On Variance Estimation for a Gini Coefficient Estimator Obtained from Complex Survey Data

Obtaining variances for the plug-in estimator of the Gini coefficient for inequality has preoccupied researchers for decades with proposed analytic formulae often cumbersome to apply, in addition to being obtained assuming an iid structure. Bhattacharya (2007, Journal of Econometrics) provides an (asymptotic) variance when data arise from a complex survey, a sampling design common with data frequently used in inequality studies. Under a complex survey sampling design, we prove that Bhattacharya’s variance estimator is equivalent to an asymptotic version of the estimator derived by Binder and Kovacevic (1995, Survey Methodology) more than a decade earlier. In addition, we show that Davidson’s (2009, Journal of Econometrics) derived variance, for the iid case, is a simplification of that provided by Binder and Kovacevic. These results are computationally useful, as the Binder and Kovacevic variance estimator is straightforward to calculate in practice. To aid applied researchers, we show how auxiliary regressions can be used to generate the plug-in Gini estimator and its asymptotic variance, irrespective of the sampling design. Health data on the body mass index for Bangladeshi women is employed in an illustration.