Estimation of Poverty Indicators at the Small Area Level in Italy

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                   Estimation of Poverty Indicators at the Small Area Level in Italy
               Monica Pratesi1; Caterina Giusti1; Stefano Marchetti2
                  University of Pisa, Dept. of Statistics and Mathematics Applied to Economics, Via Ridolfi, 10, 56124, Pisa, Italy.
                 Scuola Superiore Sant’Anna, Piazza Martiri della Liberta’, 33, 56127, Pisa, Italy.


               Available data to measure poverty and living conditions in Italy come mainly from sample surveys, such as

               the European Survey on Income and Living Conditions (EU-SILC). However, these data can be used to
               produce accurate estimates only at the national or regional level. To obtain estimates referring to smaller,
               unplanned domains, small area methodologies can be used. Among these methods, linear mixed models are
               the more popular. M-quantile models for small area estimation represent an interesting alternative, since they
               do not depend on strong distributional assumption and they are robust against outlying area values.

               Small Area Methods for Poverty Estimates
               Let xi be a known vector of auxiliary variables for each population unit i in small area j , with N j
               denoting the number of population units in area j . Assume that information for the variable of interest yij ,
               the household disposable income on an equivalised scale for unit i in small area j , is available only for the
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               n j sampled units in area j , denoted as s j . The target is to use these data to estimate the cumulative
               distribution function of the household income. For this purpose, the yij values of the N j − n j not sampled
               units in ares j , denoted as rj , need to be predicted under a given small area model. We consider linear

               mixed models, and in particular the unit level nested error regression model (Battese et al., 1988), and M-
               quantiles models, an approach to small area estimation based on the quantiles of the conditional distribution
               of the variable of study y given the covariates (Chambers and Tzavidis, 2006). When an M-quantile model
               for y holds, a bias adjusted estimator of the cumulative distribution function of y based on the proposal by

               Chambers and Dunstan (1986) is:

                                F j (t ) = N j { ∑ I ( yij ≤ t ) + n j ∑ ∑ I{ [xij β (θ j ) + ( ykj − y kj )] ≤ t}}
                                             −1                     −1          T ˆ
                                ˆ CD                                                   ˆ              ˆ                        (1)
                                                 i∈s j                 i∈r j k ∈s j

               where y kj = xT β (θ j ) is a linear combination of the auxiliary variables and θ j is an estimate of the average
                     ˆ       ij
                                ˆ ˆ                                                             ˆ
               value of the M-quantile coefficient of the units in area j . Analytic estimation of the standard error of
               estimator (1) is complex. As a consequence, we adapt to the small area problem the bootstrap procedure
               suggested by (Lombardia et al., 2003).

               Estimation of Poverty Measures in Tuscany
               Data sources for the present application are the 2004 EU-SILC survey (for the n j sampled units in area j )

           T   and the 2001 Population Census of Italy (for the rj not sampled units in area j ). The small areas of interest
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               are the 10 Tuscany Provinces plus the Municipality of Florence, considered as a stand-alone area. The
               characteristic of interest y is the household disposable income; the auxiliary variables, known for each
               household in the population, include the size of the household, the age of the head of the household, and
               other significant covariates in the models for the household income.

               Figure 1 represents the M-quantile biased adjusted estimates of the cumulative distribution function (cdf)
               obtained with (1) for the richest area, the Municipality of Florence (red solid line, on the right), and for the

               poorest one, the Province of Massa (black solid line, on the left). The direct estimates of the small area

               distribution function are in dashed lines, they are obtained simply applying the Horvitz and Thompson
               estimator to the n j sampled units in area j .

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                           Figure 1: Estimate of the cumulative distribution function of household income.

               Note that the estimation of the quantiles of the cdf allows following the behavior of the income distribution

           T   across the areas. The cumulative distribution of the Province of Massa rapidly approaches the value 1, and it
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               is always steeper than the analogous distribution of the Municipality of Firenze. Moreover, the amount of
               people immediately after the poverty line (vertical line) is bigger under the cdf of the Province of Massa than
               under the cdf of the Municipality of Firenze. us 126). In the next step of the analysis we will estimate the
               mean squared error of (1) with the proposed bootstrap methodology, in order to track a confidence interval
               around the cumulative distribution function lines in Figure 1.


               [1]   Battese, G., Harter, R. and Fuller, W. (1988). An Error-Components Model for Prediction of County

                     Crop Areas using Survey and Satellite Data. Journal of the American Statistical Association, 83, 28-
               [2]   Chambers R., Tzavidis N. (2006) M-quantile models for small area estimation, Biometrika, 93, 255-
               [3]   Chambers, R. and Dunstan, R. (1986) Estimating distribution functions from survey data, Biometrika,
                     73, 597-604.
               [4]   Lombardia, M. and Gonzalez-Manteiga, W. and Prada-Sanchez, J. (2003) Bootstrapping the
                     Chambers-Dunstan estimate of finite population distribution function, Journal of Statistical Planning
                     and Inference, 116, 367-388.


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