As the outcome variable is count data which may tend to be non-normally distributed and positively skewed, the use of ordinary least squares regression would be inadequate. Poisson regression can provide more appropriate alternatives for analysing count data. The standard Poisson probability function of household car ownership (Y) can be written as follows:

$\text{P(}Y=y\text{) = }\frac{{\mu }^y\text{ exp(-}\mu \text{)}}{y!},y=0,1,2,\dots ,\mu >0$ (1)

In the Poisson distribution, both the mean and variance of Y are equal to μ. However, the restrictive assumptions may not be suitable to handle some types of count outcomes. In practice, the variance can be either larger or smaller than the mean. When the variance exceeds the mean, it is referred as over-dispersion. The phenomenon with the variance less than the mean is under-dispersion. To deal with the estimation more flexible, Consul and Famoye [23] and Famoye [24] developed the generalized Poisson regression model that has statistical advantages over standard Poisson. It is applicable to model count data with either over-dispersion or under-dispersion. The generalized Poisson probability function of Y can be written as the following:

$\text{P(}Y=y\text{) =}{\left(\frac{\mu }{1+\alpha \mu }\right)}^{\text{y}}\frac{{\left(1+\alpha y\right)}^{y-1}}{y!}\text{exp}\left[-\frac{\mu \left(1+\alpha y\right)}{1+\alpha \mu }\right]$ (2)

The mean of Y is μ, and the variance of Y is μ(1 + αμ)². With the link function: μ = μ(x) = exp (xβ), the mean of Y is related to the explanatory variables. In the function, x is a (k- 1) dimensional vector of explanatory variables and β is a k-dimensional vector of regression parameters. In the next step, the mean parameter μ can be expressed with the log link function: log(μ) = xβ. The dispersion parameter isα . If α equals zero, the probability function reduces to the standard Poisson probability function. The positive value of α means count data with over-dispersion, while the negative value of α represents count data with under-dispersion. How to verify whether the generalized Poisson regression model is more suitable than the Poisson regression model? The null hypothesis H₀:α = 0 can be tested by using the asymptotic Wald t-statistic. The use of generalized Poisson regression model can be supported if the null hypothesis is rejected. Alternatively, the goodness of fit of two models can be assessed by the likelihood ratio test.

The Poisson and generalized Poisson regression models can reflect the effects of individual factors on household car ownership. However, these methods can estimate mean effects but not provide information on the full probability response to the change in an explanatory variable. Therefore, this study further employs the quantile count model so as to allow for different responses in different parts of the distribution. The quantile regression has been widely applied in the studies since Koenker and Bassett [25] introduced the concept of quantile regression and developed the estimation of conditional quantile functions. The typical quantile regression, based on the distribution function of a continuous random variable, can be performed for analysing the impact of the regressors on each quantile of the distribution. Consider a dependent variable Y, and the θ quantile of the distribution of a random variable Y is denoted by Q_Y (θ|X). X is a vector of observable characteristics, and θ is a number between 0 and 1. Q_Y (θ|X) can be obtained by sorting the values of Y from smallest to largest. The parameter vector β can be estimated for any quantile θ by minimizing the following expression with respect to β [25]:

$\min {\sum }_{i=1}^n\theta {(Y_i-X_i^{\text{'}}\beta )}^2\text{ for any quantile }\theta \in \left(0,1\right)$ (3)

The different parameter vectors of β for a given θ can be obtained using linear programming algorithms. However, when the dependent variable is discrete and the quantiles are not continuous, the linear programming approach of typical quantile regression cannot be used for count data. To overcome this problem, Machado and Santos Silva [21] had developed the quantile regression for count data. This approach employed a specific form of jittering technique proposed by Stevens [26] to smoothen the data. The first step is to construct a continuous variable Z where Z = Y + U. U is a random variable uniformly distributed in the interval [0, 1] and independent of Y and X. Let Q_{Z (}(θ|X) denote the θ quantile of the conditional distributions of Z:

