Valuation of Travel Time Reliability in an Extended Expected Utility Theory Framework Zheng Li David A. Hensher* John M. Rose The University of Sydney Faculty of Economics and Business Institute of Transport and Logistics Studies Newtown, NSW 2006, Australia Tel: +61 (0)2 9351 0169 Fax: +61 (0)2 9351 0088
[email protected] [email protected] [email protected] 26 November 2009 Version 3.7 *corresponding author
Abstract Valuation of travel time savings (VTTS) is a critical measure in transport infrastructure appraisal, traffic modelling and network performance. It has been recognised for some time that the traditional measure of VTTS should be complemented by a valuation of travel time reliability (VOR). An alternative approach, proposed in this paper, promotes the view that a single revised estimate of VTTS that accounts for the observed variability in travel time for a specific trip and the associated likelihood of such variations in travel times occurring, might be a more sensible way of accounting for the amount that an individual is willing to pay to save time; that is the notion of separate estimates of VTTS and VOR may be unnecessary. The most widely used approach for valuing travel time reliability is the mix of Random Utility Maximisation (RUM) and Expected Utility Theory (EUT) (i.e., linear utility specification with linear probability weighting). We extend the EUT approach (i.e., a non-linear utility specification with a linear probability weighting function), by applying a non-linear probability weighting function to accommodate choice made under risk, referred to as Extended EUT. The empirical findings recognise the extent of attitudes towards risk in estimates of ‘reliability embedded VTTS’ or REVTTS.
Keywords: travel time reliability, passenger transport, willingness to pay, choice under risk, expected utility theory, non-linear probability weighting, extended expected utility model
1 Introduction Time savings is generally recognised as the most important user benefit in transport appraisal, typically contributing over 60 percent of user benefits (Hensher 2001a). In calculating the time benefits in monetary units, a value of travel time savings (VTTS) has to be obtained. de Jong el al. (2009) amongst others have pointed out that an important user trip benefit that is often neglected in transport appraisal is the valuation of travel time reliability. This is out of line with the growing number of studies which have investigated the significance of travel time reliability in traveller behaviour (see e.g., Jackson and Jucker 1982; Small 1982; Bates et al. 2001, and Li et al. 2009 for a review)1. Some of these studies obtained higher values for reducing travel variability than for reducing scheduled journey time or for average travel time (see e.g., Asensio and Matas 2008; Batley and Ibáñez 2009). Within a choice theoretic framework, that is commonly used to obtain empirical estimates of the values of travel time savings and reliability the most popular specification assumes that an individual acts as if they are a utility maximiser, and that the inability of the analyst to observe and measure all influences on utility maximising behaviour engenders a theory of Random Utility Maximisation (RUM) (Amador et al. 2005). However, RUM assumes that the individual’s choice is made under certainty (Batley and Ibáñez 2009), despite the inability of the analyst to observe and measure all influences on utility (which engenders the randomness). In recognition that travel time variability introduces uncertainty at the attribute level, other theoretical platforms have been proposed and introduced as a way to accommodate travel time reliability. Since the early 1990s, a number of studies have incorporated Expected Utility Theory (EUT) into the representation of travel time reliability, as a way of recognising individual travel choice under uncertainty (see e.g., Senna 1994; Noland and Small 1995). This model, known as Maximum Expected Utility (MEU), involves a choice process in which the alternative with the highest value of expected utility is preferred. Since Noland and Small’s seminal paper in 1995, this has become the standard approach in travel time reliability studies (see e.g., Small et al. 1999; Bates et al. 2001; Hollander 2006; Asensio and Matas 2008). Such a willingness to pay estimate can be obtained from suitable revealed preference data, using a choice model that identifies the trading between time and other factors including monetary outlays.2 However in recent years, stated choice methods have increasingly been used and are now the dominating data paradigm, largely due to the difficulties of identifying real market situations where analysts can observe and measure the trade-offs between the attributes required to establish measures of WTP (e.g., Calfee and Winston 1998; Hensher 2001a, 2001b, 2006; Jara-Diaz and Guevara 2003; Amador et al. 2005, Hess et al. 2008). The purpose of this paper is to review previous and current methods in understanding travel time reliability, and estimating values of reliability using stated choice methods. We discuss the limitations of these approaches, and present an alternative approach 1
Similar to VTTS studies, travel time reliability studies predominantly use stated choice data, although some studies employed RP data (see e.g., Small et al. 2005; Lam and Small 2001). 2 See e.g., Brownstone et al. 2003 and Steimetz and Brownstone 2005.
