isttt-2004-0058 a model of daily time use allocation

A Model of Daily Time Use Allocation Using the Fractional Logit Methodology 1

ISTTT-2004-0058

A MODEL OF DAILY TIME USE ALLOCATION USING FRACTIONAL LOGIT METHODOLOGY Xin Ye and Ram Pendyala, Department of Civil and Environmental Engineering, University of South Florida, Tampa, Florida, USA

INTRODUCTION Studies in travel demand model and travel behavior are increasingly focusing on modelling people’s use of time and the effects of time use patterns on activity and travel characteristics. This tendency mainly arises from the shift in transportation planning emphasis from that of infrastructure construction to transportation systems management, travel demand management (TDM), and transportation control measures (TCM). The traditional trip-based travel demand models focus only on the trips made by individuals. Such a narrow perspective on activitytravel behavior is insufficient to evaluate policies aimed at managing and altering activitytravel patterns and choices. Thus trip-based models are likely to fail in capturing the full implication of various TDM-type policies (Pas, 1993; Schofer, 1993). Policies aimed at discouraging auto use may be considered beneficial from a pure vehicle trip reduction-based perspective. However, forcing solo-driving commuters to take other travel modes may not only suppress vehicle trips but also impose constraints and restrictions on their activity scheduling and travel throughout the day. Congestion may be mitigated, but a greater cost may possibly be imposed on individual commuters (Kitamura, et al., 1997). People’s time use patterns are useful reflections of the relationship between transportation and the quality of their life. The total time available each day, i.e., 24 hours per day, is identical for each person; however, the satisfaction or happiness of a person is likely to depend on how the 24 hour day is spent in terms of time allocation to various activities. For example,

2 16th International Symposium on Transportation and Traffic Theory

commuters may have to sacrifice time with family or at recreational activities when faced with a congested commute. A transportation system enhancement that relieves congestion may afford individuals the ability to spend more time participating in activities that they enjoy and derive greater satisfaction. A modelling approach will be a powerful tool for evaluating transportation planning policies if it is capable of estimating individuals’ time use allocation to travel and various activities given a set of explanatory variables, such as TDM policy indicators, demographic and socio-economic characteristics, land use characteristics, and level-of-service variables. People’s allocation of time to various activity types and to travel can have a profound impact on the frequency, timing, duration, and location of trips. Some people may like to allocate more time to activities outside the home, others may prefer allocating more time to activities inside the home, and still others may enjoy spending time travelling (inside a vehicle). These preferences and resulting time allocation patterns directly impact travel characteristics because time is a finite resource that is bounded and consumed. Thus, people who prefer to spend more time at activities (locations) outside home will have less time available for indoor activities or travel. The remainder of this paper is organized as follows. Following this introductory section, a brief review of previous work on daily time allocation modelling is provided. This is followed by a detailed description of the modelling methodology adopted in this paper. The paper then includes a brief description of the data set used in the study followed by the data description, model estimation results and sample numerical simulation results. Finally, the paper presents conclusions and directions for future research.

MODELLING DAILY TIME ALLOCATION The modelling of daily time allocation to various activity categories and travel has been of much interest to travel behavior researchers in the recent past for the reasons noted in the introductory section of the paper. Covariance-based structural equations modelling methods have been used extensively to model activity and travel time use patterns. For example, Kuppam and Pendyala (2001) applied structural equations models to analyze commuters’ daily activity and travel patterns. Commuters’ daily time allocation to work and in-home and out-of-home recreation and maintenance activities, as well as travel variables such as trip frequency by purpose, are modelled using a series of structural equations models. Trade-offs in time use allocation among various activity types and travel were explicitly identified using


the model results. Golob and McNally (1997) use structural equations to analyze activity and travel participation among household heads in multi-adult households. While structural equations approaches offer powerful methods for simultaneously modelling multiple endogenous variables, they do not guarantee that the time allocation predictions will fall within certain reasonable limits or that they will add up to the available time resource (say, 24 hours). In addition, marginal diminishing effects of exogenous variables are typically not captured in structural equations models. Bhat and Misra (1999) model allocation of weekly discretionary time between in-home and out-of-home locations and between weekdays and weekend days by developing model formulated as a continuous utility-maximizing resource allocation problem. They adopt a utility formulation that makes it possible to model the logarithm of the ratio of two time allocations as a linear regression estimated using the standard ordinary least squares (OLS) approach. Their approach is quite elegant in that it uses a well-defined parametric utility function for time use and classic microeconomic theory of utility maximization. However, the use of the logarithm of the ratio of two time allocations as the dependent variable may entail the exclusion of zero observations from the model estimation process. The exclusion of zero observations may result in bias, particularly in the context of individuals’ daily time allocation, where a considerable portion of the sample may not travel outside home and allocate zero time to out-of-home activities on the survey day. This was unlikely to be an issue in the work by Bhat and Misra (1999) because they modelled weekly discretionary time allocation, where the number of zero observations is likely to be minimal. Yamamoto and Kitamura (1999) formulated the time allocation problem as a doubly censored tobit using a dual-state utility function that accommodates both zero and non-zero time allocation observations. They apply the methodology to the modelling of discretionary time allocation to weekdays and weekend days (similar to Bhat and Misra, 1999). The doublycensored tobit approach is used to estimate a model of the ratio of discretionary time allocation between in-home and out-of-home locations and between weekdays and weekend days (also similar to Bhat and Misra, 1999). The doubly-censored tobit model has also been employed by Lee, et al., (2004) in the analysis of time use patterns within trip chains. In their work, as actual time allocations were modelled directly (as opposed to the logarithm of time allocation ratios), diminishing marginal effects of exogenous variables are not accommodated in the model. Meloni et al. (2004) applied the Nested Tobit (N-Tobit) model, with a hierarchical sequence of two equations, to describe how individuals allocate their discretionary time between inhome and out-of-home activities and between trips and activities. Similar to the work by Bhat