$Q_z\left(\theta |X\right)=\theta +\text{exp}\left[X^{\prime }\widehat{\beta }\left(\theta \right)\right]$ (4)

The additive term θ is set as the lower bound of Q_{Z (}(θ|X). In the second step, the distribution of Z is smoothed and a monotone transformation T(Z; θ) can be obtained. The transformed quantile function is linear. The quantile regression can be expressed as follows:

$Q_{T(Z;\theta )}\left(\theta |X\right)=X^{\prime }\widehat{\beta }\left(\theta \right)$ (5)

where (β) ̂(θ) is the estimated vector of parameter at the θ quantile. When the process is performed, an important necessary condition is that at least one continuous explanatory variable must exist. The monotone transformation T(Z; θ) is specified as follows:

$T(Z;\theta )={\{}_{\log (\zeta )\text{ for }Z\le \theta }^{\log (Z-\theta )\text{ for }Z\text{ > }\theta }$ (6)

where 0 < ζ< θ. The quantiles are equivariant to monotonic transformations and invariant to censoring from below up to the quantile of interest. The estimator can satisfy the property of asymptotically normal. Any inference based on the t test and Wald test can be used. Ultimately, the θ quantile of Y can be transformed from the θ quantile of Z based on the relation:

$Q_Y(\theta |X)=⌈Q_Z(\theta |X)-1⌉$ (7)

The eq. (7) means that [α] returns the smallest integer greater than or equal to a. To sum up, the estimation technique is relied on adding an extra continuous “noise” term to the counts. Then, the transformed quantile function can be estimated since the dependent variable has been smoothened. The inferences are used as conventional methods. The quantile regression for count data has not been widely applied. Only few studies have adopted this approach in the issues of the health care [27], contract duration [28], and sparrow number investigation [29]. This study extends the application of quantile regression for counts to the analysis of household car ownership. This method can help us to investigate the determinants of household car ownership with a view of full distribution and evaluate the effects of predictor variables on different quantiles of a dependent variable.

In this study, the household data are obtained from Taiwan’s Family Income and Expenditure Survey. This is a nationwide cross-sectional survey that has been conducted annually by the government, but households are not tracked. There are approximately 15,000 households involved in this survey for each year. The database of Taiwan’s Family Income and Expenditure Survey contains household information such as demographic characteristics, property and facilities, income and expenditure. The household data in 1985, 1995, 2005, and 2015 are used so as to analyse generational differences in household car ownership over the past three decades [30-33]. Table 1 reports the descriptive statistics for household car ownership. The mean value of household car ownership increased from 0.12 in 1985 to 0.71 in 2015. The share of carless households decreased from 88% in 1985 to 41% in 2015. There was no obvious difference in the distribution of car ownership between 2005 and 2015. Figure 1 shows the trend of the percentage of households that owned cars. It had increased from 5% in 1980 to 58% in 2002 and then kept at the level of 59% till 2015. These results also reflect the fact that the growth rate of car ownership had slowed down for the past decade.

Figure 1.

The trend of the percentage of households that owned cars

The explanatory variables are classified into four categories: household head characteristics, economic characteristics, demographic characteristics, and transport attributes. Household head characteristics include age, gender, education, whether the head is an employer or not, and marital status. The age of household head is a continuous variable, which captures the life-cycle stage of a household and generation effects. Gender is represented by a dummy variable that takes value 1 for male and 0 for female. Education level is an ordinal variable, measured by the highest degree that household heads obtain. This study assigns scores 1, 2, 3, and 4, respectively, to the four levels: less than junior high school, junior high school, senior high school, and bachelors or graduate degree. A dummy variable also is used to represent whether the head is an employer or not. It takes value 1 if household head is an employer and 0 if otherwise. This variable captures the difference of employment status that an employer with better economic ability and higher social-economic status may incline to own a car. Marital status is represented by a dummy variable, valued at 1 if the head has a spouse and 0 otherwise.