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that i) addresses respondent risk attitude, ii) accounts for nonlinearity in probability weighting, and iii) integrates the value of travel time savings and the value of reliability into what we refer to as a reliability embedded value of travel time savings (REVTTS). As far as we are aware, this is the first study on the valuation of travel time reliability (or variability) that estimates a model form that is nonlinear in the parameter set, specifically accommodating risk attitude in the levels of the attributes, and the perceptual processing of occurrence probabilities3 for attributes displaying varying levels over repeated trip activity for a trip with a common purpose and origin and destination (e.g. the regular weekly commute). This paper is organised as follows. The following section introduces pioneering travel time reliability studies developed within a utility maximisation framework (e.g., Jackson and Jucker 1982; Small 1982). This is followed by the contributions that focus on choosing a travel outcome with the highest expected utility (or the lowest expected disutility) (e.g., Small et al. 1999; Bates et al. 2001). Although EUT assumes a non-linear utility functional form to identify individuals’ risk attitudes, this has not been well addressed in previous travel time reliability studies. To fill this gap, a non-linear utility function is specified. Within an EU framework, we also apply a non-linear probability weighting function (referred to as extended EUT) to identify how induced probabilities in stated choice experiments are transformed. Unlike previous studies where the value of mean travel time and the value of reliability were estimated separately, we integrate the two values into a willingness to pay for total travel time experienced over repeated trip activity (mean and variability), referred to as reliability embedded value of travel time savings (REVTTS). Using a 2008 stated choice data set from Australia for commuters choosing amongst alternative trip attribute packages for car travel, we estimate a series of models based on the alternative behavioural paradigms and compare the findings. Conclusions are drawn along with major recommendations.
2 Travel Time Reliability and Random Utility Maximisation Early travel time reliability studies, developed within a utility maximising framework such as Jackson and Jucker (1982), proposed a mean-variance form in which utility, U, is defined as a function of the usual (or mean) travel time and the variance, assuming that travellers trade off time against variability (variance). They postulate that variability directly leads to disutility, similar to the mean travel time, and hence time variability can be represented by the variance or standard deviation4 of travel time (i.e., the mean-variance approach). The mean-variance model was also employed by Pells (1987) and Black and Towriss (1993).The objective is to minimise the sum of these elements (equation 1). _
U T V (T )
(1) _
where is a parameter measuring the influence of the variance in travel times; T is the usual or mean travel time; and V (T ) is the variance of travel time. 3
Referred to as under- and over-weighting in prospect theory. Some SP studies also use the coefficient of variation (i.e., standard deviation divided by mean travel time) in the utility function (see e.g., Noland et al. 1998). 4
3
Unlike the mean-variance model, in which variability is the direct source of disutility, Small (1982) introduced the concept of schedule delay, defined as the difference between actual arrival time and official start time, and he posited that utility would be decreased if arriving early (SDE) or arriving late (SDL) relative to a planned arrival time. Small proposed the scheduling model as an alternative way to understand travellers’ departure time choices in order to satisfy on-time arrival, as given in equation (2). U T SDE SDL DL
(2)
T is travel time, SDE is schedule delay early, SDL is schedule delay late, DL is a dummy variable equal to 1 when there is a SDL and 0 otherwise; and the estimated parameters ( , , , and ) are assumed to be negative. The above studies did not consider the stochastic characteristic of travel time variability. That is, given travel time variability, it is assumed that it is not possible for travellers to anticipate their travel times, and consequently different travel times have an associated probability of occurrence. Therefore, there should be a distribution of travel times rather than a fixed travel time, and hence travel choice is no longer made under certainty.
