Zipf's law for city sizes has been traditionally examined by rank size rule regression, where log of city size is regressed on log of its rank. Zipf's law approximately holds when the data is generated from a Pareto distribution with a unit tail exponent. One common way of checking Paretoness is to run an auxiliary rank size rule regression with an additional regressor of squared log (rank), and investigate if the coefficient equals to zero. However, it would be more natural to test the joint hypothesis that the coefficient of log (rank) be -1 and that of squared log (size) be 0. This paper proposes to use the F statistic, but one difficulty is that it does not have the standard distribution. We construct the critical region by simulation and apply it to empirical data.
