.28299 23.241 7282 20.00853 232.039 20.07765 227.477 20.02876 24.620 20.25993 22.564 7153 20.00627 225.486 20.05394 219.974 20.01985 23.382 20.11505 21.206 7804 20.00839 230.914 20.06049 222.068 20.01158 21.909 20.12178 21.317 8325 20.00974 230.266 20.06382 220.398 20.00937 21.377 20.06960 20.703 7156 20.00814 228.798 20.05014 217.423 20.02805 24.531 20.12845 21.468 8717 20.00960 230.127 20.05983 218.803 20.01678 22.464 20.14362 21.507 7418 20.00832 228.078 20.04694 215.528 20.00932 21.416 20.16088 21.706 7809 20.00886 229.718 20.05807 219.125 20.01801 22.777 20.08248 20.891 806

10 20.00828 225.930 20.04376 213.958 20.01674 22.500 0.02449 0.251 71511 20.00920 229.552 20.04780 216.143 20.01044 21.613 20.02255 20.238 76712 20.01002 230.740 20.06121 219.427 20.02505 23.724 20.07746 20.777 70213 20.01087 230.257 20.05826 217.120 20.01525 22.112 0.10241 0.996 676

* Travel time in scenario 1 was not subdivided into congested and uncongested components. For this scenario, the travel-time coefficient is the effectof total travel time.** Number of observations equals the number of respondents times the number of packages placed in order (513).

ECONOMETRIC ISSUES IN ESTIMATING STATED PREFERENCE DATA 703

A further specification of this model is that the coefficientvector of the random utility function takes on the formbm 5 b 1 hi, whereb represents the population mean, andhi represents individual deviations from average tastes inthe population. Then, utility may be represented asyij 5b9xij 1 hixij 1 e ij . Under this specification, the stochasticcomponent of utility,hixij 1 e ij , is correlated across alter-natives through the attributes in the model (that is, themodel does not impose IIA).

The conditional probability that individuali will choosealternativej is

O ij ~bi! 5eb9i xij

¥j51J eb9i xij

.

The unconditional probability that a given alternative willbe selected requires one to integrate the conditional proba-bility over all possible values ofbi, and it depends on theparametersu* defining the distribution ofbi. This uncon-ditional probability is

Q ij ~u* ! 5 E eb9i xij

¥j51J eb9i xij

f~biuu* !db.

Let r i represent a given rank ordering as before. Theconditional probability of observing a given rank ordering is

Pr@U~r1! . U~r2! . . . . . U~rJ!ubi# 5 Ph51

J21 eb9i xij ~rh!

¥m5hJ eb9i xij ~rm! .

The unconditional probability of observing this ranking is

Pr@U~r1! . U~r2! . . . . . U~rJ!uu* #

5 E Ph51

J21 eb9i xij ~rh!

¥m5hJ eb9i xij ~rm! f~biuu* !dbi.

The objective is to estimateu*, the parameters defining thedistribution of coefficientsbi. The log-likelihood functionto be maximized is

L~u! 5 Oi51

N

ln E Ph51

J21 eb9i xij ~rh!

¥m5hJ eb9i xij ~rm! f~biuu!dbi.

The integral has no closed-form solution. Hence, wefollow the procedure in Revelt and Train (1998) by simu-lating this probability and estimating the simulated log-likelihood function. Simulations were performed using 100draws. (Using 1,000 draws did not lead to any materialchanges in the parameter estimates.)

TABLE 4.—VALUE OF CONGESTEDTIME BASED ON DIFFERENT ESTIMATION METHODS

Scenario

Value of Congested Time (dollars/hour)(Standard errors in parentheses)a Value of Congested Time as a Proportion of Hourly Wage

Ordered ProbitBased on Ratings

Ordered ProbitBased on Rankings

Ordered LogitBased on Rankings

Ordered ProbitBased on Ratings

Ordered ProbitBased on Rankings

Ordered LogitBased on Rankings

1* 2$97.07** $2.92 $3.61 2499% 15% 19%(125.74) (.34) (.21)

2 2$7.92** $5.21 $5.47 236% 24% 25%(14.79) (.54) (.37)

3 2$7.86** $5.47 $5.16 239% 27% 26%(10.29) (.68) (.48)

4 $13.49 $4.48 $4.33 63% 21% 20%(5.25) (.50) (.36)

5 $47.80** $4.02 $3.93 273% 23% 22%(71.08) (.48) (.35)

6 $17.65 $3.69 $3.70 90% 19% 19%(5.25) (.56) (.38)

7 $19.47 $3.55 $3.74 82% 15% 16%(9.34) (.51) (.35)

8 $16.10 $3.35 $3.38 83% 17% 18%(6.85) (.50) (.38)