and Misra (1999) and Yamamoto and Kitamura (1999), the logarithm of time allocation ratios are treated as dependent variables in the models. The models taking log-ratios as dependent variables have a common disadvantage in that it is computationally difficult to recover the expected value of the dependent variable, which is important for model applications. Consider a log-odds model where ln(y /1-y) = xβ + v, where two time fractions, y and 1 − y , are involved. By back-calculating y, one

obtains y =

exp( xβ + v ) , where v is the random error term in the linear model. 1 + exp( xβ + v )

Then, E ( y | x ) = ∫

∞

−∞

exp( xβ + v ) f ( v | x ) dv . A distributional assumption on v and intensive 1 + exp( xβ + v )

computations are required to approximate the generalized integral for recovering the expected value of y (Papke and Wooldridge, 1996). To overcome the limitations noted above, this paper develops and applies the fractional logit modelling methodology in the context of time use analysis. Papke and Wooldridge (1996) proposed this approach to analyze the participation rate in pension plans, where there are only two fractions: a participating fraction and a non-participating fraction where both fractions are bounded in the interval from 0 to 1 and must sum up to 1. By analogy, a binary fractional logit modelling approach can be extended to accommodate multiple fractions as dependent variables. Sivakumar and Bhat (2002) extended the fractional logit modelling approach from its original bi-fractional situation to the multi-fractional situation and applied it to analyze statewide commodity flows. If treated as fractions of individuals’ daily time resource, the time spent on travel and various activities can be reasonably modelled within the framework of the fractional logit approach, where all dependent fractional variables are bounded in the interval from 0 to 1 and must sum up to 1. The quasi-maximum likelihood method, as opposed to the traditional non-linear least squares method, is employed to estimate efficient model parameters. In addition, the modelling methodology allows one to easily recover the expected values of dependent variables, accommodate zero observations, and capture the diminishing marginal effects of exogenous variables on the limited time resource. In this paper, two fractional logit models – one for workers and one for non-workers – are estimated to analyze person-level daily time allocation patterns to out-of-home activities, outof-home travel, and in-home activities. This trinomial allocation problem is modelled using person-level data derived from the 2000 Switzerland Microcensus data in which a sample of individuals completed 24 hour travel diaries documenting all of the trips and activities outside home that they pursued during the travel diary day.


MODELLING METHODOLOGY As mentioned in the previous section, time allocation to travel and various activities may be treated as fractions of the total daily time resource that is available to an individual. Modelling such fractional dependent variables can be conveniently done within the I

framework of the fractional logit modelling methodology. Let 0 ≤ yqi ≤ 1 and ∑ yqi = 1, i =1

where q is an index that represents the individual and i is an index that represents the activity or travel type (q = 1, 2, ……N where N is the sample size and i = 1, 2, ……I where I is total number of various activity or travel types for each individual). The population model is assumed to be E(yqi|xq1, xq2,…, xqI) = G(β, xq1, xq2,…, xqI), where xqi is a vector of explanatory variables for yqi and β are vectors of model parameters. For ensuring E(yqi|xq) to lie between 0 and 1 and sum of yqi across i to be exactly 1, the multinomial logit may be used as a convenient functional form for G( ). Thus, one may have,

E ( y qi | xqi ) = G( y qi | x qi ) =

exp( x qi β i ) I

∑ exp( xqj β j )

(1)

j =1

Alternatively, one may have,

y qi =

exp( x qi β i ) I

∑ exp( x j =1

qj

βj)

