The aim of this work is to test empirically the validity of Gibrat’s Law -also known as the Law of Proportional Growth, which establishes that the growth rate of a variable is independent of its initial size- in the growth of cities, using data for all the twentieth century of the complete distribution of cities (without any size restrictions) in three countries: the US, Spain and Italy. To do that, we use parametric and non-parametric methods. Starting from the following general expression of the growth equation, ln(Sit)-ln(Sit-1)=μ+βln(Sit-1)+Uit, where Sit is the size of city i at the time t and Uit is a random variable representing the random shocks which the growth rate may suffer (which we shall suppose to be identically and independently distributed for all cities), if β=0 Gibrat’s Law holds and we obtain that growth is independent of the initial size. The results of the OLS estimation of β for the three countries considering all the cities, without size restrictions, show that is β always significantly different to zero (we have calculated the t-ratios using White’s Heteroskedasticity-Consistent Standard Errors, because the residues resulting from these regressions are usually heteroskedastic), for any period and in the three countries, and the estimated parameter is always positive (except in the period 1970-1980 in the US), so that the three exhibit divergent behavior throughout the 20th century. On considering the entire size distribution of cities, we find a tendency to divergence.
However, this fact does not impede, whether from an empirical or a theoretical point of view, that city size distribution can be adequately approximated with a lognormal distribution. Also, the conclusions which can be obtained as to fulfilment or not of Gibrat’s Law depend, first, on the sample size -we again estimate the growth equation for different sample sizes: 50, 100, 200, 500, 1000 and so on, adding groups of 500 cities at a time until they are all considered, starting with the largest cities (from upper tail) and the smallest cities (from lower tail), and find what we have called the critical sample size for the US, Spain and Italy, the size from which we reject the null hypothesis β=0 with a significance level of 5%- and, second, on the size of the cities being considered (large or small) -we use a non-parametric method to estimate the growth rate conditional on city size as a local mean, and again the results show that cities of different sizes do not present significant differences in their growth rates, which should lead us to accept Gibrat’s Law, with the evidence being somewhat more favourable in the US than in Spain and Italy (in these last two countries, especially in Spain, the growth rate presents some tendency to growth according to city size)-, which means that the results of any study which does not use all the distribution will be relative.