9 2$1.14** $3.83 $3.93 25% 17% 17%(2.09) (.50) (.37)

10 $10.51 $3.49 $3.17 48% 16% 14%(4.46) (.54) (.40)

11 2$3.06 $3.40 $3.12 215% 16% 15%(2.39) (.47) (.34)

12 $21.64 $3.47 $3.67 102% 16% 17%(10.52) (.51) (.34)

13 2$1.65 $3.06 $3.21 28% 14% 15%(1.47) (.45) (.33)

Average $2.15 $3.84 $3.88 10% 18% 19%

* Travel time in scenario 1 was not subdivided into congested and uncongested components. For this scenario, the travel-time coefficient is the effectof total travel time.** These values of time estimates are based on price and/or time coefficients that are not statistically different from zero.a Standard errors for the ratio of two normally distributed random variables are based on an approximation given by M. G. Kendall and A. Stuart,Advanced Theory of Statistics(1977).

THE REVIEW OF ECONOMICS AND STATISTICS704

We assumed that the utility function for the populationtook on log-normal distributions for the coefficients on priceand congested travel time. These attributes represent thesource of potential error correlation across alternatives. (Theparameters for uncongested travel time and trucks on theroad were assumed to be fixed.) The motivation for assum-ing log-normal distributions was that the parameters wouldbe strictly positive (note the parameters were premultipliedby 21 to reflect the fact that they actually measure disutilityas in the conventional rank-ordered logit model) and that theratio of these parameters would have a log-normal distribu-tion and be strictly positive.14 The population distribution ofwillingness to pay therefore has a well-behaved distribution.

The estimated parameters and the WTP based on themixed logit specification are presented in table 5.15 Note thatthe table presents estimates of the mean,m, and standarddeviation, s, associated with the random parameters forprice and congested travel time. The estimates ofm have theexpected sign (recall these have been premultiplied by21)and are statistically reliable. The estimates ofs tend to berelatively small and in some cases are not statisticallysignificantly different from zero, indicating little variation incommuters’ valuation of price and congested travel time.The estimates for uncongested travel time are statisticallyinsignificant, but the estimates for truck restrictions, incontrast to previous findings, are statistically significant inseveral scenarios.

The estimates of WTP based on the estimated coefficientsof the mean of the random parameters for price and con-gested travel time are very similar to the WTP estimatesbased on the conventional rank-ordered logit and orderedprobit models using rankings, indicating that the “spacings”restriction imposed by the conventional ordered models

may not lead to serious imprecisions if one is primarilyinterested in estimating the ratios of particular coefficients.16

As before, the low estimates of travelers’ WTP—roughly$4.00 per hour—imply that the average commuter does notappear willing to pay much to reduce automobile traveltime. The small coefficients of the standard deviation ofprice and congested travel time indicate few travelers in thesample are willing to pay considerably more than theaverage. The WTP distribution, summarized by the 90thpercentile and 10th percentile estimates of WTP, is fairlytight. Indeed, the 90th percentile WTP exceeds $7.00 in onlyone scenario. An important implication of this finding is thatthe attraction of high-occupancy toll (HOT) lanes, intro-duced in southern California and Houston, Texas, for mo-torists with the greatest WTP, may be very limited if tollsgreatly exceed the average motorist’s WTP.17

VI. Partially Ranked Models

Stated preference models assume that respondents care-fully rank all alternatives instead of paying closer attentionto their top choices. Hausman and Ruud (1987) havepointed out that, if respondents pay less attention to lowerranks, this could introduce noise into the data that will resultin inconsistent parameter estimates. If this has occurredhere, the parameter and WTP estimates should be sensitiveto the number of ranks used in estimation.

14 Our findings were not affected when we assumed that the randomparameters took on a normal instead of a log-normal distribution.

15 The model was estimated in GAUSS using a GAUSS program writtenby Kenneth Train.

16 A similar conclusion is reached by Brownstone and Train (1999).17 High-occupancy toll lanes allow a single-occupant vehicle to pay a

toll to drive in a less-congested lane that is normally reserved for carpoolers. The San Diego HOT lane does attract some commuters who payas much as $4.00 to save ten minutes, which implies their hourly WTP is$24. This value is not necessarily inconsistent with our findings. It mayreflect the WTP of higher-income travelers who face unusual time pres-sure on that particular day. (On other days, their WTP may be muchlower.) In any case, given that most commuters do not choose to use HOTlanes and that the vast majority who do pay much less than $4.00, thisvalue represents the very upper tail of the WTP distribution of allautomobile commuters who face congestion.