+ ε qi

where

E (ε qi | xqi ) = 0

(2)

and xqiβi represents a time allocation function for activity type i. In this context, it should be noted that Gliebe and Koppelman (2002) developed a proportional shares model of daily time allocation and applied it to analyze the joint activity participation between two household members. Although the proportional shares model appears similar to the fractional logit model, it is not the same. In the proportional shares model, the gumbel-distributed error terms, εi, are a part of the utility functions, Vi = Xβ + εi. In the fractional logit model, the random error term appears solely outside the logit formula and takes no specific probabilistic distributional form other than having a conditional expectation of zero given the set of exogenous variables. The random error term here does not have a behavioral interpretation and serves purely as a random disturbance in the model formulation analogous to Bhat and Misra (1999). The non-linear least squares (NLS) method is one approach that may be used to estimate the model parameters β. However, Papke and Wooldridge (1996) have shown that NLS estimates are not statistically efficient when applied to the fractional logit model. They state that the


quasi-maximum likelihood method, as proposed in Gourieroux, et al. (1984), needs to be applied to estimate efficient model parameters. The Bernoulli log-likelihood function may be given by (Papke and Wooldridge, 1996 and Sivakumar and Bhat, 2002),

     exp( x qi β i )  LL = ∑ ∑ y qi log  I (3)  q =1 i =1  ∑ exp( x qj β j )   j =1  The log-likelihood function depicted in equation (3) offers a convenient functional form for maximization due to the logit specification used to represent time allocation fractions. In addition, this function is a member of the linear exponential family (LEF) and hence the quasi-maximum likelihood estimator (QMLE) is statistically consistent and asymptotically normally distributed provided equation (1) holds, regardless of the distribution of yqi conditional on xqi. N

I

It is worth noting that QMLE differs from conventional maximum likelihood estimates (MLE). MLE procedures entail maximizing the exact log-probability or the log-probability density function (with respect to model parameters), which is usually derived from the distributional assumption on the phenomenon under investigation. For instance, in the traditional multinomial logit model, the exact probability function for random discrete observations is derived from the assumption of gumbel-distributed random utility and the notion of utility maximization. On the other hand, QMLE procedures entail maximizing the N

equation ∑ log{l[ y q , f ( x q , β )]} , where y q = f ( x q , β ) + ε q and l(u,m) belongs to the Linear q =1

Exponential Family (LEF). There is no specific distributional assumption on the random term, εq. If l(u,m) is not a member of the Linear Exponential Family (LEF), QMLE will be invalid because parameters can not be consistently estimated (Gourieroux, et al. 1984). For the QMLE procedures, Gourieroux et al. (1984) show that N ( βˆ − β ) ~ Normal(0, A −1 BA−1 ) as N → ∞ . On the other hand, the asymptotic property of MLE is given by ( βˆ MLE − β ) ~ Normal (0, A −1 ) , where, β stands for true value of model parameters, βˆ for quasi-maximum likelihood estimator, A-1 for negative inverse of Hessian matrix, B for outer product of first derivative of the quasi-log-likelihood function, and N for sample size. The log-likelihood function of the fractional logit model is the same as that of the multinomial logit model when dealing with fractional data (continuous values between 0 and 1). The NLOGIT 3.0 (Greene, 2002) program can be utilized to estimate the parameters of the fractional logit model by replacing the discrete dependent variable with fractional data.

A Model of Daily Time Use Allocation Using the Fractional Logit Methodology 7 However, Aˆ −1 Bˆ Aˆ −1 associated with asymptotic variance estimates are not directly provided; instead, NLOGIT 3.0 provides Aˆ −1 , that can only be used to obtain non-robust variance of the

parameter estimates. and

As

∂LL N = ∑ xqi ( y qi − Gqi ) , then, ∂β i q =1

∂LL ∂LL ' N 2 = ∑ [ xqi ( y qi − Gqi ) 2 ] ∂β i ∂β i q =1

∂ 2 LL N = ∑ [− x qi2 G qi (1 − Gqi )] . One can then derive that covariance-variance matrix of βˆi 2 ∂β i q =1

ˆ − y )2 N (G 2 ˆ −1 −1 −1 qi qi −1 −1 2 −1 ˆ ˆ ˆ ˆ ˆ ˆ is σ i Ai , where Ai and Bi are partitions of A and B for βi, σ i = N ∑ , ˆ ˆ q =1 G qi (1 − G qi )

and Gˆ qi is estimated expectation of yqi , i.e., G( βˆ , xq). To obtain the robust variance for βˆ i , −1 one only need calculate σˆ i2 and then scale up Aˆ i . Diagonal elements extracted from this

scaled matrix are the variances of βˆ i . Therefore, in this study, model parameters were estimated first and the residuals Gˆ qi − yqi were saved. Then the weighted standard errors, σˆ i2 , were calculated and the non-robust variances were scaled up to finally obtain the robust variances and robust t-statistics.

