Mind the Income Gap: Behavior of Inequality Estimators from Complex Survey Small Samples

Income inequality measures are biased in small samples leading generally to an underestimation. After investigating the nature of the bias, we propose a bias-correction framework for a large class of inequality measures comprising Gini Index, Generalized Entropy and Atkinson families by accounting for complex survey designs. The proposed methodology is based on Taylor's expansions and Generalized Linearization Method, and does not require any parametric assumption on income distribution, being very flexible. Design-based performance evaluation of the suggested correction has been carried out using data taken from EU-SILC survey. Results show a noticeable bias reduction for all measures. A bootstrap variance estimation proposal and a distributional analysis follow in order to provide a comprehensive overview of the behavior of inequality estimators in small samples. Results about estimators distributions show increasing positive skewness and leptokurtosis at decreasing sample sizes, confirming the non-applicability of classical asymptotic results in small samples and suggesting the development of alternative methods of inference.Comment: 29 pages, 5 figures. Submitted for publicatio

Small Area Estimation of Inequality Measures using Mixtures of Betas

Economic inequalities referring to specific regions are crucial in deepening spatial heterogeneity. Income surveys are generally planned to produce reliable estimates at countries or macroregion levels, thus we implement a small area model for a set of inequality measures (Gini, Relative Theil and Atkinson indexes) to obtain microregion estimates. Considering that inequality estimators are unit-interval defined with skewed and heavy-tailed distributions, we propose a Bayesian hierarchical model at area level involving a Beta mixture. An application on EU-SILC data is carried out and a design-based simulation is performed. Our model outperforms in terms of bias, coverage and error the standard Beta regression model. Moreover, we extend the analysis of inequality estimators by deriving their approximate variance functions.Comment: 28 pages, 7 figures, 2 tables, 2 pages of supplementary materia

Comparing alternative distributional assumptions in mixed models used for small area estimation of income parameter

Linear Mixed Models used in small area estimation usually rely on normality for the estimation of the variance components and the Mean Square Error of predictions. Nevertheless, normality is often inadequate when the target variable is income. For this reason, in this paper we consider Linear Mixed Models for the log-transformed income (which require back-transformation for prediction of means and totals on the variable’s original scale) and a Generalized Linear Mixed Model based on the Gamma distribution. Various prediction methods are compared by means of a simulation study based on the ECHP data. Standard predictors obtained from Linear Mixed Model for the untrasformed income are shown to be preferable to the considered alternatives, confirming their robustness with respect to the failure of the normality assumption

Mind the income gap: bias correction of inequality estimators in small-sized samples

Income inequality estimators are biased in small samples, leading generally to an underestimation. This aspect deserves particular attention when estimating inequality in small domains. After investigating the nature of the bias, we propose a bias correction framework for a large class of inequality measures comprising Gini Index, Generalized Entropy and Atkinson index families by accounting for complex survey designs. The proposed methodology is based on Taylor’s expansions and does not require any parametric assumption on income distribution, being very flexible. Design-based performance evaluation of our proposal has been carried out using data taken from the EU-SILC survey, showing a noticeable bias reduction for all the measures. Lastly, a small area estimation exercise shows the risks of ignoring prior bias correction in a basic area-level model, determining model misspecification

Small area estimation of inequality measures using mixtures of betas

Economic inequalities referring to specific regions are crucial in deepening spatial heterogeneity. Income surveys are generally planned to produce reliable estimates at countries or macro region levels, thus we implement a small area model for a set of inequality measures (Gini, Relative Theil and Atkinson indexes) to obtain microregion estimates. Considering that inequality estimators are unit-interval defined with skewed and heavy-tailed distributions, we propose a Bayesian hierarchical model at area level involving a Beta mixture. An application on EU-SILC data is carried out and a design-based simulation is performed. Our model outperforms in terms of bias, coverage and error the standard Beta regression model. Moreover, we extend the analysis of inequality estimators by deriving their approximate variance functions

The Small Area Estimation of Economic Security: A Proposal

The objective of this work is to propose a small area estimation strategy for an economic security indicator. In the last decade the interest for the measurement of economic security or insecurity has grown constantly, especially since the financial crisis of 2008 and the pandemic period. In this work, economic security is measures through a longitudinal indicator that compares levels of equivalized household income over time. To solve a small area estimation problem, due to possible sample sizes too low in some areas, a small area estimation strategy is suggested to obtain reliable estimates of the indicator of interest. We consider small area models specified at area level. Besides the basic Fay-Herriot area-level model, we propose to consider some longitudinal extensions, including time-specific random effects following an AR(1) process or an MA(1) process. A simulation study based on EU-SILC data shows that all the small area models considered provide a significant efficiency gain with respect to the Horvitz-Thompson estimator, especially the small area model with MA(1) specification for random effects