107 research outputs found
Gibrat's law for cities: uniformly most powerful unbiased test of the Pareto against the lognormal
We address the general problem of testing a power law distribution versus a
log-normal distribution in statistical data. This general problem is
illustrated on the distribution of the 2000 US census of city sizes. We provide
definitive results to close the debate between Eeckhout (2004, 2009) and Levy
(2009) on the validity of Zipf's law, which is the special Pareto law with tail
exponent 1, to describe the tail of the distribution of U.S. city sizes.
Because the origin of the disagreement between Eeckhout and Levy stems from the
limited power of their tests, we perform the {\em uniformly most powerful
unbiased test} for the null hypothesis of the Pareto distribution against the
lognormal. The -value and Hill's estimator as a function of city size lower
threshold confirm indubitably that the size distribution of the 1000 largest
cities or so, which include more than half of the total U.S. population, is
Pareto, but we rule out that the tail exponent, estimated to be ,
is equal to 1. For larger ranks, the -value becomes very small and Hill's
estimator decays systematically with decreasing ranks, qualifying the lognormal
distribution as the better model for the set of smaller cities. These two
results reconcile the opposite views of Eeckhout (2004, 2009) and Levy (2009).
We explain how Gibrat's law of proportional growth underpins both the Pareto
and lognormal distributions and stress the key ingredient at the origin of
their difference in standard stochastic growth models of cities
\cite{Gabaix99,Eeckhout2004}.Comment: 7 pages + 2 figure
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