Statistical Inference via Bootstrapping for Measures of Inequality

Abstract

In this paper we consider the use of bootstrap methods to compute interval estimates and perform hypothesis tests for decomposable measures of economic inequality. The bootstrap potentially represents a significant gain over available asymptotic intervals because it provides an easily implemented solution to the Behrens-Fisher problem. Two applications of this approach, using the PSID (for the study of taxation) and the XLSY (for the study of youth inequality), to the Gini coefficient and Theil's entropy measures of inequality, are provided. The results suggest that (i) statistical inference is essential even when large samples are available, and (ii) the bootstrap appears to perform well in this setting. 'Cowell(1989a) is a notable esception. 2See, for example, Gastwirth (19i4), Gastwirth et al. (1986) and Cowell (1989a). (1994) provides a thorough review and some examples of use of asymptotic results. 1 Maasoumi small sample properties of these intervals are not known. Further, all the decomposable inequality measures used in the literature are bounded (e.g. the Gini coefficient lies in the [0, l] ' t m erval), so that application of standard asymptotic results may lead to estimated intervals that extend beyond the theoretical bounds of a particular measure (e.g. a negative lower bound for Gini). An alternative method for computing probability intervals is to bootstrap. The bootstrap provides interval estimates drawn from the small sample distribution. These interval estimates have been shown to be superior to asymptotic intervals both the-3 oretically and in a variety of applications.. Bootstrap intervals are computationally inexpensive and easy to calculate, the same method applies to all the inequality measures used in the literature, and the bootstrap method automatically takes into account any bounds that, apply to a particular measure. Further, since bootstrap intervals computed using the percentile met.hod have a clear Bayesian interpretation, they provide a straightforward solution to the Behrens-Fisher problem of comparing means from two distributions (see section 3).

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Extensions to decomposition of the redistributive effect of health care finance

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Estimating the empirical Lorenz curve and Gini coefficient in the presence of error with nested data

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