3 Recognising Expected Utility Expected Utility Theory (EUT) has been extensively applied in a number of fields such as experimental economics, environmental economics, health economics, and in travel time reliability studies after the 1990s. Unlike RUM models, which typically assume a linear-additive utility function for the observed or representative consumer component (i.e., U k ( k xk ) , where k are the estimated parameters and xk are the attributes that underlie individual preferences), EUT models postulate a non-linear functional form, for example, U x r where r is an estimate parameter which explains respondents’ attitudes towards risk ( r 1 : risk averse; r 1 : risk neutral (which implies linear functional form); r 1 : risk loving) (see Harrison and Rutström 2009). A basic EUT model is given in equation (3).
E (U ) m ( pm xmr )
(3)
where E (U ) is the expected utility; m (=1,…,M) are the possible outcomes for an attribute and m 2 ; pm is the probability associated with the mth outcome; and xm is the value for the mth outcome. Noland and Small (1995) extended Small’s scheduling model to accommodate travel time variability through the incorporation of EUT. Travel time (T) is no longer deterministic but has a distribution dependent on departure time ( th ) (Bates et al. 2001). Hence, the expected utility of the scheduling model is expressed as equation (4), where the possible delay or early arrival with respect to the preferred arrival time are modelled separately, and their consequences are measured by separate
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parameters5. That is, expected utility ( U (th ) ) is a function of the expected travel time ( E[T (th )] ), the expected schedule delay early ( E[ SDE (th )] ), the expected schedule delay late ( E[ SDL(th )] ), and the probability of experiencing a late arrival ( PL (th ) ). E[U (th )] E[T (th )] E[ SDE (th )] E[ SDL(th )] PL (th )
(4)
Senna (1994), Polak (1987) and Small et al. (1999) also used an expected utility, E(U) framework to analyse traveller responses to travel time variability, within the meanvariance framework developed by Jackson and Jucker (equation 1). A typical meanvariance specification under EUT is shown in equation (5). E (U ) T E (T r ) SD SD(T ) C C
(5)
where E (U ) is expected utility, T , SD and C are the estimated parameters for the expected travel time ( E (T ) ), the standard deviation of travel time ( SD(T ) ), and travel cost (C) respectively and r is the risk attitude parameter. Despite the appeal of EUT, in most travel time reliability studies which adopt EUT, a linear functional form was used (i.e., r = 1; see Small et al. 1999; Bates et al. 2001; Hollander 2006; Asensio and Matas 2008; Batley and Ibáñez 2009) with equal occurrence probabilities for each described level of travel time. Senna (1994) used a non-linear utility specification to investigate travel choice; however he imposed an assumption (rather than estimated the relevant parameter) on this non-linearity (i.e., the value of r): 0.5 for commuters with fixed arrival time, 1.4 for commuters with flexible arrival time, and 1.4 for non-commuters. Whether those assigned values are able to reflect respondents’ true attitudes is unknown. Polak (1987) and Polak et al., (2008) also applied EUT in investigating travel choice in the face of travel time variability, using Bates et al.’s data in the 2008 paper within an MNL framework with a constant absolute risk aversion (CARA), U (1 e ax ) / a . Polak (1987), cited in Senna (1994), used the form U e ax .
