Deliverable 2.2. Small Area Estimation of Indicators on Poverty and Social Exclusion

Transcription

1 Deliverable 2.2 Small Area Estimation of Inicators on Poverty an Social Exclusion Version: 2011 Risto Lehtonen, Ari Veijanen, Mio Myrsylä an Maria Valaste The project FP7-SSH AMELI is supporte by European Commission funing from the Seventh Framewor Programme for Research.

2 II Contributors to Deliverable 2.2: Chapter 1: Risto Lehtonen, Ari Veijanen, Mio Myrsylä an Maria Valaste, University of Helsini. Chapter 2: Risto Lehtonen, Ari Veijanen, Mio Myrsylä an Maria Valaste, University of Helsini. Chapter 3: Risto Lehtonen, Ari Veijanen, Mio Myrsylä an Maria Valaste, University of Helsini. Chapter 4: Risto Lehtonen, Ari Veijanen, Mio Myrsylä an Maria Valaste, University of Helsini. Chapter 5: Mio Myrsylä, University of Helsini. Chapter 6: Risto Lehtonen, Ari Veijanen, Mio Myrsylä an Maria Valaste, University of Helsini. Main responsibility Risto Lehtonen, University of Helsini Data provision an commenting Timo Alano, Pauli Ollila, Marjo Pyy-Martiainen, Statistics Finlan; Rui Selja, Statistics Slovenia; Kaja Sõstra, Statistics Estonia. Evaluators Internal evaluator: Matthias Templ, Vienna University of Technology. AMELI-WP2-D2.2

3 III Aim an Objectives of Deliverable 2.2 There is increasing user eman for regional or sub-population official statistics within the EU. In many countries, statistics on poverty an social exclusion are base on sample surveys, such as the SILC survey. One of the aims state for the AMELI project was to investigate the aaptation of moern small area an omain estimation (SAE) approaches for selecte inicators on poverty an social exclusion (Laeen inicators). At-ris-of poverty rate, the Gini coefficient, relative meian at-ris-of poverty gap an quintile share ratio were selecte for consieration. Estimation approaches examine in Wor Pacage 2 involve the use of auxiliary population ata an statistical moels for borrowing strength for regional (e.g. area sizes below NUTS3) an small area estimation purposes. The methos inclue esign-base moel-assiste estimators an moel-base estimators. The relative merits an practical applicability of the methos was assesse by simulation experiments using real register an survey ata. It was consiere important to cover a broa variety of typical practical estimation settings existing in ifferent EU countries. Therefore, the methos were investigate uner various statistical infrastructures, sampling esigns, omain compositions an outlier contamination schemes. In many cases, the methos assume access to unit-level auxiliary population ata. This option is becoming increasingly realistic in statistical infrastructures of the EU countries, where opportunities to use aministrative registers an population census ata for statistical purposes are improving. Methos were also evelope that use aggregate-level auxiliary ata, which option is useful for countries where aggregate auxiliary ata are available for example from official statistics sources. The accompanying R programs coes were provie for practical application of the methos. In the prouction of Deliverable 2.2 on small area statistics methoology, the aim was to combine expertise from acaemic research with expertise from Official statistics proucers. NSIs involve inclue Statistics Finlan, Statistics Estonia an Statistics Slovenia. University of Helsini has the main responsibility of the prouction of the eliverable

4 IV Contents 1 Introuction Objectives Basic approaches Estimation approaches Report structure Planne an unplanne omain structures Direct an inirect estimators Estimation of poverty inicators The role of moels an auxiliary ata The role of moels The role of auxiliary information Estimation uner outlier contamination 11 2 Basic properties of omain estimators 12 3 Moels an estimators Moels an auxiliary ata Design-base estimators Horvitz-Thompson estimator Generalize regression estimator Moel calibration Moel-base estimators Synthetic estimator EBLUP an EBP estimators Transformations of preictions Frequency-calibrate preictors calculate using nown omain marginal totals of auxiliary variables Composite estimators Simulation-base methos 36 AMELI-WP2-D2.2