DATA SET AND DATA PREPARATION The data set used for analysis and model estimation is extracted from the Swiss Travel Microcensus 2000. A very detailed description of the survey and the survey sample can be found in Ye and Pendyala (2003). Only a very brief description of the survey sample is provided in this paper in the interest of brevity. The survey respondent sample consists of 27,918 households from 26 cantons in Switzerland. The survey sample was formed by randomly selecting one person over 6 years old from each household with less than 4 household members and two persons over 6 years old from each household with 4 or more members. As a result of this sampling scheme, the person respondent sample consisted of 29,407 persons. All of the persons in the sample were asked to report their travel in a one-day trip diary. The resulting trip data set includes 103,376 trips reported by 29,407 interviewed persons including the possibility of some respondents making zero trips on the survey day. Those zero-trip reporting persons, accounting for 11% of the total sample, are assumed to spend 24 hours on in-home activities. Separate time use expenditure models are estimated for commuters and non-commuters to recognize the differences between these two market segments. In this paper, a commuter is defined as a respondent who reported at least one work trip on the travel survey day while a


non-commuter is defined as one who did not report any work trips. In the remainder of this paper, the term commuter and worker will be used synonymously; likewise, the terms noncommuter and non-worker will be used synonymously. Also, for purposes of this analysis, the total time spent at work is considered exogenous. Although one could argue that an individual chooses the nature of his or her employment and may choose to work full time, part time, etc., the amount of time spent at work is generally fixed by the employer once this choice is made by an individual. Thus, for commuters, the available daily time resource is considered to be equal to 24 hours minus the total time spent at work. On the other hand, for non-commuters, the total time resource available is considered to be equal to the full 24 hours of the day. Thus, the three time use fractions are calculated based on 24 hours minus the total time spent at work for workers. For non-workers, the three time use fractions are calculated based on the full 24 hours. As most children do not necessarily have the freedom to allocate their daily time use and often depend on adults for mobility, models were only developed for the adult sample consisting of those 18 years or older. Table 1. Person Characteristics of Swiss Travel Microcensus 2000 Characteristic Sample Size

Swiss Sample Workers Non-Workers 7978 17057

Age (in years) Young (18~29) Middle (30~59) Old (≥60)

41.5 (Mean) 18.4% 75.6% 6.0%

52.8 (Mean) 13.1% 46.1% 40.8%

#Trips/day Work trips Non-work trips

4.68 1.61 3.07

2.92 0.00 2.92

Daily Time Use (min) Travel duration Work School Shopping Recreation Home Other

101.3 465.4 4.2 11.6 88.5 748.2 2.9

93.4 0.0 41.3 23.1 169.9 1100.0 4.0

Notes: Workers are those who made at least one work trip on the travel survey day. All others are non-workers.

The Swiss Travel Microcensus 2000 is a traditional household travel survey in which respondents were asked to report their trips rather than the full daily time use pattern. However, the time use allocation pattern may be reasonably derived from the trip diary data using information on trip timing and trip purpose. Also, there are instances where the


reported trips are not strictly made within a 24-hour daily cycle. The following rules were applied to derive time use fractions from the trip diary data: 1. The trip ending time is considered to be the starting time of the subsequent activity. The trip start time is considered to be the ending time of the previous activity. The activity type is determined by the purpose of the previous trip (i.e., the trip leading up to the activity). A trip to “work” indicates that the subsequent activity is work and trip to “return home” indicates that an in-home activity follows the trip. 2. The daily cycle is defined as the period from 3:00 AM to 3:00 AM on the next day. If travel is reported to start before 3:00 AM or end after 3:00 AM on the next day, it is assumed to start at 3:00 AM or end at 3:00 AM on the next day respectively. The trips or activities (and their respective durations) recorded outside this period are not taken into account for daily time use analysis in this paper. 3. If the purpose of the first trip is not “return home”, duration from 3:00 AM to the starting time of the first trip is considered as in-home activity time. If the purpose of the first trip after 3:00 AM is “return home”, then the duration from 3:00 AM to the arrival time at home is considered as out-of-home non-work activity. 4. If the purpose of the last trip is not “return home”, then the duration from the end time of the last trip to 3:00 AM on the next day is considered as out-of-home activity time. Table 2. Average Fractional Time Allocations for Workers and Non-Workers Variable In-Home Activity Time Fraction Travel Time Fraction Out-of-Home Non-work Activity Time Fraction

Workers (N=7978) Mean Std Dev 0.777 0.184 0.102 0.081 0.122 0.154

Non-Workers (N=17057) Mean Std Dev 0.777 0.210 0.066 0.082 0.157 0.179

Using the rules noted above, activity and travel durations can be computed and aggregated across three categories (travel, in-home activity, and out-of-home activity) for each individual. Household and person demographic and socio-economic characteristics and work-related information for workers (e.g., total work time, working timing, and commute distance) are merged into this person-level dataset for model development and estimation. Table 1 provides a summary of the person characteristics of the adult respondent samples by workers and non-workers that are used for model estimation.