TABLE 5.—MIXED LOGIT PARAMETER ESTIMATES BASED ON RANKINGS

Scenario

ln Price (cents) ln Congested Time (min.) UncongestedTime (min.) Trucks on Road Mean

Willingnessto Pay

90thPercentile

WTP

10thPercentile

WTPmSt. dev.

m sSt. dev.

s mSt. dev.

m sSt. dev.

s Coeff. St. Dev. Coeff. St. Dev.

1* 1.592 0.078 0.178 0.087 1.075 0.112 0.203 0.155 0.399 0.055 $3.71 $5.06 $2.532 1.532 0.050 0.121 0.074 1.433 0.067 0.001 0.473 0.048 0.089 0.297 0.035 $5.47 $6.34 $4.663 1.187 0.095 0.061 0.327 0.999 0.139 0.347 0.074 0.049 0.094 0.294 0.035 $5.29 $7.81 $3.164 1.834 0.075 0.268 0.069 1.529 0.091 0.159 0.162 0.080 0.153 0.357 0.046 $4.64 $6.59 $2.975 1.962 0.065 0.250 0.067 1.495 0.087 0.165 0.156 0.063 0.107 0.450 0.055 $3.93 $5.52 $2.566 1.699 0.078 0.068 0.242 1.184 0.136 0.360 0.115 0.048 0.089 0.596 0.081 $3.83 $5.73 $2.247 1.868 0.074 0.179 0.100 1.420 0.113 0.190 0.124 0.053 0.090 0.587 0.074 $3.97 $5.36 $2.748 1.786 0.076 0.251 0.095 1.160 0.142 0.451 0.132 0.059 0.105 0.530 0.068 $3.66 $6.21 $1.669 1.843 0.066 0.127 0.118 1.445 0.089 0.149 0.17720.018 0.028 20.082 0.009 $4.11 $5.18 $3.13

10 1.814 0.092 0.249 0.084 1.180 0.153 0.205 0.26420.016 0.025 0.024 0.003 $3.35 $4.81 $2.1011 1.854 0.069 0.139 0.122 1.183 0.127 0.213 0.20120.010 0.016 20.023 0.002 $3.17 $4.25 $2.2112 1.983 0.063 0.230 0.069 1.488 0.093 0.118 0.17220.025 0.037 20.000 0.008 $3.78 $5.09 $2.6313 2.077 0.090 0.332 0.090 1.489 0.110 0.061 0.39220.015 0.021 0.102 0.010 $3.53 $5.13 $2.16

m ands represent the estimated parameters for the coefficients specified to be randomly distributed;s is not to be confused with the standard deviation of the parameter estimatem. The standard deviations ofthe parameter estimatesm ands are recorded in the columns headed “St. dev.”

* Travel time in scenario 1 was not subdivided into congested and uncongested components. For this scenario, the travel-time coefficient is the effectof total travel time.

ECONOMETRIC ISSUES IN ESTIMATING STATED PREFERENCE DATA 705

Our parameter estimates are based on respondents’ rank-ings of thirteen packages. We reestimated a conventionalrank-ordered logit model based on respondents’ rankings oftheir eight and five most-preferred packages, and found thatrespondents’ WTP is not sensitive to the depth of therankings included in estimation. (See table 6.) By includingall the rankings data, we have increased the precision of,rather than biased, our parameter estimates.18

VII. Conclusion

In the absence of market transactions that reveal con-sumer preferences, economists are increasingly using datafrom stated preference surveys to estimate those prefer-ences. We have raised a number of methodological issuesrelated to the econometric analysis of these data and ex-plored them in the context of estimating the value ofautomobile travel time. Using conventional ordered econo-metric models and mixed logit, we have found that theaverage WTP to save automobile travel time is low and doesnot exhibit much variation among motorists. Although ourfindings using data on respondents’ rankings of alternativeswere robust, we found that extreme caution should be usedin estimating stated preferences based on respondents’ ratings.

REFERENCES

Beggs, Steven, Scott Cardell, and Jerry Hausman, “Assessing the PotentialDemand for Electric Cars,”Journal of Econometrics17 (Septem-ber 1981), 1–19.

Bhat, Chandra, “A Heteroscedastic Extreme Value Model of IntercityTravel Mode Choice,”Transportation Research B29 (December1995), 471–489.

Brownstone, David, and Kenneth Train, “Forecasting New Product Pen-etration with Flexible Substitution Patterns,”Journal of Economet-rics 89:1–2 (1999), 109–129.

Calfee, John E., and Clifford Winston, “The Consumer Welfare Effects of Lia-bility for Pain and Suffering:An ExploratoryAnalysis,”Brookings Paperson Economic Activity: Microeconomics,no. 1 (1993), 133–196.

———, “The Value of Automobile Travel Time: Implications for Con-gestion Policy,” Journal of Public Economics69 (September1998), 83–102.

Greene, William H.,Econometric Analysis,2nd ed. (Englewood Cliffs,NJ: Prentice Hall, 1993).

