A Discrete Choice Model of Household Car Ownership

To formulate a Random Utility model of car ownership, the first step is to determine the decision maker and choice set. In the current study, the decision makers are the household i within cohort c. For a car ownership model, the complete choice set is the number of car owned: 0 car, 1 car… n Cars. Due to smaller sample size for households with 3 or more cars, we limit the choice set of our car ownership model to 0 car, 1 car and 2+ cars.

As mentioned above, the different assumptions on the distribution of random residuals give rise to discrete choice model of different forms. The most common model is multinomial logit model (MNL), which assumes the random terms (e_a,it in the current study) are distributed IID (independently and identically distributed) Gumbel. Its popularity lies in the low computation costs of parameter estimation. However, it exhibits the Independence of Irrelevant Alternatives (IIA) property, i.e. the ratios of any two probabilities is necessarily the same no matter what other alternatives are in the choice set or what the characteristics of other alternatives are. This property is clearly inappropriate in certain situation, and the MNL has the danger of failure in the

presence of correlated alternatives (e.g. the red bus, blue bus problem). As the two options of owning one car and two plus cars are correlated, MNL might not be appropriate for the car ownership model either. Nevertheless, such structure is popular in the empirical work, especially as part of the bigger modelling system (e.g. Train, 1986; HCG, 2000).

Multivariate probit model relaxes the IID assumption of the random components of the utility. Instead, the random residuals are assumed to be distributed jointly normal, with a general variance-covariance matrix. Because of the assumption on the random utility components is completely general, probit model successfully tackles two problems confronting MNL model: non-independence from irrelevant alternatives as well as taste variation among individuals. The choice probabilities of the probit model are quite complex (multiple integrals with no closed form), and it can only be estimated using alternative approaches such as Clark approximation and Monte Carlo Simulation (Daganzo, 1979; Train, 2003). The application of probit models to the nonlinear pseudo panel model of (9) is even more complicated. More specifically, there are two sources of unobserved heterogeneity, one choice specific and one cohort specific. To distinguish these two sources of heterogeneity, while maintaining the flexible correlation structure of the composite random term e_a,it , would make consistent estimation of the model extremely difficult.

MNL and probit are “flat” models, which can be illustrated by graph (a) in Figure 6-1.

Given the drawbacks of MNL and complexity of probit, a hierarchical model structure becomes an attractive alternative, which is illustrated by graph (b). It involves estimating two binary choice models in steps.

The hierarchical model of car ownership involves two binary choice models: the first is the choice between zero car and one plus cars (noted as Model 1+ hereafter); then conditional on owning at least one car, choice between owning exactly one car and two plus car (noted as Model 2+|1+ hereafter). It should be noted that the hierarchical model of (b) is not a standard Nested Logit Model, which would have the same complication as the multivariate probit model; instead, it consists of two separate binary choice models. For each binary choice model, it does not require the IIA

Figure 6-1 Two Structures of multiple car ownership modelling

assumption and the assumption on the random term can be general. Moreover, such formulation facilitates the consistent estimation of the unobserved heterogeneity and dynamic effect, thanks to a growing econometric literature in this field. It also has the advantage of choice probability (of higher car ownership) increases monotonically with income. As a result, the hierarchical model structure is adopted for the current project, similar to other car ownership models such as NRTF (1997), Whelan (2001) and RAC (2002b).

After determining the decision maker, choice set and model structure, the empirical model of car ownership can be readily identified. For Model 1+, the utility of owning no car is normalized to zero: U₀ =0; on the other hand, the utility of owning at least one car can be expressed as a linear function of:

it c

ct e

x

U1₊ = ′β+λ + (10)

where x_ct′ is a vector of explanatory variables for cohort c in year t, including car purchase price, car running costs, income and other relevant household characteristics²⁹. While all the households in cohort c have the same mean deterministic utility (x′ctβ) and unobserved cohort heterogeneity (λ_c), they have different composite error term e_it. This reflects the essence of the Random Utility Model: given the same observed deterministic utility, decision makers behave differently due to the unobserved random

29 For dynamic models, lagged dependent variable y_c,t−1 could also be included, although it could

0 Car 1 Car 2+ Cars

0 Car 1+ Cars

2+ Cars 1 Car

(a) “Flat” model of MNL and probit

(b) Hierarchical model structure

error. In the current study, this is manifested in the fact that only a proportion of households in a cohort choose to own car(s). Note the household in cohort c owning at least one car in year t is noted as y¹_ct⁺ =1, then:

Assuming the distribution of e_it is IID logistic, the probability of household i in cohort c owning at least one car is that of a familiar logit model:

) practical reasons, and will be investigated in the empirical section later this chapter.

Model 2+|1+ would be estimated using a reduced pseudo panel dataset of car owning households, while having the identical formulation of Model 1+. The utility of owning exactly one car will be normalized to zero, and the utility of owning two or more cars will be defined in a similar fashion. The choice probability of household owning two or more cars conditional on owning at least one car (P2+|1+) will also be similar to (12) based on the IID logistic assumption of the composite random term.

Finally, we discuss the interpretation of the linear utility of owning a car, i.e.

expression (10). Households derive utility of car ownership from driving the car; as a result, it seems odd that income, price and other household characteristics are appropriate explanatory variables to be included in the utility function. This is a recognized issue and in the literature of joint car ownership/use model, the utility function of (10) is interpreted as the linear approximation of the conditional indirect utility function. The conditional indirect utility function is the function that gives maximum utility achievable at given prices and income, conditional on the choice of a certain alternative. As shown by Varian (1992), consumers’ preferences can be equivalently represented by a direct utility function or an indirect utility function.

Starting with the latter, it is relatively straightforward to derive the demand function of

car use using Roy’s identity³⁰ (Train, 1986). Although the current study deals with car ownership only, it is more appropriate to interpret (10) as the (linear approximation of) conditional indirect utility function, and the extension to car use would also become straightforward.