The Use of a Three-LevelM-Quantile Model to Map Poverty at Local Administrative Unit 1 in Poland

Abstract

Summary A three‐level M‐quantile model for small area estimation is proposed. The methodology represents an efficient alternative to prediction by using a three‐level linear mixed model in the presence of outliers and it is based on an extension of M‐quantile regression. A modified method of the traditional M‐quantile (two‐level) approach for poverty estimation is also proposed. In addition, an estimator of the mean‐squared prediction error is described, which is based on a bootstrap procedure. The methodology proposed, as well as the three‐level empirical best predictor, are applied to Polish European Union Survey on Income and Living Conditions and census data to estimate poverty at local administrative unit 1 level in Poland, i.e. the level for which the Central Statistical Office of Poland has not published any official estimates to date.

A Comparison of Methods for Poverty Estimation in Developing Countries

“…Thus, MQ estimator of FGT indicators can be written as where , and because the model is based on log per capita income or expenditure, the estimate of per capita expenditure for the k ‐th non‐sampled HH is where e ik is a sample residual defined by the MQ fitted model to . Marchetti et al () proposed an alternative procedure to calculate following an MC simulation approach parallel to the EBP approach, and Marchetti et al () have extended the usual two‐level MQ model to a three‐level model.…”

“…The small area‐specific regression coefficients are then used to predict response values for the individuals belonging to the area. In this sense, the MQ method also incorporates a form of the between‐area variation and ignores the between‐cluster variation, except when a three‐level model is applied as in Marchetti et al ().…”

“…A set of six papers, all of which use EBP or MQ, illustrate the underlying data‐dependent issues. Tzavidis et al (), Giusti et al (), Marchetti et al (), Tzavidis et al (), Marchetti et al () and Marhuenda et al () all use some form of unit record census data. Marhuenda et al () use EBP with categorical data and model‐based unit record census data and consider a three‐level model that includes variation at domain and subdomain level and seek small area estimates at both these levels.…”

“…Unlike ELL and EBP, MQ is distribution‐free and automatically provides outlier robust inference and relaxes assumptions of random errors in the traditional unit level nested error regression model. MQ has been used in poverty mapping studies by Tzavidis et al () and been applied to estimate poverty indicators at local administrative level in European countries including Albania (Tzavidis et al , ), Tuscany (Marchetti et al , ), Italy (Giusti et al , ) and Poland (Marchetti et al , )…”

“…Unlike ELL and EBP, MQ is distribution-free and automatically provides outlier robust inference and relaxes assumptions of random errors in the traditional unit level nested error regression model. MQ has been used in poverty mapping studies by Tzavidis et al (2010) and been applied to estimate poverty indicators at local administrative level in European countries including Albania (Tzavidis et al, 2008), Tuscany , Italy (Giusti et al, 2011) and Poland (Marchetti et al, 2018) M-quantile does not specify small area or cluster level variation explicitly because it puts aside boundary issues when developing the model. ELL and EBP can suffer from model misspecification if there is an untenable parametric specification of random errors or under the influence of marked outliers.…”