EUT not only changes the utility function for travel time reliability, it also leads to significant challenges in the way that stated choice (SC) experiments have to be designed to capture travel time variability. In studies that do not incorporate a EUT probability weighting function, travel time variability is typically presented as the extent and frequency of delay relative to normal travel time (which we refer as a Type 1 experiment). For example Jackson and Jucker (1982) ask respondents to make a choice between a journey that always takes 30 minutes and a journey which has a shorter time, but a possibility of 5-minute delay once a week (Table 1). Table 1: Stated Choice task from Jackson and Jucker (1982), Type 1 Card 1
5
Usual time: Possible delays:
Route 1 30 minutes None
Route 2 20 minutes 5 minutes a week
The travel time destination needs to be assumed for estimating the values of parameters, which is often assumed to be equi-probable (see e.g., Small et al. 1999; Bates et al. 2001).
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In recognising that travel time does vary, a series of arrival times (normally five or 10 levels), rather than the extent and frequency of delay, have been considered in recent SC experiments (referred to as type 2) (see, e.g., Senna 1994; Noland and Small 1995; Small et al. 1999; Hollander 2006; Asensio and Matas 2008; Batley and Ibáñez 2009). An example of the type 2 design is given in Figure 1, which has five outcomes related to the travel time attribute, giving information to calculate the expected value.
Figure 1: Stated Choice task from Small et al. (1999), Type 2
The two types of models6 (equations 4 and 5) dominate the current transport literature on valuation of travel time reliability. The value of reliability (VOR) is defined as the travellers’ WTP for a unit reduction in variability (shown as the standard deviation) in travel time (i.e., SD / C ). An important output from these studies is the reliability ratio, defined as the marginal rate of substitution between average travel time and travel time variability (i.e., SD / T , when r = 1). The estimated reliability ratios vary across studies, with some as high as 2.1 (Batley and Ibáñez 2009) and others as low as 0.1 (Hollander 2006). de Jong et al. (2009) suggest that the ratio for car travel is 0.8 and 1.4 for public transport. Bates et al. (2001) suggest that the ratio should be around 1.3 for car travel and no more than 2.0 for public transport. The monetary values derived from the scheduling model are often referred to as scheduling costs for arriving early and late respectively.
4 The Appeal of Non-Expected Utility Models In EU models, the probabilities of different outcomes presented in a choice experiment are directly used to weight utility. EUT can be criticised for its failure to account for the way in which the probabilities offered in experiments are transformed by respondents in recognition of the perceptual processing of such probabilities, which entails elements of over and under-weighting, especially at the extremes of the 6
A third type model is the mean lateness model, which is fast becoming the ‘standard’ approach for analysing reliability for passenger rail transport in the UK (Batley and Ibáñez 2009), where travel unreliability is measured by the mean lateness at departure and/or arrival, while the mean earliness (i.e., negative lateness) is not considered.
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occurrence distribution. Given this, non-linear probability weighting was introduced into a number of non-EU models either cumulatively (e.g. Rank-Dependent Utility Theory (RDUT) and Cumulative Prospect Theory (CPT)) or separably (e.g., Original Prospect Theory (OPT) as an instrument to explain the violation of the independence axiom7 of EUT revealed by the Allais paradox (Allais 1953), i.e., the induced probabilities in experiments can be over(under)weighted. A probability weighting function widely used in behavioural economics (Tversky and Kahneman 1992) is given in equation (6).
pm
w( pm )
1
(6)
[ pm (1 pm ) ]
w( pm ) is the probability weight function; pm is the probability associated with the mth outcome for an alternative with multiple outcomes, and is the probability weighting parameter. If 1 , then w( p ) p , which implies EUT linear probability weighting. A common finding from controlled laboratory experimental studies is that people tend to overweight outcomes with lower probabilities, and underweight outcomes with higher probabilities (see e.g., Tversky and Kahneman 1992; Camerer and Ho 1994; Tversky and Fox 1995). In our empirical application, we incorporate a non-linear probability weighting function (shown in equation (6)) into the EU framework (see equation (3)) separably, and refer to the EU model with non-linear probability weighting as an Extended EU (EEU) model, shown in equation (7).8
EE (U ) m [ w( pm ) U ]
(7)
5. Empirical Assessment The empirical focus herein is on estimating the non-linear probability weighed travel time reliability profiles, and deriving the willingness to pay for reliability embedded value of travel time savings (REVTTS). The data are drawn from a study undertaken in Australia in the context of toll vs. free roads, which utilised a stated choice (SC) experiment involving two SC alternatives (i.e., route A and route B) pivoted around the knowledge base of travellers (i.e., the current trip). The trip attributes associated with each route are summarised in Table 2.