Economic characteristics contain household income, house ownership status, and the number of parking lots. An important variable to capture economic ability is household disposable income, which includes both regular and non-regular income after expenses, such as taxes, housing rent, interest and insurance payment. Household disposable income is transformed into a real term and deflated by the consumer price index, whose base year is 2016. The literature indicated that the car ownership level is positively related to household income [34], [35]. Rith et al. [16] pointed out that, as compared with the other socio-economic attributes, household income is found as the main factor of household car ownership. House ownership status is measured by a dummy variable, valued at 1 if the house is owner occupied and 0 otherwise. Owner-occupied households who possess property rights may have higher economic ability to own cars. The number of parking lots self-owned by a household is included in the model because easy car parking may be also a key point for households to own cars.

The variables of demographic characteristics capture the effects of household structure, including household size, the number of elderly members, and the number of children. Household size is defined as the number of household members, which may be positively associated with the demand for cars. The number of elderly members can represent the seniority effect on car ownership since the travel patterns may be different between order and younger generations. The elderly are defined as household members aged 65 years and older. Similarly, the number of children is considered as an explanatory variable. Due to generational differences, the households with children may have different travel patterns that influence their decision to buy cars. The children are defined as household members aged under 12 years.

Transport attributes include the number of motorcycles, public transit expenditure, and whether the household lives in the urban areas. In Taiwan, motorcycles are important means of transport due to the characteristics of relatively low prices and time-saving. The relationship between motorcycle and car ownership may be negative because household motorcycle ownership may substitute car ownership, however, since motorcycles also play a role to supplement the mobility, the relationship between household motorcycle and car ownership may be positively related [35]. Besides, previous studies demonstrated that the increase in public transport use can decrease vehicle ownership [18], [36]. Public transit expenditure can be used for measuring the dependency on public transit. As households depend on public transit much more, they may reduce their demand for cars. Public transit expenditure is deflated using the consumer price index, whose base year is 2016. Lastly, the model also considers the spatial effect and includes a dummy variable to represent whether the household lives in the urban areas. It takes value 1 if the household lives in the urban areas and 0 otherwise. The urban areas have different types of land development and high population density. Thus, households living in the urban areas may have different travel patterns and higher living standards. On the other side, households in the urban areas may own fewer cars if the public transit networks are well-developed. Previous studies found that people incline to have fewer cars if they are living in a high-density built environment [16], [37].

The descriptive statistics for explanatory variables are reported as Table 2. The statistics show that the age of household head, the number of elderly members, the number of motorcycles, the ratio of female-headed households, and the education level of household head exhibited an upward trend, while household size, the number of children and the ratio of household heads with a spouse gradually declined. It is worth noting that household income increased from 608 thousand NT dollars in 1985 to 1,225 thousand NT dollars in 2005, and then decreased to 1,179 thousand NT dollars in 2015. This phenomenon revealed the stagnation of real income in Taiwan for the past decade.

Firstly, the Poisson and generalized Poisson models are used to estimate the effects of possible factors on household car ownership. This study focuses on the household datasets in 1985, 1995, 2005, and 2015, which represent four time points from different decades. Table 3 reports the estimation results of Poisson and generalized Poisson models. The results show that the dispersion parameters are significantly negative in all of the models, implying that count data have a feature of under-dispersion. These results support that the generalized Poisson regression model is preferred to the Poisson regression model. Since the adequacy of the generalized Poisson regression over the Poisson model has been confirmed, the following interpretation of predictors would be based on the results of generalized Poisson regression.

Among the household head characteristics, the variables of age, education level, whether the head is an employer, and whether the head has a spouse had significant effects for all of the four datasets. The age effects on household car ownership were negative, suggesting that the households headed by the younger generation tended to own more cars than the households headed by the older generation. The effects of education level were positive, implying that household heads with higher education attainment would have more cars due to their better social-economic status. Besides, household heads who are employers would own more cars than those who are employees. Household heads with a spouse would have more cars than household heads without a spouse.

