Dynamic Model of One plus Car

The explanatory variables in the utility function include the lagged dependent variable and other exogenous variables, including income and other household characteristics (household size, number of person in work, number of children), location of household as proxy for accessibility, costs of car ownership and use, and finally the second polynomial of average age of household head in the cohort.

In the initial tests, unobserved heterogeneity is not modelled and the models estimated are pooled logit or probit. We compared models with different representation of the

household characteristics, using either average household demographic statistics (household size etc.) or split of eight household types. We also compared models with different representation of the household locations, using either the full five location categories or the three compressed location categories. Regarding household income and costs of car ownership and use, variables of both linear form and logarithm form have been tested. Finally, different functional forms of logit and probit have been investigated. Table 7-1 summarises the various models mentioned above and compares their degree of fit based on log likelihood.

Table 7-1 Summary of Initial specification search Functional

Form

HH Characteristic Variables

Location Variables

Income/Cost Variables

No. of Variables

Log Likelihood Model 1 Logit Ave No. of people* Compressed Linear 12 -65911

Model 2 Logit Household types Full Linear 18 -65879

Model 3 Logit Household types Compressed Linear 16 -65879

Model 4 Logit Ave No. of people* Full Linear 14 -65910

Model 5 Logit Ave No. of people* Compressed Log 12 -65901

Model 6 Logit Household types Compressed Log 16 -65877

Model 7 Probit Ave No. of people* Compressed Linear 12 -65914 Model 8 Probit Household types Compressed Linear 16 -65881 (* Average number of people within the household, in work and under the age of 16)

The null log likelihood of the logit model is -81240, so the Adjusted Likelihood Ratio Index (also called Rho bar square) of Model 1 to 6 vary between 0.1885 and 0.1889.

Using proportions of eight household types (Model 3) instead of average number of people (Model 1) as explanatory variables, the model loses 4 degree of freedom but has the log likelihood increased by 32 and the Adjusted Likelihood Ratio Index increased by 0.0004. On the other hand, there is almost no change of log likelihood when the five household location variables (Model 2) are compressed into three (Model 3), indicating no loss of fit. From these analysis, it appears that household characteristics are better represented by the eight-way categorization of household type, while locations are better represented by three area types, i.e. metropolitan areas, least populated rural areas and others.

Model 5 transforms the income and cost variables in Model 1 to the logarithm form, and the log likelihood is increased by 10. Similarly, Model 6 is the logarithm version of Model 3, while the log likelihood is increased by 2. As the degree of freedom is the

same between each pair of linear and logarithm models, the higher log likelihood suggests that models with log transformed income and costs variables are preferred.

Finally, Model 7 and 8 are the probit version of Model 1 and Model 3 respectively.

Because the null log likelihood for the probit model is slightly different, the log likelihood of Model 7 and 8 is not directly comparable to that of other models.

However, the marginal effects in the each pair of logit and probit models are very similar, suggesting that the robustness of the logit specification.

The above results are similar to those reported in the previous chapter for the static model. Models 1 to 8, while including the lagged dependent variable, do not consider the unobserved heterogeneity. Given the importance of accounting for unobserved effects in the dynamic model, both fixed effect models and random parameter models are also investigated.

The fixed effect version of Models 1 to 6 was estimated. With the presence of cohort dummy variables, the degree of freedom is reduced by 14, while the log likelihood is increased by 7 to 16 depending on the specific models. It should be noted that the fixed effects are treated as parameters in the model, and the finite T bias would be present for the model coefficients estimated using maximum likelihood method, although the bias might not be significant due to the relatively long sample period. Table 7-2 reported the results of two fixed effect model with log income and costs variables.

Comparing Model 13 and 14 with their pooled logit version of Model 5 and Model 6, the log likelihood increases by 10 and 15 respectively. The likelihood ratio test of fixed effects, with 14 degree of freedom, is not statistically significant for Model 13 but significant at 5% level for Model 14. This is not surprising as none of the coefficients for the cohort dummy in Model 13 is significant. In Model 14, the slope coefficient and marginal effect are larger for the younger cohorts, indicating a higher propensity of car ownership for the younger generations. Figure 7-1 illustrates the marginal effects of the cohort dummy variables, which has a clear linear trend except for the youngest cohorts.

This result is similar to those reported for the linear models in Chapter 4 and 5 and those reported in Dargay and Vythoulkas (1999).

Table 7-2 Fixed Effect Models with Log Income and Cost Variables (t-stat in parenthesis)

Table 7-3 Short run elasticity derived from FE models with log income and price variables

Model 13 Model 14

Figure 7-1 Marginal Effects of Cohort Dummies (Model 14)

Marginal Effects of Cohort Dummies

0.00 0.05 0.10 0.15 0.20 0.25

C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 Cohort

Marginal Effect

One concern about the fixed effect model is the much smaller coefficient (and smaller t-stat) of the income variable, especially for Model 14, suggesting some explanatory power of income has been taken away by the cohort dummies. This would have some negative impacts on forecasts, as income is the exogenous variable that we can more easily control, while the fixed effects for the new cohorts are difficult to predict. Table 7-3 reports the short run elasticity of income, car purchase price and running costs for low, median and high income cohort, which have been calculated as El=(∂P ∂x)/P.