5 V 4 Estimators for poverty inicators an results of Monte Carlo simulation experiments Introuction Experimental esign Register-base population from Western Finlan Amelia population Quality measures Contamination schemes Estimators At-ris-of poverty rate HT-CDF estimator Methos base on poverty inicators Simulation results The Gini coefficient Poverty gap Quintile share ratio S20/S Classifying omains by poverty 69 5 Case stuy: Estimation of poverty rate an its variance Introuction Design Estimators Poverty rate estimators Variance estimators Results Poverty rate estimators Variance estimators 80 6 Discussion of results General New preictors Comparison of outlier an contamination mechanisms 84 References

6 VI Annex 1. Manual for R coes 93 Annex 2. AMELI WP 2 Estimation: Summary of SAE methos 98 Annex 3. Technical summary of selecte estimator types 100 AMELI-WP2-D2.2

7 1 1 Introuction 1.1 Objectives There are increasing nees in the society for accurate statistics on poverty an social exclusion (poverty inicators for short) prouce for ifferent population subgroups or omains such as regional areas an emographic groups. One of the aims of the AMELI project was to investigate the current (stanar) methos for omain an small area estimation of poverty inicators an evelop new methos where appropriate. This report presents the methoological evelopments an summarizes our main finings on statistical properties of propose estimators. Properties of estimators of selecte poverty inicators (so-calle Laeen inicators as agree in Laeen European Council in December 2001) were stuie by simulation experiments. The stuy ha the following objectives: 1. Investigation of statistical properties (bias an accuracy) of stanar irect estimators of the selecte poverty inicators for population omains an small areas. Stanar estimators o not use auxiliary ata or moelling. 2. Introuction of alternative estimators, which use statistical moels an auxiliary ata at the unit level, an investigation of bias an accuracy of the new estimators. 3. Introuction of estimators that use auxiliary ata at an aggregate level an investigation of bias an accuracy of these estimators. 4. Implementation of points 1 to 3 uner equal an unequal probability sampling schemes. 5. For stuying robustness of methos, the implementation of points 1 to 4 uner various outlier contamination schemes. 6. Stuy of applicability of a metho incorporating a novel transformation of preictions. 7. Implementation of points 1 to 5 for populations from two ifferent ata sources, register-base ata maintaine by Statistics Finlan (the Western Finlan population) an sample survey ata from EU-wie SILC survey (the Amelia population)

8 2 1.2 Basic approaches Estimation approaches This report presents the research one at University of Helsini in the context of AMELI Wor Pacage 2 on the estimation of selecte inicators on poverty (monetary Laeen inicators) for omains an small areas. Domain estimation of poverty has been recently stuie by D Alo et al. (2006), Fabrizi et al. (2007a, 2007b), Srivastava (2009), Molina an Rao (2010), an Haslett et al. (2010). Verma et al. (2010) reports empirical results for regional estimation using EU-SILC ata. The inicators consiere in this report are the following: At-ris-of poverty rate The Gini coefficient Relative meian at-ris-of poverty gap Quintile share ratio (S20/S80 ratio). The inicators are typically nonlinear an are base on non-smooth functions such as meians an quintiles, which maes the estimation a non-trivial tas. This hols especially for the estimation of the inicators for omains an small areas. In this report, both esign-base an moel-base or moel-epenent methos are evelope an investigate for the estimation of the selecte poverty inicators for omains an small areas. Design-base methos are chosen because of the ominance of the framewor in official statistics prouction. Moel-base approaches are important to be covere because in many small area estimation situations, moelbase methos provie a realistic solution. Design-base estimation for finite population parameters refers to an estimation approach where the ranomness is introuce by the sampling esign. In esign-base estimation, it is emphasize that estimators shoul be esign consistent an, preferably, nearly esign unbiase at least in omains with meium-size samples (an estimator is nearly esign unbiase if its bias ratio bias ivie by stanar AMELI-WP2-D2.2

