Rank-1/2: A Simple Way to Improve the OLS Estimation of Tail Exponents
Rank-1/2: A Simple Way to Improve the OLS Estimation of Tail Exponents
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DOI：
10.1198/jbes.2009.06157
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发表时间：
2011-01-01
影响因子：
3
通讯作者：
Ibragimov, Rustam
中科院分区：
文献类型：
--
作者：
Gabaix, Xavier;Ibragimov, Rustam
尽管有更复杂的方法可用，但估计帕累托指数的流行方法仍然是运行OLS回归：log（Rank）= a - B log（Size），并将B作为帕累托指数的估计值。这种流行的原因可以说是这种方法的简单性和鲁棒性。不幸的是，这种方法在小样本中有很大的偏差。我们为这种偏差提供了一个简单实用的补救措施，并建议，如果想使用OLS回归，应该使用秩-1/2，运行log（秩-1/2）= a - B log（大小）。1/2的移位是最佳的，并且将偏差减小到领先阶。帕累托指数xi上的标准误差不是OLS标准误差，而是渐近（2/n）（1/2）xi。数值结果表明，所提出的方法的优势，在标准的OLS估计程序，并表明，它表现出良好的依赖重尾过程表现出偏离幂律。考虑的估计程序说明了使用经验应用Zipf定律为美国城市规模分布。
Despite the availability of more sophisticated methods, a popular way to estimate a Pareto exponent is still to run an OLS regression: log(Rank) = a - b log(Size), and take b as an estimate of the Pareto exponent. The reason for this popularity is arguably the simplicity and robustness of this method. Unfortunately, this procedure is strongly biased in small samples. We provide a simple practical remedy for this bias, and propose that, if one wants to use an OLS regression, one should use the Rank - 1/2, and run log(Rank - 1/2) = a - b log(Size). The shift of 1/2 is optimal, and reduces the bias to a leading order. The standard error on the Pareto exponent xi is not the OLS standard error, but is asymptotically (2/n)(1/2)xi. Numerical results demonstrate the advantage of the proposed approach over the standard OLS estimation procedures and indicate that it performs well under dependent heavy-tailed processes exhibiting deviations from power laws. The estimation procedures considered are illustrated using an empirical application to Zipf's law for the United States city size distribution.