Introduction Poverty rate M-quantile model Results Summary A three-level M-quantile model for poverty mapping in Poland Maciej Beręsewicz 1 , Stefano Marchetti 2 , Nicola Salvati 2 , Marcin Szymkowiak 1 , Łukasz Wawrowski 1 1 Poznan University of Economics and Business 2 University of Pisa 19-08-2016 SAE 2016 Maastricht - The Netherlands
Introduction Poverty rate M-quantile model Results Summary Outline 1 Introduction Introduction 2 Poverty rate Poverty rate definition Aim of the presentation Teritorial aggregation of Poland Review of using SAE in Poland – poverty mapping 3 M-quantile model M-quantile model A three level M-quantile model 4 Results Results 5 Summary SAE 2016 Maastricht - The Netherlands
Introduction Poverty rate M-quantile model Introduction Results Summary Introduction Acording to Central Statistical Office 1 (CSO) in Poland, the poverty indicator for the whole country in 2011 based on EU- SILC survey amounts to 17.7%. CSO does not publish information about 2 the poverty indicator for the lower level of spatial aggregation (NUTS 2 and lower levels). This information is available only at the level of the whole country and at the regional level (NUTS 1). Figure: The poverty rate at the level of regions (NUTS 1) SAE 2016 Maastricht - The Netherlands
Introduction Poverty rate M-quantile model Introduction Results Summary Introduction In Poland there is a huge demand for information about poverty indicators at 3 lower level of spatial aggregation. Poverty maps are required in Poland by Ministry of Infrastructure and Devel- 4 opment, Ministry of Labour and Social Policy, local authorities and many other organizations. Given the small sample size in the relevant cross classifications of the EU-SILC 5 survey, it is necessary to use chosen techniques of indirect estimation draw on alternative data sources to estimate the parameters of interest at low levels of spatial aggregation with acceptable precision. SAE 2016 Maastricht - The Netherlands
Introduction Poverty rate definition Poverty rate Aim of the presentation M-quantile model Teritorial aggregation of Poland Results Review of using SAE in Poland – poverty mapping Summary Poverty rate Definition of poverty rate Percentage of persons with an equivalised disposable income (after social transfers) below the at-risk-of-poverty threshold set at 60% of the national median of equivalised disposable income. Formula for poverty rate ARPR = N u N · 100 , (1) where: N u — number of households living in poverty, N — number of all households. SAE 2016 Maastricht - The Netherlands
Introduction Poverty rate definition Poverty rate Aim of the presentation M-quantile model Teritorial aggregation of Poland Results Review of using SAE in Poland – poverty mapping Summary Aim of the presentation Estimation of at-risk-poverty indicator (poverty rate, head count ratio - HCR) at LAU 1 level (in Polish poviats) using a three-level M-quantile model. SAE 2016 Maastricht - The Netherlands
Introduction Poverty rate definition Poverty rate Aim of the presentation M-quantile model Teritorial aggregation of Poland Results Review of using SAE in Poland – poverty mapping Summary Teritorial aggregation of Poland 7 regions (NUTS 1) 16 voivodships (NUTS 2) 66 subregions (NUTS 3) 379 poviats (NUTS 4/LAU 1) 2478 gminas (NUTS 5/LAU 2) SAE 2016 Maastricht - The Netherlands
Introduction Poverty rate definition Poverty rate Aim of the presentation M-quantile model Teritorial aggregation of Poland Results Review of using SAE in Poland – poverty mapping Summary Teritorial aggregation of Poland 7 regions (NUTS 1) 16 voivodships (NUTS 2) 66 subregions (NUTS 3) 379 poviats (NUTS 4/LAU 1) 2478 gminas (NUTS 5/LAU 2) SAE 2016 Maastricht - The Netherlands
Introduction Poverty rate definition Poverty rate Aim of the presentation M-quantile model Teritorial aggregation of Poland Results Review of using SAE in Poland – poverty mapping Summary Teritorial aggregation of Poland 7 regions (NUTS 1) 16 voivodships (NUTS 2) 66 subregions (NUTS 3) 379 poviats (NUTS 4/LAU 1) 2478 gminas (NUTS 5/LAU 2) SAE 2016 Maastricht - The Netherlands
Introduction Poverty rate definition Poverty rate Aim of the presentation M-quantile model Teritorial aggregation of Poland Results Review of using SAE in Poland – poverty mapping Summary Teritorial aggregation of Poland 7 regions (NUTS 1) 16 voivodships (NUTS 2) 66 subregions (NUTS 3) 379 poviats (NUTS 4/LAU 1) 2478 gminas (NUTS 5/LAU 2) SAE 2016 Maastricht - The Netherlands