In this paper, daily time expenditures are classified into three broad categories: out-of-home travel time, in-home activity time, and out-of-home activity time. In the trinomial fractional logit models presented in this paper, these three categories of time use are considered as three fractions of the total available time resource. Table 2 shows the mean and standard deviation of the three time use fractions for 7978 adult workers and 17057 adult non-workers. It should be noted that the fractions for workers are computed based on a denominator equal to 24 hours minus the total time spent at the work activity. For example, if the total work time is assumed to be 8 hours, the mean in-home activity time is (24 – 8) × 0.777 = 12.43 hours, the mean travel time is (24 – 8) × 0.102 = 1.63 hours, and the mean out-of-home non-work activity time is (24 – 8) × 0.122 = 1.95 hours. Similarly, on average, non-workers spend 24 × 0.777 = 18.65 hours on in-home activity, 24 × 0.066 = 1.58 hours on travel, and 24 × 0.157 = 3.77 hours on out-of-home activity. Interestingly, the fractions of available time resources allocated by workers and non-workers to in-home activities are found to be virtually identical (i.e., 0.777). On average, this may be interpreted as non-workers spending around 6 hours more on in-home activities and around 2 hours more on out-of-home activities when compared with workers. This accounts for the typical 8-hour mandatory daily work activity duration. It is also noteworthy that the average daily travel durations of workers and nonworkers are quite similar.

MODEL ESTIMATION RESULTS This section presents results of the model estimation effort. First, model estimation results are presented for the commuter sample and second, for the non-commuter sample.

Model Estimation Results for Workers The model estimation results for the worker sample are presented in Table 3. For purposes of identification, the out-of-home non-work activity is treated as the base fraction and the other two fractions are determined relative to the base. The estimation results yielded results that are generally consistent with expectations. The constant term in the function for in-home activity time allocation is positive implying that inhome activity time allocation (fraction) is the largest of the three fractions. On the other hand, the constant associated with the travel time fraction function is negative reflecting that travel


generally accounts for the smallest fraction of time expenditure among the three categories considered in this paper. The estimated function for in-home activity time allocation fraction indicates that working males are likely to allocate less time to in-home activities relative to working females. This finding may reflect the tendency for female workers to bear a greater share of in-home activities and obligations. Older workers show a positive propensity to allocate larger time fractions to in-home activities and to travel as evidenced by the positive coefficients in these two functions relative to the base alternative. This is consistent with expectations in that older workers may have greater household obligations and hence stay more time in-home and more time travelling to take care of household obligations and chauffeuring. Similarly, married workers and workers with children are found to allocate more time to inhome activities and travel. Auto ownership displays a negative coefficient in both the functions associated with in-home activity time allocation and travel time allocation. In general, greater auto ownership may be associated with higher income levels and the ability to spend more time at out-of-home non-work (shopping and recreational) activities while travelling faster than non-auto owning individuals. Similarly, the variable representing high income shows negative coefficients in both time allocation functions. As expected, Fridays are associated with greater out-of-home non-work activity time allocation as evidenced by the negative coefficients in both equations associated with in-home activity and travel time allocation. The total time spent at work has a positive impact on both in-home activity time allocation and daily travel time allocation. This implies that workers who spend more time at work are likely to spend less time at out-of-home non-work activities as they spend their non-work activity time primarily travelling and at home. Commute distance is found to be negatively impacting in-home activity time allocation and positively impacting travel time allocation. These findings are consistent with expectations as workers with long commutes are likely to spend more time travelling and less time at home. The model also included several variables representing the work schedules of workers. Although these variables were not necessarily yielding significant coefficients, they were retained because they offered plausible interpretations and provided potential policy sensitivity (e.g., flex work hours). In general, these variables offer negative coefficients implying that working full time is associated with shorter in-home activity and travel time allocations. However, it should be noted that the variable representing working hours from 6 AM to 4 PM exhibits the most negative coefficient among the three work schedule variables in both time allocation equations.

Out-of-Home Non-work Activity Time Function = 0. a significant at the 0.10 level; b insignificant at the 0.10 level. All the other variables appear significant at the 0.05 level.

