An analysis of weekly out-of-home discretionary activity participation and time-use behavior

Doherty et al.’s study suggests that activity-travel behavior may be guided by an underlying activity scheduling process that is associated with multiple time horizons that range from a week (or, perhaps, more than that) to within a day.

Bhat et al. (2004, 2005) base their conclusion of weekly rhythms on a visual inspection of the hazard profile and confine attention to the participation decision without attention to time allocation, while Habib et al. (2008) base their conclusion of weekly rhythms in participation/time-use on the stability of model parameters estimated separately on each of six weeks of data. In both these studies, while there may be some suggestion of weekly periodicity of activity participation in relatively coarsely defined discretionary activities, there is no quantification whatsoever of the within-individual week-to-week variability and between-individual variability.

The use of this classification system is motivated by the differences in the activity-travel dimensions (participation rates, durations, time-of-day of participations, accompaniment arrangement, etc.) associated with episodes of each type. For instance, earlier time-use studies have provided evidence that participation rates in social and leisure (window shopping, making/listening music, etc.) activities tend to be higher than in other discretionary activities. Also, when participated in, episodes of these activities are participated for long durations. However, social activity episodes are mostly pursued with friends and family, while leisure activities are mostly pursued alone (see, for example, Kapur and Bhat 2007). The basis for the other activity types is provided in the next section.

The inclusion of the ‘other’ activity in the analysis enables the analyst to endogenously estimate the total time-investment in the first six types of OH discretionary activity purposes. In the presentation of the model structure later in this section, we will label this “other” activity purpose as the first alternative for presentation convenience.

All individuals in the sample participate for some non-zero amount of time in ‘other’ activities, and hence this alternative (which we will consider as the first alternative) constitutes the “outside alternative” that is always consumed (see Bhat 2008 for details). The term “outside alternative” refers to an alternative that is “outside” the purview of the choice of whether to be consumed or not. The rest of the (K-1) “inside” alternatives that are “inside” the purview of whether to be consumed or not correspond to the OH discretionary activities. Thus the first element of x _qt should always be positive, while the second through Kth elements of x _qt can either be zero or some positive value. Whether or not a specific x _qtk value (k = 2, 3,…, K) is zero constitutes the discrete choice component, while the magnitude of each non-zero x _qtk value constitutes the continuous choice component. In this paper, the terms “time investments” and “time use” are used interchangeably to refer to these discrete-continuous x _qtk values.

Some other utility function forms were also considered, but the one specified here provided the best data fit while allowing for estimation of all the parameters without any identification problems. For conciseness, these alternative forms are not discussed. The reader is referred to Bhat (2008) for a detailed discussion of alternative utility forms. The reader will also note the implicit assumption in the formulation that there is utility gained from investing time in OH discretionary activities. This is a reasonable assumption since individuals have the choice not to participate in such activities. Also the reader will note that the inclusion of the IH and OH maintenance and IH discretionary activities as the “outside good” (the first alternative) allows the analyst to endogenously estimate the total amount of time invested in OH discretionary pursuits.

The constraints that γ _k  > 0 (k = 2, 3,…, K) are maintained through appropriate parameterizations (see Bhat 2008). Also, the γ parameters are subscripted only by activity purpose k (unlike the ψ parameters that are subscripted by q, t, and k) because specification tests in our empirical analysis did not show statistically significant variation in these parameters based on individual specific or time-specific observed/unobserved characteristics.

Multiple discrete-continuous extreme value models (whether MDCEV or MMDCEV) require identification restrictions analogous to single discrete choice (i.e., multinomial logit, whether mixed or not) models, because the probability expression for the observed optimal time investments is completely characterized by the (K-1) utility differences (Bhat, 2008). Thus, the MMDCEV model requires the usual location normalization of one of the alternative-specific constants/variables to zero (this is the reason for the absence of a θ ₁ term and a β′z ₁ term in Eq. 1). Further, as with the current context, when there is no price variation across alternatives, the scale of the utility is normalized by standardizing the type I extreme-value distributed error terms ε _qtk . While one can, subject to some identification considerations, allow the choice-occasion specific error terms ε _qtk to have different variances across alternatives, and allow choice-specific covariances across alternatives, we assume that these error terms are identically and independently distributed. Also, appropriate identification restrictions need to be imposed on the third and fourth components of the utility components in the main text above. These two components generate the individual-level variance-covariance matrix of the overall individual-level error terms affecting the logarithm of the alternative-specific baseline preferences. The identification conditions can be derived in a straightforward manner by examining the variance-covariance matrix of the implied error term differences in a manner similar to that for a cross-sectional model (see Bhat 2008). In our empirical specification, we apply restrictions on the individual-level variance-covariance matrix that are more than sufficient for identification. We should also note here that we considered individual-level unobserved heterogeneity for the outside alternative (i.e., the first alternative), which can indeed be estimated in a panel setting subject to appropriate identification restrictions on the covariance terms. However, this term did not turn out to be statistically significantly different from zero, and so we did not introduce unobserved inter-individual heterogeneity terms in Eq. 2 for the outside alternative.