Introduction Poverty rate definition Poverty rate Aim of the presentation M-quantile model Teritorial aggregation of Poland Results Review of using SAE in Poland – poverty mapping Summary Teritorial aggregation of Poland 7 regions (NUTS 1) 16 voivodships (NUTS 2) 66 subregions (NUTS 3) 379 poviats (NUTS 4/LAU 1) 2478 gminas (NUTS 5/LAU 2) SAE 2016 Maastricht - The Netherlands
Introduction Poverty rate definition Poverty rate Aim of the presentation M-quantile model Teritorial aggregation of Poland Results Review of using SAE in Poland – poverty mapping Summary Review of using SAE in Poland – poverty mapping In 2013 the Center for Small Area Estimation, which is a special unit at the Statistical Office in Poznan, in cooperation with Central Statistical Office of Poland and the World Bank prepared a poverty map of Poland at the level of subregions (NUTS 3) using the Fay-Herriot approach. By implementing the Fay-Herriot area level model it was possible to produce estimates of the HCR in Poland at the level of subregions, i.e. at a lower level of aggregation than the direct estimates published by official statistics so far. This has increased the scope of information about poverty: it is now available at the level of 66 subregions. SAE 2016 Maastricht - The Netherlands
Introduction Poverty rate definition Poverty rate Aim of the presentation M-quantile model Teritorial aggregation of Poland Results Review of using SAE in Poland – poverty mapping Summary Review of using SAE in Poland – poverty mapping A preliminary analysis of the poverty map 1 created using the SAE methodology has revealed a difference between Central and Eastern Poland (with a higher poverty rate) and Western Poland, characterized by a lower poverty rate. It was the first step to use the SAE 2 methodology to estimate poverty rate in Poland. The next step is to create poverty map in 3 Poland at LAU 1 level using data from EU- SILC, Polish Local Databank and Polish National Census of Population and Hous- ing 2011 and a three level M-quantile Figure: The poverty rate at the level of model. subregions (NUTS 3) SAE 2016 Maastricht - The Netherlands
Introduction Poverty rate M-quantile model M-quantile model A three level M-quantile model Results Summary M-quantile model The classical regression model summarizes the behavior of the mean of a random variable at each point in a set of covariates. Instead, quantile regression summarizes the behavior of different parts of the conditional distribution f ( y | x ) at each point in the set of the x ’s. A linear regression model for the q conditional quantile of f ( y | x ) is: Q y ( q | X ) = X β ( q ) , (2) where X is a design matrix and y j is a scalar response variable corresponding to a realization of a continuous random variable with unknown continuous cumulative distribution function. Estimates of β ( q ) are obtained by minimizing the following function: n � | y j − x T j β ( q ) |{ qI ( y j − x T j β ( q ) > 0 ) + ( 1 − q ) I ( y j − x T j β ( q ) ≤ 0 ) } . (3) j = 1 SAE 2016 Maastricht - The Netherlands
Introduction Poverty rate M-quantile model M-quantile model A three level M-quantile model Results Summary M-quantile model The M-quantile regression is a ”quantile-like” generalization of regression based on influence functions (M-regression). The M-quantile q of the conditional density f ( y | x ) , denoted by m , is defined as the solution to the estimating equation: � ψ q ( y − m ) f ( y | x ) dy = 0 , (4) where ψ q is an asymmetric influence function that is the derivative of an asymmet- ric loss function, ρ q . When a linear relation between M-quantile m and auxiliary variables holds, then the M-quantile regression model is: m y ( q | X ) = X β ψ ( q ) . (5) SAE 2016 Maastricht - The Netherlands
Introduction Poverty rate M-quantile model M-quantile model A three level M-quantile model Results Summary M-quantile model An estimate of β ψ ( q ) is obtained by minimizing: n � ρ q ( y j − x T j β ψ ( q )) . (6) j = 1 The minimization of (6) reduces to the following estimating equation: n � 2 ψ q ( s − 1 ( y j − x T j β ψ ( q ))) = 0 , (7) j = 1 where ψ q is the derivative of the Huber loss function ρ q , and it is equal to: � q ψ ( u ) if u > 0 , ψ q ( u ) = (8) ( 1 − q ) ψ ( u ) if u ≤ 0 , with ψ ( u ) = uI ( | u | ≤ c )+ sgn ( u ) cI ( | u | > c ) and c is a tuning constant. Provided that the tuning constant c is strictly greater than zero, estimates of β ψ ( q ) are obtained using iterative weighted least squares (IWLS). SAE 2016 Maastricht - The Netherlands
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