Adj ρ2

ρ2

Variable Constant Age in years Age2 Number of vehicles in household / household size Distance between residence and work place (km) Household Size Monthly household income is above Fr 10000 Person lives in rural area Person lives in a household with children Person is of Swiss nationality Person is highly educated with BA/MS/PhD degree Person is married Person is male Person is retired Person is of Swiss nationality Season is winter Travel is on Friday Travel is on weekend (Saturday or Sunday) Total daily work time in minutes Work starts at 6:00 and ends at 16:00 Work starts at 7:00 and ends at 17:00 Work starts at 8:00 and ends at 18:00 Est. σ1 Est. σ2 Est. σ3

Time Allocation Function

Workers (Sample Size = 7978) In-Home Activity Travel Time Function Time Function Coeff. Robust T Coeff. Robust T 0.8843 11.135 -0.7291 -11.307 0.0149 11.180 0.0083 8.121 -----0.2401 -7.092 -0.1633 -5.930 -0.0009 -1.162b 0.0068 12.509 -----0.1740 -4.480 -0.1049 -3.311 ----0.2101 4.129 0.1727 4.432 --------0.0858 3.146 ---0.2883 -12.015 -------0.2418 -5.528 -0.1186 -3.364 -----0.3018 -8.209 -0.1606 -5.286 ----0.0021 25.053 0.0009 13.128 -0.1023 -1.470b -0.1395 -2.418 -0.0175 -0.378b -0.0594 -1.568b -0.0130 -0.191b -0.0258 -0.468b 0.4185 0.2502 0.4623 0.3902 0.3892

Non-workers (Sample Size = 17057) In-Home Activity Travel Time Function Time Function Coeff. Robust T Coeff. Robust T 1.1752 24.316 -1.3893 -16.353 0.0138 15.591 0.0276 8.522 ---0.0003 -8.315 -0.1897 -8.307 ---------0.0292 -3.248 -0.0863 -2.758 -----0.1024 -4.485 0.0895 2.638 --0.0841 3.091 -0.0674 -2.324 --0.1279 4.506 0.1104 5.186 ---0.1962 -10.296 ---0.0754 -2.260 ---0.1699 -5.967 -----0.1066 -4.776 --0.1509 4.863 -0.0353 -1.830a ------------------0.4959 0.3162 0.4893 0.4013 0.4009

Table 3. Model Estimation Results for Workers and Non-workers



This appears to imply that working earlier hours in the day may contribute to a greater time allocation to out-of-home non-work activities relative to the other later working hours.

Model Estimation Results for Non-Workers Table 3 also provides estimation results for the non-worker sample. Similar to the worker model, the constant term in the function for in-home activity time allocation takes on a positive value whereas the constant term in the function for travel time allocation takes a negative value. Thus, the findings suggest that the greatest fraction of the day is allocated to in-home activities while the smallest fraction is allocated to travel. As in the worker model, out-of-home activity time allocation is set as the base fraction. Several explanatory variables take the same sign as those in the worker model. For example, male non-workers are found to allocate less time to in-home activities relative to female nonworkers. This is once again consistent with expectations that females may be taking on more of the in-home tasks. Fraction of time allocation to in-home activities and travel increases with age, although the fraction allocated to travel decreases beyond a certain peak travel age as evidenced by the negative coefficient associated with the square of the age. Auto availability and high income are associated with a smaller in-home time allocation. Low income individuals are found to allocate less time to travel, possibly because they do not have the disposable income to travel and participate in various activities. The low income may also be representative of low car ownership. Married individuals and those with children are found to allocate more time to in-home stay, possibly due to family obligations. As expected, those who filled the travel diary survey on weekends showed a smaller propensity to allocate time to in-home activities, a finding that is suggestive of the discretionary travel orientation of weekends. The variable representing Friday has a positive coefficient in the daily travel time allocation function. This is consistent with expectations that people will likely engage in more travel on Fridays relative to other weekdays when other household members may be obligated to work and school activities. A few other variables that offered plausible interpretations include education, season of travel, and type of residential location. In the function representing travel time allocation, higher education is associated with greater travel time allocation. On the other hand, individuals appear to allocate less time to travel in the winter season. This is likely due to the more cumbersome nature of travelling during the colder winter months in Switzerland. Also, those residing in rural country areas show a negative propensity to allocate time to travel. This may be reflective of the fewer destination opportunities that are easily accessible for engaging in


out-of-home activities and travel in such areas. This may also be due to lower traffic congestion levels in rural country areas, thus contributing to lower travel times. Overall, it is found that the fractional logit models presented in this paper offer very plausible interpretations and high level of sensitivity to a host of socio-economic, demographic, and work-related policy variables. The robust t-tests show that the variables are, for the most part, statistically significant and capable of explaining time use allocation patterns of individuals. The adjusted ρ2 values are also reasonably consistent with what one might expect from disaggregate models of this type.

SAMPLE NUMERICAL SIMULATION This section presents a small sample numerical example to demonstrate how the fractional logit models presented in this paper can be used to estimate changes in time allocation patterns of individuals in response to changes in explanatory factors and some policy-related variables of interest. Consider a married male who is 40 years old and lives with his spouse and one child. Also, suppose that household income is high and there is one automobile in the household. The auto availability ratio is then 1/3. His expected time use allocation on a work day can be estimated by the fractional logit model presented in the left block of Table 3 including various work-related attributes such as work timing, total work time, and commute distance. The changes in the time use allocation pattern in response to changes in the work schedule may be suggestive of the changes in activity travel patterns that might result from implementation of TDM policies aimed at mitigating peak-period traffic congestion by shifting commuters’ work schedules. Table 4 presents the results of the scenario analysis.