As for the effects of economic characteristics, the variables of household income, house ownership, and the number of parking lots would have significantly positive effects for all of the four datasets. The level of household car ownership would rise as household income and the number of parking lots increased. Owner-occupied households inclined to own more cars since they had higher economic ability.

The effects of demographic characteristics were evident and important. Household size had significantly positive effects for all of the four datasets. The increase of family members would induce the demand for cars. Table 3 shows that, in the generalized Poisson model, the coefficient of household size had increased from 0.0841 in 1985 to 0.1647 in 2015. It is worth noting that the marginal effect of household size had increased with time. This result reflects the fact that household size had gradually decreased. Thus, the effects of household size on household car ownership would be stronger as the number of household members decreased. In addition, the number of elderly and children had significantly negative effects in 1995, 2005 and 2015, implying that households would own fewer cars if the dependent members increased. These results may be attributed to two possible reasons. First, the households with more elderly and children may have higher economic burden. Furthermore, the elderly and children are not the working group as a result of a lack of demand for cars.

Regarding the effects of transport attributes, the number of motorcycles had significantly negative effects in 1985 and 1995, but the effects were positive in 2005 and 2015. The negative relationship between motorcycle and car ownership indicated that household motorcycle ownership would substitute car ownership. According to Taiwan’s Family Income and Expenditure Survey, during the period of 1980-2000, about 75% of households had at least one motorcycle, while only 29% of households had at least one car. Households would be able to own motorcycles but not afford to buy cars due to cost-saving incentives. However, as the income level gradually increased, household were able to own cars and motorcycles. In 2015, around 83% of households had at least one motorcycle, and 60% of households had at least one car. Since motorcycles could play a role to supplement the mobility, the relationship between household motorcycle and car ownership became complementary. Besides, the variable of public transit expenditure had negative effects on household car ownership, suggesting that the higher dependence on public transport would contribute to reduce the level of car ownership. In terms of the spatial effects, the urban variable had positive effects on household car ownership, indicating that households living in the urban areas would tend to own more cars than households living in the rural areas. The higher level of household car ownership in the urban areas may possibly be related to the higher living standards and meeting the demand to commute.

Accounting for different responses in different parts of the distribution, this paper further employs the quantile count model to investigate whether the responses would be different between households with a high level of car ownership and households with a low level of car ownership. The quantile estimation is repeated for the 1985, 1995, 2005, and 2015 datasets. The results can help us to know how the effects of explanatory variables have evolved over time and understand which types of households own more or fewer cars. Table 4 reports the estimation results of quantile regression for counts. Since some households may not own any car, it would be meaningless to compute the model at a low quantile where the value of the outcome variable is zero. Therefore, the quantile regressions for the 50^th, 75^th, and 90^th quantiles are estimated.

As for the effects of household head characteristics, the age effects were significantly negative in the all quantiles, indicating that families in a later stage of the family life cycle have fewer car demand than do younger families. The effects of education level of household head were significantly positive. Households with higher-educated household heads owned more cars than households with lower-educated household heads. It is worth noting that the education effects were greater in 1985 than those in other periods. The education effects were strongest in the 90^th quantile of the 1985 dataset. However, for the 1995, 2005, and 2015 datasets, the education effects in the 90^th quantile were weaker than the effects in other quantiles. These results mean that, in the early stages of economic development, the increase of education level may generate a stronger contribution to enhance economic ability and induce households’ high demand for cars. But, along with income growth and education popularization, these effects may be weaker than before. Besides, the result also shows that households headed by an employer owned more cars than households headed by an employee. The household head with a spouse would tend to have more cars than the household head without a spouse.

Similar with pervious results of the generalized Poisson models, the variables of household income, house ownership, and the number of parking lots had significantly positive effects in all quantiles of these four datasets. The increase of income levels and parking lots would induce households to have more cars. Owner-occupied households may own more cars than non-owner-occupied households. This study further verifies that the marginal effects of household income and house ownership were larger in 1985 than the effects in other periods, indicating that responses of car demand to the income level or economic ability had declined with time. As the effects among the different quantiles are compared, the results show that the marginal effects of house ownership and parking lots variables were the strongest in the 50^th quantile, while the income effects showed small variations across quantiles.

