4

# log V = a + b, log Y + b2 D (log Y - log Y*)

where V is the dependent variable, Y is per capita income, Y* is the benchmark income level that separates the countries into low-income and high-income groups, and D is a dummy variable that takes the value of zero for low-income countries and one for highincome countries. With this specification, b, is the elasticity for the low-income group, and (b, + b2) is the elasticity for high income groups. Results testing for non-linearity effects of income using piece-wise linear specifications for vehicle ownership, road length, and the vehicle-road ratio (presented in Appendix Tables 6 and 7) are also discussed in the next section. In these equations, we used $5,500 (in 1987 constant US dollars) as the benchmark income level to separate developing countries from high-income countries.

# Controlling Fixed Effects

Both the national and urban data sets contain observations from a wide range of country income levels, but data are available at only two or three points in time, usually at the beginning of decades or when decennial censuses are taken. Ingram and Liu (1997) presented charts indicating that the cross-section and time-series relations among variables such as vehicle ownership and income are sometimes very similar, but sometimes differ dramatically--particularly for road length at the urban level. The cross-section regressions with the use of year dummy variables, reported in Ingram and Liu (1997), do not really exploit the panel nature of the data. In those regressions, the _levels_ of income often explain much of the variations in the _levels_ of the dependent variable. It is often interesting to know how _changes_ in the dependent variable over a given period of time correspond to the _changes_ of the explanatory variables over the same period of time. In addition, it is often desirable to control for country or urban fixed effects which may be associated with the level of the dependent variable. This can be done by estimating the first difference regressions in the functional form shown below: log V2 - log VI = bo + E bi (log Xi2 -log Xil) k); where subscripts 1 and 2 denote two points in time; Vt, the dependent variable; X,t, the ith explanatory variable; and bo and bi, parameters. The first-difference regression results are shown in Appendix Table 8 for the national level and Appendix Table 9 for the urban level.

# Analyses with Urban Area Data

In Ingram and Liu (1997), income and population density were often used as explanatory variables in the same regression equation. As Figure 1 indicates, urban per capita income and population density are strongly and nonlinearly related. This crosssection correlation between income and population density at the urban level makes it difficult to interpret the coefficients of these two variables and their effects on urban area motorization when both are used as independent variables. In this study, we attempt to deal explicitly with this problem.

Figure 1 shows that urban population densities vary widely in high income cities. This variation in population density in high income cities reflects the impact of historical paths of development on urban form. Cities that have experienced much of their