Small Area Estimation of the Relative Median Poverty Gap


SAE in welfare and wellbeing studies

Enrico Fabrizi (Università Cattolica del S. Cuore)
Maria Ferrante (Department of Statistics, University of Bologna)$star
Carlo Trivisano (Department of Statistical Sciences "Paolo Fortunati", University of Bologna, Italy)

The reduction of poverty in Europe is a milestone in the Europe 2020 strategy. The most frequently used indicator in Europe, the At Risk of Poverty Rate (ARPR), measures the share of people below the poverty threshold but it has a serious shortcoming: it neglects how the left tail of the income distribution is shaped and thereby it misses to give indication of the depth of poverty. The relative median poverty gap (RMPG) tries to complement the information provided by the APRR by measuring the relative distance between the median income of the poor and the poverty threshold. Specifically, it is defined as the difference between the poverty threshold and the median income of individuals below the threshold, relative to the threshold. Again, to limit the attention to the ARPR can be misleading in this sense: ARPT can be unaffected by poverty relief policies that increase the incomes of all those in poverty without necessarily raising anyone above the threshold. We focus on the RMPG, that is based on median income of the poor, instead of the mean poverty gap, as the latter is more sensitive to extremely low and negative incomes. Moreover, the RMPG is one of the ‘Laeken Indicators’, a set of common European statistical indicators on poverty and social exclusion , selected by the European council in 2001. Eurostat publishes estimates of the RMPG only at national level since the EU-Survey of Income and Living Condition (EU-SILC) doesn’t reach a sample size large enough to produce reliable estimates at sub-national level. In this research we propose a small area estimation model for the RMPG.
As far as estimation is concerned, we adopt a Hierarchical Bayesian approach implemented by means of a MCMC computational method. We discuss an application based on data from the Italian sample of the EU-SILC survey.
The literature of small area estimation of poverty indicators only recently has devoted attention to the estimation of the poverty gap. Molina et al. (2014), Rao and Molina (2016) adopt models for the estimation of mean poverty gap, specified at the unit level. When specified on a transformation of the income variable, as it is often the case, small area estimation based on these model requires unit-level auxiliary data for each non sampled unit in the population. We work within the “area level” model framework, where models are estimated starting from design based estimates, so they easily incorporate information on sampling design and on nonresponse adjustments. The design based estimates of RMPG at small area level are very imprecise. Denoting with na the small sample size in area a, these estimates are obtained using incomes of those who are poor among the na belonging to the sample, that can be very few in small in areas with a limited ARPR and small na. Moreover, in our experience the auxiliary information that can be used to explain RMPG has a really weak predictive power.
To overcome these problems we propose a small area model based on a suitable income distributional assumption. We exploit the advantages of parametric estimation of income distributions that allows to explicitly express poverty and inequality measures as functions of the parameters of the assumed distribution. Specifically, in order to describe the equivalized income we assume a Generalized Beta distribution of the second kind (GB2), a four parameter distribution, that includes as special cases or as limiting cases a number of distributions frequently used to describe income (Kleiber and Kotz, 2003). Moreover, empirical studies on income (Jenkins, 2009) tend to show that the GB2 outperforms other four-parameter distributions for modeling income. In order to obtain reliable small area estimates of the RMPG, we did not make use of direct estimators of the same quantity but we express the RMPG as functions of the parameters characterizing the GB2 distribution. The direct estimators are in fact not only characterized by a very large variance for the reason discussed above but they are also biased since standard survey weighted estimators of the median are biased in small samples when the underlying distribution is skewed.
We checked the robustness of estimators obtained as posterior summaries with respect to the distributional assumption. The estimates obtained for RMPG in small areas are characterized by a very satisfying coefficient of variation. The proposed estimators are also design consistent: the difference between model based and direct estimators go to zero as the domain specific sample size increases.

S. P. Jenkins, Distributionally-Sensitive Inequality Indices and the GB2 Income Distribution, Review of Income and Wealth (2009) 55, 2, 392–8. C. Kleiber, S. Kotz, Statistical Size Distributions in Economics and Actuarial Sciences, John Wiley & Sons, Hoboken (2003). I. Molina, B. Nandram, J.N.K. Rao, Small Area Estimation of General Parameters with Application to Poverty Indicators: A Hierarchical Bayes Approach, Annals of Applied Statistics (2014), 8, 852-885. J. N. K. Rao, I. Molina, Empirical Bayes and Hierarchical Bayes Estimation of Poverty Measures for Small Areas, in Analysis of Poverty Data by Small Area Estimation (ed M. Pratesi), Wiley (2016).
Keywords: Relative median poverty gap, Generalized Beta distribution of the second kind

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