Base Scenario (0) The base scenario assumes that the individual works for 8 hours (480 minutes) with work starting in the period of 7:00~7:59 and ending in the period of 17:00~17:59. This was found to be the most typical work schedule in the worker sample accounting for 13 percent of the observations. The distance between residence and work place is assumed to be 10 km. In this base scenario, the model indicates that he is expected to spend 99 minutes on travel, 105 minutes on out-of-home non-work activity and 756 minutes on in-home activities. These three figures add up to 960 minutes in accordance with the daily disposable time [1440 – 60 × 8


(work time)]. Time allocations obtained from a series of alternative scenarios are compared with the indications from this base scenario.

Scenarios (1) and (2): Effect of Work Schedule In the first scenario, the worker is assumed to start and end work about one hour earlier in the day due to a flexible work hour strategy implemented at the work place. The model developed in this paper can be applied to analyze the policy implications of such an action. Table 4 shows the results of this analysis. As a result of this shift in work time, it is plausible to expect the person to have more discretionary time available in the post-work period (in the evening). This increased time availability may lead to increased level of non-work activity participation in that period. Indeed, the application of the fractional logit model shows the person spending eight additional minutes on out-of-home non-work activities. Seven of these minutes come from in-home activity time allocation and one minute from travel time. The total travel time allocation may decrease modestly, possibly due to reduced exposure to congested traffic as a result of the shift in work schedule. While one may argue that the flex work hour strategy did not really yield much in the way of travel time savings (reduction in travel), one could also argue that the person is able to spend more time at non-work activities which presumably provide some satisfaction and benefit to the individual. In the second scenario, the work schedule is shifted later in the day by about one hour. As expected, this scenario does not yield much change in the total time allocation pattern at all. There is a reduction of two minutes in in-home activity time allocation and a reduction of one minute in out-of-home non-work activity time allocation. The travel time allocation increases by three minutes.

Scenarios (3) and (4): Effect of Total Work Time The third and fourth scenarios help identify the potential impacts of total work duration on time allocation patterns. Many flexible work hour strategies provide workers the ability to spend 9 hour or 10 hour work days with the option of getting a part of or the entire Friday off. In the third scenario, the total work time is increased by one hour while in the fourth scenario, the total work time is decreased by one hour. The 60 minutes of additional time spent at work in the third scenario have to naturally come from time spent travelling, in-home, and out-ofhome at non-work activities. In-home activity duration decreases by 33 minutes, travel time decreases by 11 minutes, and non-work activity time decreases by 16 minutes.

Scenario Variable AGE AUTO_RT DIS_WORK (km) FRIDAY HIGH_INC MARRIED PMALE SWISS WITH_KID WORK_TM (min) WORK6_16 WORK7_17 WORK8_18 In-home activity time (min) Travel time (min) Out-of-home non-work activity (min)

1 40 0.33 10 0 1 1 1 1 1 480 1 0 0 749 (-7) 98 (-1) 113 (+8)

Base 0 40 0.33 10 0 1 1 1 1 1 480 0 1 0 756 99 105

104 (-1)

40 0.33 10 0 1 1 1 1 1 480 0 0 1 754 (-2) 101 (+3)

2

89 (-16)

40 0.33 10 0 1 1 1 1 1 480+60 0 1 0 724 (-33) 88 (-11)

3

124 (+19)

40 0.33 10 0 1 1 1 1 1 480-60 0 1 0 786 (+30) 110 (+11)

4

105 (0)

40 0.33 10+5 0 1 1 1 1 1 480 0 1 0 753 (-3) 102 (+3)

5

Table 4. Sample Numerical Example of Model Application for Scenario Analysis


105 (0)

40 0.33 10-5 0 1 1 1 1 1 480 0 1 0 760 (+3) 95 (-3)

6


The fourth scenario offers very similar results in the opposite direction. Of the 60 minutes less spent at work, a person allocates about one-half to in-home activities as evidenced by the 30 minute increase in in-home activity duration. The total travel time increases by 11 minutes and the out-of-home non-work activity time increases by 19 minutes.

