Empirical Results of Car Ownership Model with Saturation

The specification search is similar to that described in section 7.3.1 for models of one plus car and section 7.3.2 for models of two plus cars. While the consideration of saturation improves the goodness of fit, it does not change the comparative

46 The Dogit Model has the probability distribution as:

∑ ∑

performance between models with different forms of explanatory variables and unobserved heterogeneity.

For model of one plus car, the model with the best fit is the fixed effect model, where the household characteristics are represented by the eight-way categorization of household types. While this model has the highest log likelihood after taking into account the degree of freedom, there are practical difficulties in using it for forecasts.

One is the fixed effect for new cohorts in future years, which can not be estimated from existing data. The other difficulty is forecasting the split of households for the eight household types in each cohort, which, unlike the variables of household size and number of children/employed, can not be controlled using the forecasts published by the planning authority. Both issues would lead to additional uncertainty in the forecasts.

Furthermore, the income elasticity implied by the fixed effect model is lower than the range identified in the literature, which is another cause of concern.

For these reasons, we also report the result of the “second best” model with alternative household characteristic variables. Initially, the model estimated was a random parameter (mixed logit) model; but as none of the random parameter has standard deviation significantly different from zero, the random parameter specification was abandoned and all parameters were treated as fixed. Table 7-9 reports the results of the fixed effect model and the “second best” pooled logit model for household owning at least one car, which will be used for forecasting in the next chapter.

The log likelihood of the fixed effect model with household type split is -65859, higher than the pooled logit model with average demographic statistics by about 40. The former has higher adjusted likelihood ratio index of 0.1889, which suggests better model fit after allowing for degree of freedom adjustment. As mentioned before, there are practical issues in applying the fixed effect model in forecasts, so it would be interesting to compare the forecasting results based on both models later on.

The coefficient for the linear modifier S^* is similar for both models. They translate to the saturation level of 0.9212 and 0.9437 respectively. However, the income elasticity differs significantly between these two models, with that for the pooled logit model being much higher. Table 7-10 compared the short run and long run income elasticity

Table 7-9 Forecasting model of one plus cars (t-statistic in parentheses)

Slope Coefficient Marginal Effect Fixed Effect Pooled Logit Fixed Effect Pooled Logit

ONE -4.8731 (-2.31) -0.9628 **

Table 7-10 Short run and long run income elasticity of one plus car model Short Run Income Elasticity Long Run Income Elasticity Income Fixed Effect Pooled Logit Fixed Effect Pooled Logit

Low 0.198 0.550 0.238 0.925

Middle 0.082 0.240 0.111 0.413

High 0.065 0.191 0.069 0.252

for these two models. The impacts of different income elasticity on car ownership forecast will be examined in the next chapter.

Regarding the model of household owning two or more cars conditional on owning the first car, specification search shows that fixed effect models do not have better goodness of fit. While using the five-area household location split improves model fit, there is no significant loss of fit when using average demographic statistics rather than eight-way household type split if degree of freedom is taken into account. The average household size variable is not significant and subsequently dropped, so the household characteristics are described by the average number of children and people in work per household. This leads to the model of best fit reported in Table 7-11. It should be noted that the model initially estimated was a random parameter model; as the standard deviations of the random parameters were not significantly different from zero, all parameters are treated as fixed in the final model.

Table 7-11 Model of Car 2+|1+ (t-stat in parenthesis) Slope Coeff Marginal Effect ONE -10.4365 (-3.08) -2.1587 ***

LagY 2.3361 (5.23) 0.4832 ***

LnInc 1.0649 (4.63) 0.2203 ***

Child -0.1362 (-3.67) -0.0282 ***

Worker 0.1844 (2.26) 0.0381 **

AREA2 2.2701 (2.77) 0.4695 ***

AREA3 1.1823 (1.45) 0.2446 ' AREA4 1.6324 (2.11) 0.3376 **

AREA5 1.0771 (1.44) 0.2228 ' LnPrice -0.6078 (-1.88) -0.1257 * LnRunCst 0.6191 (2.42) 0.1280 **

Age 0.0769 (5.15) 0.0159 ***

AgSq -0.0840 (-5.08) -0.0174 ***

S* -0.7891 (-3.07) -

Log Like'd -47147 Null LL -56288 Adj. LRI 0.1621

***: Significant at 1% level;

**: Significant at 5% level;

*: Significant at 10% level;