7
That is, if two acts (alternatives) have the same consequence given a particular state, the preference between those two acts is independent of that state with the common consequence. 8 We are not implementing any prospect theoretic model with referencing, but simply using the nonlinear probability weighting idea in an extended version of EUT. Some authors (e.g., Hess et al. 2008 and Jou et al. 2009) have implemented the referencing feature of prospect theory but ignored the decision weighting and also assumed risk neutral attitudes.
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Table 2: Trip Attributes in Stated Choice Design9 Routes A and B Free flow travel time Slowed down travel time Stop/start/crawling travel time Minutes arriving earlier than expected Minutes arriving later than expected Probability of arriving earlier than expected Probability of arriving at the time expected Probability of arriving later than expected Running cost Toll Cost
Each alternative has three travel scenarios - ‘arriving x minutes earlier than expected’, ‘arriving y minutes later than expected’, and ‘arriving at the time expected’. Each is associated with a corresponding probability10 of occurrence to indicate that travel time is not fixed but varies from time to time. For all attributes except the toll cost, minutes arriving early and late, and the probabilities of arriving on-time, early or late, the values for the SC alternatives are variations around the values for the current trip. Given the lack of exposure to tolls for many travellers in the study catchment area, the toll levels are fixed over a range, varying from no toll to $4.20, with the upper limit determined by the trip length of the sampled trip. The variations used for each attribute are given in Table 3. Table 3: Profile of the Attribute range in the SC design Attribute Free Flow time Slowed down time Stop/Start time Min. Early Min. Late Prob arriving Early Prob arriving On-time Prob arriving Late Running costs Toll costs
Level 1 -40% -40% -40% 5% 10% 10% 20% 10% -25% $0.00
Level 2 -30% -30% -30% 10% 20% 20% 30% 20% -15% $0.60
Level 3 -20% -20% -20% 15% 30% 30% 40% 30% -5% $1.20
Level 4 -10% -10% -10% 20% 40% 40% 50% 40% 5% $1.80
Level 5 0% 0% 0% 60% 15% $2.40
Level 6 10% 10% 10% 70% 25% $3.00
Level 7 20% 20% 20% 80% 35% $3.60
Level8 30% 30% 30% 45% $4.20
A survey was designed and implemented in late 2008 to capture a large number of travel circumstances, to determine how each individual trades-off different levels of travel times and trip time reliability with various levels of proposed tolls and vehicle running costs in the context of tolled and non-tolled roads. Sampling rules were imposed on three trip length segments: 10 to 30 minutes, 31 to 45 minutes, and more than 45 minutes (capped at 120 minutes). Sampling by the time of day that a trip commences was also included, defining the peak11 as trips beginning during the period 7-9 am or 4.30-6.30pm. All non-peak trips are treated as off peak in the internal quota counts. There are three version of the experimental design depending on the trip length, with each version having 32 choice situations (games) blocked into two subsets of 16 9
The descriptive statistics for the time and probability variables are given in Appendix A. The probabilities are designed and hence exogenously induced to respondents, similar to other travel time reliability studies. 11 The way we handle trips that are partly in the peak: a trip is peak if 60 percent or more of the trip falls within the peak period. 10
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choice situations. In generating the designs, the free flow, slowed and stop/start times were set to five minutes if the respondent entered zero for their current trip. It is important to understand that the distinction between free flow, slowed down and stop/start/crawling time is solely to promote the differences in the quality of travel time between various routes – especially a tolled route and a non-tolled route, and is separate to the influence of total time. An example of a choice scenario is given in Figure 2. The first alternative is described by attribute levels associated with a recent trip; with the levels of each attribute for Routes A and B pivoted around the corresponding level of actual trip alternative. In total, 300 commuters were sampled for this study. The experimental design method of D-efficiency used herein is specifically structured to increase the statistical performance of the models with smaller samples than are required for other lessefficient (statistically) designs such as orthogonal designs (see Rose and Bliemer 2008 and Rose et al. 2008).