Scenario (5) and (6): Effect of Commute Distance Analyzing the impacts of commute time reduction that may occur in response to an improvement in the transportation system is of much interest due to the potential for induced out-of-home activity and travel engagement. Ideally, one would like to treat commute time as an exogenous variable in the time use allocation modelling context. However, in most travel surveys, it is difficult to isolate pure commute travel time from travel time to and from other activities because of the prevalence of trip chaining. Then, the home-to-work distance may serve as a useful proxy variable for commute time to measure its influence on commuters’ time use allocation behavior as longer commute distances generally entail longer commute times. The application of the fractional logit model to these scenarios shows that there is a three minute trade-off between in-home time allocation and travel time due to the 5 km increase or decrease in commute distance. When commute distance increases by 5 km, the individual allocates three additional minutes to travel, all of which are taken from in-home time. Similarly, when commute distance decreases by 5 km, the individual spends 3 minutes less on travel and 3 additional minutes at home. There is no change in out-of-home non-work activity time allocation. Indeed, one might expect the change in commute distance to affect travel time allocation without affecting the actual time spent at non-work activities. Whereas the individual may have to spend 10-20 additional minutes travelling to and from work, the individual may trip chain more efficiently or choose a faster mode so that the total travel time changes only modestly (at about 3 minutes) leaving the non-work activity time allocation largely unchanged. The simple numerical example presented in this section shows that the fractional logit model is capable of serving as a powerful time use allocation analysis tool that is responsive to explanatory factors that may influence time use patterns. Combined with other models of activity and time use such as activity frequency, duration, timing, and type choice models, the fractional logit model would be able to serve as a powerful framework for analyzing activitytravel behavior while truly reflecting daily time resource constraints.


CONCLUSIONS In this paper, the fractional logit modelling methodology has been employed to represent time use allocation fractions and patterns of workers and non-workers. The samples used for the modelling effort have been drawn from the 2000 Swiss Travel Microcensus in which detailed socio-economic, demographic, commute, and one-day travel characteristics were collected. Separate models were estimated for worker and non-worker samples to recognize the differing constraints under which these market segments engage in activities and travel. Time use analysis has been receiving increasing attention in the field due to the increased interest in understanding how transportation investments, enhancements, or policies may affect people’s lifestyles and quality of life. A robust time use model is needed to ensure that the policy analysis results are consistent with time constraints under which people operate and make decisions and choices. The fractional logit modelling methodology offers key advantages in the context of time use analysis, as summarized below: 1) Fractional logit model can guarantee predictions of time allocations between 0 and 24 hours or any available time resource. 2) Fractional logit model can guarantee the sum of all time use allocations to be exactly equal to 24 hours or any available total time resource. 3) Fractional logit model accommodates the non-linear relationship between explanatory variables and dependent time use allocations. It is believed that a non-linear relationship is more reasonable than a linear relationship because the marginal effect of an explanatory variable ought to diminish. This property recognizes that time is a finite resource and can not vary infinitely. 4) Fractional logit model parameters can be estimated easily within the simplistic logit formula using the Quasi Maximum Likelihood Method. The iterative process for maximization of the log-likelihood function is fast. 5) Using the fractional logit model, one can include zero observations without additional mathematical transformations. 6) Using the fractional logit model, one can easily recover the expectation of dependent variables without making any distributional assumption on the error terms. Despite the advantages of the fractional logit modelling methodology, there are some notable limitations that should be addressed in future research efforts. The analysis conducted in this paper was also limited by the data available from the Swiss Microcensus travel diary survey. The travel diary survey does not provide any data on specific in-home activities such as sleep,


eat, recreation, home-care, child-care, etc. A greater understanding of time use allocation behavior could potentially be obtained by using a true time use diary data set that provides specific information about in-home activities. Such data would allow an explicit identification of in-home activity types and trade-offs that might exist in allocating time between in-home and out-of-home activities. An application of the fractional logit model to a time use data set remains a future research task. The fractional logit modelling methodology is also subject to violations of the well-known IIA (Independence from Irrelevant Alternatives) assumption. The IIA property implies that time use allocations to travel and various activities are mutually independent. In the models estimated in this paper, the change in the allocation function for one fraction will not influence the ratio of the other two fractions. However, this assumption may be violated when two fractions are correlated with one another. In general, one would expect people to engage in additional travel if they spend more time at activities outside home. If the assumption that additional out-of-home activities result in additional travel, then travel time allocation is inevitably correlated with out-of-home activity time. Then, the use of the multinomial logit formulation is inappropriate. One possible way to solve this problem is to define a nested logit formulation within the fractional logit modelling framework. With the correlated time use fractions specified into respective nests, the fractional nested logit model would serve as a potential method to overcome the IIA limitation. The development and estimation of such a model system remains a future research effort.

ACKNOWLEDGEMENTS The authors thank Dr. Jeffrey S. DeSimone, Assistant Professor in Department of Economics at the University of South Florida for his assistance and guidance in this research. In addition, the authors benefited from discussions with Abdul Rawoof Pinjari who was a graduate research assistant in the Department of Civil and Environmental Engineering at the University of South Florida when this research was performed. Finally, the authors thank Dr. Giovanni Gottardi of Jenni + Gottardi, AG, Zurich and the Swiss Ministry of Transport for providing the data set used in this study.


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