': Not statistically significant

Similar to the unconstrained Car 2+|1+ model reported in Table 7-7, there are two parameters with unexpected sign. The coefficient of average number of children in the household is negative and significant, but it might be due to the correlation between that variable and the average number of people in work. The latter is significant at 5%

level, with marginal effects on the conditional choice probability of 0.038. While the coefficient for log of real purchase price is negative and significant, that for the log of real running costs is significant but of wrong sign. This result was previously identified for the static and unconstrained dynamic model of Car 2+|1+ and might be caused by the concurrent substantial rise of car running costs and ownership of two plus cars in the second half of 1990s. In terms of household location, if the proportions of households living in metropolitan and rural areas (Area type 2 to Area type 4) increase at the expense of that in Greater London (the base case of Area type 1), the conditional probability of household owning two or more cars would also increase. Finally, the coefficients for the average age of household head and age square (dividing by 100) are positive and negative respectively, indicating a peak of car ownership during the household life cycle.

The estimated linear utility modifier S^* is -0.7891, implying a saturation level of 0.6876.

We have also calculated the short run and long run income and costs elasticity for cohorts with low, median and high income level, which is reported in Table 7-12, The income and purchase price elasticity are higher than those for models of one plus car, which is as expected. On the other hand, the running cost elasticity is shown in italic due to its unexpected sign.

Table 7-12 Income and cost elasticity of Car 2+|1+

Short Run Long Run

Income

Income Elasticity

Price Elasticity

Running Cost Elasticity

Income Elasticity

Price Elasticity

Running Cost Elasticity

Low 0.95 -0.54 0.55 1.23 -0.70 0.72

Middle 0.78 -0.45 0.45 0.94 -0.53 0.54

High 0.62 -0.35 0.36 0.68 -0.39 0.39

7.5 Conclusion

In this rather long chapter, we tackle two important issues that motivate the estimation of non-linear pseudo panel model in the first place: the consideration of dynamics and saturation in the car ownership choice. Since the 1980s, there have been a growing number of researchers that recognised the importance of dynamics and applied it in transport studies (e.g. Hensher and Wrigley, 1986; Kitamura, 1990; Mears et al. 1990;

Goodwin et al. 1990; Goodwin, 1997; Long, 1997). In the current study, we focus on the methodological aspects of the nonlinear dynamic models. In particular, the first two

sections deal with the development and consistent estimation of the dynamic discrete choice pseudo panel model. We have shown in the previous chapter that the utility function of the pseudo panel model is a direct transformation from its cross-sectional counterpart, and if the measurement error can be ignored, these two types of models have the similar probability function, albeit with different scale (random term). This result facilitates the discussion in this chapter, where the behaviour models are more conveniently developed for individual decision maker before transforming into pseudo panel model.

In developing a dynamic car ownership model, we follow a general to specific approach, starting from a structural model with three forms of true state dependence.

The general model encompasses three specific models: standard state dependence model, propensity dependence model and dynamic optimisation model. In propensity dependence model, the lagged effect is captured by the previous tendency to select a state (choice), which is unobservable and will lead to additional uncertainty in forecasts.

Model of dynamic optimisation, despite its promise of enriching choice dynamics, requires a fundamental shift of car ownership model from holding model to transactions model. As a result, the standard state dependence model is chosen as the preferred model of dynamic car ownership choice.

The estimation of the nonlinear dynamic pseudo panel model is examined theoretically and empirically. We first review the various fixed effect, random effect and semi-parametric estimators proposed for the genuine panel data in the literature;

subsequently, the fixed effect model and random parameter model of pseudo panel are proposed for the current study. In the empirical section, separate models have been estimated for households owning at least one car and those owning two or more cars conditional on the ownership of the first car. Both fixed effect models and random parameter models are tested, although the standard deviations of the random parameters are not significantly different from zero in all models.

The other important issue investigated in this chapter is the specification and estimation of car ownership model with saturation, which is a key consideration for the use of discrete choice method. In the framework of random utility model, saturation implies that some households are constrained not to own a car. Accordingly, the choice set

faced by the decision makers has been expanded to include the new alternatives of

“constrained zero cars”. To facilitate estimation, the model is specified with a “tree logit” structure and instead of directly estimating the saturation level S (a nonlinear term in the probability function), we estimate a linear modifier S^* in the utility function.

The constrained dynamic model is subsequently implemented in the empirical study of car ownership. The estimated models of Car 1+ and Car 2+|1+ will then be used to forecast car ownership in Britain to year 2021, which will be the subject of next chapter.