The short run income elasticity reported in Table 7-3 is lower than that derived from the pooled logit model (Model 5 and 6 in Table 7-1). While Model 14 seems to have better goodness of fit, the income elasticity is lower than those commonly found in the literature. Furthermore, the income elasticity becomes lower than the price elasticity in Model 14, inconsistent with earlier findings in Dargay and Vythoulkas (1999). On the other hand, the income and cost elasticity derived from Model 13 is more sensible, even though it has worse goodness of it.

One advantage of using dynamic model is the ability to capture long run relationship under equilibrium and the possibility of estimate long run elasticity. However, unlike the case of linear model, the long run elasticity can not be easily derived for dynamic discrete choice model. In the current study, we use Taylor expansion to derive approximate results. The formula used in the calculation of long run elasticity using

Taylor expansion is reported in Appendix 3 and the results for Model 13 and 14 are reported in Table 7-4.

Table 7-4 Long run elasticity derived from FE models with log income and price variables

Model 13 Model 14

Table 7-5 Results of Random Parameter Model (t-stat in parenthesis)

Mean of Param Std Dv of Param Marginal Effect ONE -1.72791 (-0.88) -0.00009 (-0.01) -0.34581 ' Vythoulkas (1999), and is also lower than those derived from linear dynamic model in Chapter 5. These results are clearly unsatisfactory.

As discussed in Section 2, the fixed effect estimators are fixed T biased due to the incidental parameter problem. We then proposed to extend the random effect model to random parameter model, which relaxes the orthogonality assumption between the unobserved effects and the explanatory variables and accommodates lagged dependent variable. For the ownership model of one plus car, we estimate the random parameter version of Models 1 to 6 reported in Table 7-1. We report the results of the model with best fit in Table 7-5.

The model log likelihood of the random parameter model is -65877 and the adjusted likelihood ratio index is 0.1887, slightly lower than the fixed effect model. The residual plot (Figure 7-2) does not reveal any severe problems of auto-correlation and heteroskedasticity.

Figure 7-2 Residual plot of the Random Parameter model

Residual against Obs ID (by cohort and by year for each cohort)

-0.12 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10

0 50 100 150 200 250 Rsd(car1+)

The log variables of purchase price and running costs index are the same in any year for all cohorts, so their coefficients are treated as fixed parameters. The coefficients of other variables are assumed to be normally distributed. However, none of the estimated standard deviations of the random parameters (the square root of the diagonal elements of Γ in equation 16) are statistically different from zero. Regarding the mean of the random parameters, the constant term and log running cost parameter are not significant, while lagged dependent variable, log income, log purchase price, age parameters are all significant at 1% level. Only some of the household characteristic

variables and location variables have significant coefficients. Based on the mean of the random parameter, the short run and long run elasticity is reported in Table 7-6.

Table 7-6 Short Run and Long Run elasticity based on the mean of random parameters

Short Run Long Run

Income

Income Elasticity

Price Elasticity

Running Cost Elasticity

Income Elasticity

Price Elasticity

Running Cost Elasticity

Low 0.39 -0.29 -0.06 0.61 -0.46 -0.10

Middle 0.17 -0.13 -0.03 0.23 -0.18 -0.04

High 0.13 -0.10 -0.02 0.17 -0.13 -0.03

The running cost elasticity is not significant so is shown in italic. The short run income elasticity ranges from 0.13 for high income household to 0.39 to low income household, while the long run income elasticity ranges from 0.17 to 0.61. The purchase price elasticity varies between -0.10 and -0.29 in the short run while the range is between -0.13 and -0.46 in the long run. The income and price elasticity for high income households is about one third of that for low income households, and such difference is much bigger than that reported in Dargay and Vythoulkas (1999). This might suggest that the logit functional form adopted here better accounts for the impact of saturation.

Finally, it is worthy to provide some tentative explanations of why none of the random parameters has standard deviation significantly differently from zero. The first possibility is that the unobserved heterogeneity is no longer significant after the individual data are aggregated into cohorts. However, this argument is not supported by results of the fixed effect models, where the fixed cohort effects are significant for most of the cohorts under certain specification. Nevertheless, the improvement of the goodness of fit for the fixed effect model is not spectacular given the loss of degree of freedom (likelihood ratio test is significant at 5% but not at 1% level), which might suggest that the significance of heterogeneity is limited for cohort data. The second possibility is that the pseudo panel sample size might be too small to reliably estimate the distribution of parameters representing unobserved heterogeneity. Although the Family Expenditure Survey has thousands of observations each year, after they are aggregated into pseudo panel, we only have observations for 16 cohorts covering 19 years. This small sample size might not be enough to establish the distribution of the random parameters. The third possibility is that the assumption of normal distribution of parameters is inappropriate. However, alternative assumptions of uniform, triangular

and log-normal distribution all yield almost identical results so the assumption on parameter distribution is less likely to be the problem here.