Figure 2: Illustrative stated choice screen
Fieldwork took place in November 2008, sampling residents of a Metropolitan area in Australia. Table 4 shows the quotas for each trip type, and the number of interviews achieved, after data cleaning. All segments, with the exception of long distance peak hour commuters were close to, or exceeded the specified quota. This outcome reflects the nature of random sampling and is not a bias. The socioeconomic profile of the data is given in appendix A, together with a descriptive overview of choice experiments attributes. Table 4: Quotas, recruitments and final achievement numbers Peak Hours
Quota Off Peak Hours
Total
Peak Hours
Achieved Off Peak Hours
Total
10 to 30 minutes
60
40
100
61
50
111
31 to 45 minutes
60
40
100
71
32
103
46 to 120 minutes
60
40
100
51
15
66
Total
180
120
300
183
97
280
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Across the 16 choice scenarios that each respondent evaluated we varied the probability of early, on-time and late arrival (see Table 2). In contrast in designs such as Small et al. (1999) and Asensio and Matas (2008), the probability is fixed as five travel times for an alternative, each with an occurrence probability of 0.2; and in Bates et al. (2001) and Hollander (2006) the occurrence probability is not mentioned, implicitly assuming that travel times are equally distributed..
6. Model Estimation and Valuation of Reliability Embedded Travel Time Savings Instead of treating mean travel time and reliability separately, we develop models that integrate these two components of a travel time distribution, based on the alternative theoretical frameworks set out in previous sections. We present a partial EU model (a linear utility function form, i.e., r = 1), a full EU model and the Extended EU model, beginning with a linear form for the utility function and the probability weights, extending it by introducing nonlinearity in the utility function and the probability weights. All models are multinomial logit.12 6.1 Linear Utility Function with Linear Probability Weighting (Model 1)
A linear utility function with a linear probability weighting function is given in equation (8).
E (U ) On ( POn OnT ) E ( PE ET ) L ( PL LT ) Cost Cost
Age Age TollascTollasc
(8)
POn is the probability of arriving on time, shown to respondents; OnT is the on-time travel time; PE is the probability of arriving early shown to respondents; ET is the actual travel time for early arrival scenario; PL is the probability of late arrival shown to respondents; LT is the actual travel time for late arrival; and age is a person’s age in years13. Tollasc is the dummy variable to indicate whether a specific alternative is a tolled road. On , E , L , age and tollasc are parameters to be estimated. The modelling (Multinominal logit (MNL) model)14 results are given in Table 5.
12
Models 2 and 3 are highly nonlinear and complex to estimate, even within an MNL framework. Ongoing research is investigating extensions to incorporate preference and scale heterogeneity. 13 We investigated a number of socioeconomic effects (e.g., income, gender) but did not find any statistically significant except age). 14 In this paper, we only used simple MNL models. More sophisticated models such as random parameter models will be used in our future papers.
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Table 5: Model 1: Linear utility function with EUT probability weighting Variable Reference OnTime Early Late Cost Tollasc Age No. of observations Information Criterion AIC15 Log-likelihood
Parameter 0.5285*** -0.5861*** -0.9160*** -0.7853*** -0.2543*** -0.3162*** 0.0049*
(t-ratio) (4.3) (-12.6) (-13.2) (-13.0) (-11.7) (-3.3) (2.0) 4480 6835.5 -3410.764
Note: ***, **, * = Significance at 1%, 5%, 10% level.