Hausman, Jerry, and Paul Ruud, “Specifying and Testing EconometricModels for Rank-Ordered Data,”Journal of Econometrics34(1987), 83–104.

Miller, Ted, “The Value of Time and the Benefit of Time Saving,” UrbanInstitute working paper (1989).

MVA Consultancy, Institute of Transport Studies, University of Leeds, andTransport Studies Unit, University of Oxford,The Value of TravelTime Savings,Policy Journals, Newbury, Berks, U.K. (1987).

Revelt, David, and Kenneth Train, “Mixed Logit with Repeated Choices:Households’ Choices of Appliance Efficiency Level,” thisREVIEW80:4 (1998), 647–657.

Small, Kenneth A.,Urban Transportation Economics(Philadelphia: Har-wood Academic Publishers, 1992).

APPENDIXFORTY-MINUTE MAXIMUM PACKAGES:

PackagePrice

(One Way)Travel Time

(Minutes One Way)Trucks onthe Road?

1 $0.00 40 Trucks2 $0.35 30 Trucks3 $0.70 30 Trucks4 $1.00 30 Trucks5 $0.35 20 Trucks6 $0.70 20 Trucks7 $1.35 20 Trucks8 $2.00 10 Trucks9 $3.35 10 Trucks

10 $0.35 40 No trucks11 $0.70 30 No trucks12 $1.75 20 No trucks13 $4.00 10 No trucks

PackagePrice

(One Way)

Travel Time(Minutes One Way)

Trucks onthe Road?

CongestedTime

UncongestedTime

TotalTime

1 $0.00 30 10 40 Trucks2 $0.35 20 10 30 Trucks3 $1.35 20 10 30 Trucks4 $0.35 10 20 30 Trucks5 $0.70 10 20 30 Trucks6 $0.35 10 10 20 Trucks7 $1.35 10 10 20 Trucks8 $2.00 0 10 10 Trucks9 $3.35 0 10 10 Trucks

10 $0.35 30 10 40 No trucks11 $0.70 20 10 30 No trucks12 $1.75 10 10 20 No trucks13 $4.00 0 10 10 No trucks

SIXTY -MINUTE MAXIMUM PACKAGES:

PackagePrice

(One Way)Travel Time

(Minutes One Way)Trucks onthe Road?

1 $0.00 60 Trucks2 $0.50 45 Trucks3 $1.00 45 Trucks4 $1.50 45 Trucks5 $0.50 30 Trucks6 $1.00 30 Trucks7 $2.00 30 Trucks8 $3.00 15 Trucks9 $5.00 15 Trucks

10 $0.50 60 No trucks11 $1.00 45 No trucks12 $2.50 30 No trucks13 $6.00 15 No trucks

18 If inconsistent estimates do arise from respondents’ inattention tolower ranks, then one can obtain consistent estimates using an estimatordeveloped by Hausman and Ruud (1987). Bhat (1995) has also developeda method that can address this problem.

TABLE 6.—ORDERED LOGIT WILLINGNESS TO PAY; PARTIAL RANKINGS

Scenario Fully Ranked Top 8 Ranks Top 5 Ranks

1* $3.62 $3.72 $4.312 $5.43 $5.35 $5.263 $5.17 $5.20 $5.884 $4.34 $4.92 $5.175 $3.84 $4.20 $4.176 $3.72 $3.81 $4.337 $3.85 $3.92 $3.918 $3.35 $3.41 $3.939 $3.88 $4.11 $4.77

10 $3.22 $3.61 $3.9911 $3.10 $3.45 $3.8512 $3.71 $4.01 $3.7713 $3.19 $3.33 $3.63

* Travel time in scenario 1 was not subdivided into congested and uncongested components. For thisscenario, the travel-time coefficient is the effect of total travel time.

THE REVIEW OF ECONOMICS AND STATISTICS706

APPENDIX—(CONTINUED)

PackagePrice

(One Way)

Travel Time(Minutes One Way)

Trucks onthe Road?

CongestedTime

UncongestedTime

TotalTime

1 $0.00 45 15 60 Trucks2 $0.50 30 15 45 Trucks3 $2.00 30 15 45 Trucks4 $0.50 15 30 45 Trucks5 $1.00 15 30 45 Trucks6 $0.50 15 15 30 Trucks7 $2.00 15 15 30 Trucks8 $3.00 0 15 15 Trucks9 $5.00 0 15 15 Trucks

10 $0.50 45 15 60 No trucks11 $1.00 30 15 45 No trucks12 $2.50 15 15 30 No trucks13 $6.00 0 15 15 No trucks

ECONOMETRIC ISSUES IN ESTIMATING STATED PREFERENCE DATA 707