The results of the demographic effects reveal that, in 1985, the effects of household size were only significantly positive in the 50^th quantile, while the effects were significantly positive for all of the quantiles in 1995, 2005, and 2015. The positive effects of household size support that an additional family member would generate higher car demand. Moreover, the marginal effects of household size may differ across the quantiles. In 1995, 2005, and 2015, the effects were greater in 75^th quantile than theses in other quantiles. Besides, the number of elderly members had negative effects on car ownership, suggesting that households with more elderly members would reduce car ownership. However, the effects of the number of children were positive in 1985 but negative in 1995, 2005, and 2015. The effects of the number of children may be complex. Households may need cars to pick up their children, but they also face greater economic burden that would crowd out the budget for cars if they have children. Thus, for the results of the 1995, 2005 and 2015 datasets, the later effect may dominate the former one.

With respect to the effects of transport attributes, the number of motorcycles had significantly negative effects in 1985 and 1995, but the effects were significantly positive in 2015. These results are similar with our previous findings. During the 1980s and 1990s, the relationship between car and motorcycle ownership was substitutive. Households may own motorcycles for cost-saving but not afford to buy cars. However, as households have higher economic ability, the relationship between motorcycle and car ownership may be complementary. Households may be able to own both cars and motorcycles since motorcycles can satisfy their need for mobility. The variable of public transit expenditure had negative effects on car ownership in all quantiles for all the four datasets, implying that public transit use could bring about the benefit of alleviating the growth of car ownership. As for the spatial effect, the finding shows that the urban variable had positive effects on household car ownership, which means that households living in the urban areas would own more cars than households living in the rural areas. The higher car demand in the urban areas may be related to higher living standards and meeting the demand to commute. The spatial effects are contrary to the typical anticipation that car ownership is generally lower in urban cities. Our result is in line with the findings of Zhao and Bai [37] and Le Vine et al. [38]. Their studies evidenced that, in China, car ownership in rural areas was lower than in urban areas. Zhao and Bai [37] suggested that this result is attributed to the income effect and increasing income inequality between urban and rural households has augmented the gap in car ownership, while Le Vine et al. [38] suggested that the reasons is inconclusive and further research will be needed. The higher level of car ownership in the urban areas also implies that the public transit network in the urban development may fail to suppress car demand. The fundamental strategy is to improve the quality and convenience of public transport services. More incentives for public transportation should be provided to attract car users.

As the effects across the different quantiles are compared, many explanatory variables, such as the household head characteristics and household income, would have stronger effects in the 90^th quantile in 1985. This result may be attributed to the high ratio of carless households. Thus, the effects of possible factors would be greater at the higher tail of the distribution. However, as the level of household car ownership had gradually increased, most households owned at least one car. Thus, in 1995, 2005, and 2015, the effects of possible factors may be greater at the 50^th or 75^th quantiles of the distribution. Households with one car would be more sensitive to the change of determinants and more likely to buy another car than those households with two cars.

For the past three decades, Taiwan had experienced dramatic changes in the demographic structure, such as population aging, number of births declining, and household size decreasing. Besides, household income level ever exhibited a high annual growth rate of 5.1% during the period 1980-1999. However, during the period 2000-2017, the annual income growth rate decreased to 0.3%, displaying a phenomenon of income stagnation. Therefore, this study further explores how the changes in the demographic structure and income level may bring about generational differences in household car ownership. Taking account that the impacts of explanatory variables may not be linear, the variables of household income, the ages of household heads and demographic structure are respecified. First, all households can be ranked according to household disposable income and divided into five groups. Thus, five dummy variables are used to represent households from the lowest to the highest income levels: Income1, Income2, Income3, Income4, and Income5. In the model, the lowest income group is used as the reference category. Second, households are classified into five groups according to the age of household heads: under 29 (Age1), 30-39 (Age2), 40-49 (Age3), 50-59 (Age4), and above 60 years old (Age5). Thus, the age of household heads can be transformed from a continuous variable to five dummy variables. The group headed by under 29 years old is used as the reference category. Third, the model considers that the effects of household structure may depend on the shares of different age groups. Thus, the model includes the proportion of children aged under 12 (Childrate), and the proportions of family members aged 30-39 (Age30rate), 40-49 (Age40rate), 50-64 (Age50rate), and above 65 (Age65rate). Specifically, the quantile regression model can be estimated after the variables of household income, age of household heads, and household structure are replaced. The definitions of other variables are the same as these in the previous section. Due to the limited space, Table 5 only reports the estimation results of the respecified variables.