All parameters are significant at the 99 percent confidence interval, with the exception of Age (90 percent). The estimated parameter for the Reference (status quo) specific constant is positive, which suggests, after accounting for the observed influences, that sampled respondents prefer their current trip relative to two stated choice alternatives, with this tendency stronger as the age of a respondent increases (0.0049). Tollasc is negative, which indicates that, on average after accounting for the time and cost of travel, other factors bundled into the idea of a ‘toll road quality bonus’ are less desirable for a tolled route than a non-tolled route, mainly due to the lack of exposure to tolls for our sampled respondents. The marginal utilities of travel time associated with arriving early, later and on time are given in equations (9)-(11): E (U ) E PE ( ET )
(9)
E (U ) L PL ( LT )
(10)
E (U ) On POn (OnT )
(11)
Hence, the willingness to pay for total time including mean travel time and variability, or reliability embedded VTTS (REVTTS), can be estimated as equation (12). E (U ) E (U ) E (U ) ( ET ) ( LT ) (OnT ) E PE L PL On POn REVTTS E (U ) Cost Cost
15
(12)
Akaike information criterion: AIC=-2×log-likelihood + 2×K , where K is the number of parameters. The smaller AIC indicates a better model fit.
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The value of REVTTS is influenced by the probabilities of arriving early, late and on time. The REVTTS has a mean of Au$16.95 per person hour and a standard deviation of Au$0.98 per person hour. 6.2 Non-linear Utility Function with linear probability weighting (Model 2)
Model 2 jointly estimates all the parameters in the value function containing the attribute parameters and the risk attitude parameter. For the non-linear utility specification, we adopt the constant relative risk aversion (CRRA) model form rather than CARA (i.e., exponential specification). CRRA postulates a power specification (e.g., U x r ), which has been widely used in behavioural/experimental economics and psychology (see e.g., Tversky and Kahneman 1992; Holt and Laury 2002; Harrison and Rutström 2009) and often delivers “a better fit than alternative families” (Wakker 2008, p.1329). We estimate the constant relative risk aversion (CRRA) model form as a general power specification (i.e., U x1 r /(1 r ) ), more widely used than the simple xr form (Andersen et al. 2009; Holt and Laury 2002), as given in equation (13). E (U ) On [ POnOnT1 r /(1 r )] E [ PE ET1 r /(1 r )] L [ PL L1T r /(1 r )] Cost Cost Age Age TollascTollasc (13)
Compared with equation (8), one extra parameter needs to be estimated, the risk attitude parameter. If (1-r) = 1, equation (13) collapses to a linear utility function (i.e., equation 8). The model results are summarised in Table 6. Table 6: Model 2: Non-linear utility functional form under EUT with risk attitude Variable Reference Risk OnTime Early Late Cost Tollasc Age No. of observations Information Criterion: AIC Log-likelihood
Coefficient 0.5038*** 0.2873** -0.1828** -0.2517** -0.2314** -0.2552*** -0.3130*** 0.0053**
(t-ratio) (4.1) (2.4) (-2.1) (-2.3) (-2.3) (-11.8) (-3.3) (2.1)
4480 6831.5 -3407.751
Note: ***, **, * = Significance at 1%, 5%, 10% level.
All estimated parameters are significant at the 95 or 99 percent confidence interval. This non-linear model delivers similar behavioural responses to the previous linear model. However, in terms of model fit, the non-linear model is slightly better than the previous linear model (AIC: 6831.5 vs. 6835.5), although the log-likelihood improves from -3410.764 to -3407.751 (with 1 degree of freedom difference). The risk attitude parameter is 0.2873 (with a t-ratio of 2.4). (1-r)