Although the age effects were negative in the results of Table 4, the findings in Table 5 showed that the age effects may differ across the age levels. In 1985, the age effects were not significant in all quantiles. In 1995 and 2005, households with heads aged 50-59 would tend to have more cars than the reference group. In 2015, households with heads aged 40-49 would tend to have more cars than the reference group. These findings imply that as household heads were in their middle age, they would have better economic ability to own cars. Besides, in 2005, the results showed that households with heads aged above 60 would have fewer cars than the reference group, implying that the seniority effects would reduce the demand for cars.

As for the income effects, in 1985, only the highest and second highest income groups significantly owned more cars than the reference group in the 75^th quantile. In 1995, 2005, and 2015, the income variables were significantly positive for all quantiles, suggesting that households with a higher income level would own more cars than households with a lower income level. Particularly, the income effects would be greater for high-income households than low-income households. In addition, the income effects were stronger at the middle quantile than these at the higher quantile. Thus, the responses of car demand to the income level would be more sensitive for households with one car than those households with two cars.

With respect to the effects of household structure, in 1985 and 1995, the results revealed that the proportion of family members aged 50-64 and the proportion of elderly members had significantly negative effects on household car ownership, suggesting that the seniority effects would lead to the decrease of car demand. In 2005 and 2015, the finding that the higher share of the elderly would discourage car ownership can also be verified. However, different from the results in 1985 and 1995, the proportion of family members aged 50-64 had significantly positive effects on household car ownership in 2005 and 2015, which may be due to their better economic ability and postponed retirement. The proportion of children only had significantly positive effects in 1985. One important reason would be attributed to the declining of the proportion of children, which decreased from 0.23 in 1985 to 0.07 in 2015. Hence, these results indicated that the effects of household structure may change along with time.

On the whole, our results verify that the socio-demographic characteristics had significant impacts on household car ownership. The seniority effects would lead to a decrease of household car ownership. Two results can be observed. First, households in the later stages of the life cycle would own fewer cars. Second, households with a higher proportion of elderly members would have a lower level of car ownership. On the other hand, household in the middle stage of the family life cycle may tend to have more cars, which is owing to better economic ability. The findings in 1995 and 2005 showed that as household heads were in their middle age, they would own more cars. Besides, in 2005 and 2015, the increase of the share of family members aged 50-64 would incline to own more cars due to their better economic ability and postponed retirement. The results manifested that car ownership decisions would be associated with the life course, household structure, and economic ability. The important of the life course in car ownership decisions had been found in the study of Guo et al. [39]. They indicated that the temporal interdependencies between life course events would be involved in car ownership decisions. Furthermore, it is evident that travel patterns of different age groups would be diversified [40], [41]. Due to different roles in the life stages, travel purposes change during the life course.

Thus, our findings confirm that household car ownership would vary across different generations. The reasons would be related to the life course, household structure, and income effects. The future trend of car ownership may depend on demographic changes and income effects. The policy implication means that the vehicle management policies should pay attention to decoupling the income effects with car ownership. The demographic changes and generational differences in travel preferences should be considered in urban transportation planning. Moreover, considering the increasing life expectancy and aging population, an important concern in transport planning is to enhance the safety and accessibility of transport services for the elderly.