9 3 eviation approaches zero with orer 1/2 On ( ) when the total sample size n tens to infinity (Estevao an Särnal, 2004)). For a nearly esign unbiase estimator, the esign bias is, uner mil conitions, an asymptotically insignificant contribution to the estimator s mean square error (Särnal, 2007, p. 99). This property is inepenent of the choice of the assisting moel. Generalize regression (GREG) type estimators an calibration type estimators are examples of nearly esign unbiase estimators. Moel-assiste GREG estimators are constructe such that they are robust against moel mis-specification. GREG an moel-free calibration are iscusse in Särnal, Swensson an Wretman (1992) an Särnal (2007). Lehtonen an Veijanen (2009) iscuss GREG an moelfree calibration in the context of omain estimation. In calibration, we concentrate on moel calibration estimators, introuce in Wu an Sitter (2001). Moel calibration has been evelope for omain estimation in Lehtonen, Särnal an Veijanen (2009). In GREG an moel calibration we often employ estimators that use nonlinear assisting moels involving ranom effects in aition to the fixe effects. Design-base estimators for omains an small areas are usually constructe so that the complexities of the sampling esign, such as stratification an unequal inclusion probabilities, are accounte for. For example, it is customary that esign weights are incorporate in a esign-base estimation proceure. This oes not necessarily hol for moel-base or moel-epenent methos. In this respect, a conceptual separation of moel-base an moel-epenent methos can be helpful. In strict moelepenent methos, the estimation is consiere to rely exclusively on the statistical moel aopte. For example, esign weights o not play any role in a moelepenent estimation proceure. For esign consistency, variables that capture (at least some) of the sampling complexities, such as stratification variables an PPS size variable, can be inclue in the unerlying moel. In moel-base methos, esign weights can be incorporate in the estimation proceure to account for unequal probability sampling, leaing to esign consistent pseuo synthetic, pseuo EBLUP (empirical best linear unbiase preictor) an pseuo EBP (empirical best preictor) type approaches (see e.g. Rao, 2003; You an Rao, 2002; Jiang an Lahiri, 2006). The methos coincie uner equal probability sampling. In this report, we use moel

10 4 base as a general concept unless it is instructive to treat separately the two approaches. Moel-base estimators can have esirable properties uner the moel but their esign bias oes not necessarily ten to zero with increasing omain sample size (Hansen, Hurvitz an Maow, 1978; Hansen, Maow, an Tepping, 1983; Särnal, 1984, an Lehtonen, Särnal an Veijanen, 2003). Moel-base methos for small area estimation inclue a variety of techniques such as synthetic (SYN) an composite estimators, EBLUP an EBP type estimators an various Bayesian techniques, such as empirical Bayes an hierarchical Bayes. The monograph by J.N.K. Rao (2003) provies a comprehensive treatment of moel-base small area estimation (SAE). Mixe moels that are commonly use in SAE are iscusse for example in Jiang an Lahiri (2006). Moel-base small area estimation methoology was extensively stuie in the context of the EU s FP6 research project EURAREA (Enhancing Small Area Estimation Techniques to meet European Nees, ), see The EURAREA Consortium (2004). EURAREA concentrate mainly on the estimation of small area totals an means an recommene the moel-base methos for official statistics prouction for small areas (e.g. area sizes below NUTS3). In AMELI we exten the SAE methoology to consierably more complex statistics incluing the Gini coefficient, relative meian at-ris-of poverty gap an quintile share ratio. In aition to moel-base methos, avance esign-base methos are evelope Report structure The report inclues the escription of the estimators evelope for the selecte poverty inicators an the results of the Monte Carlo simulation experiments on the statistical properties (bias an accuracy) of the estimators. The report is organize as follows. The remainer of this section covers the efinition of the basic concepts an introuces the estimators of the poverty inicators to be examine as well as the role of moels an auxiliary ata in the construction of the estimators. Section 2 summarizes the basic properties of the various estimator types for omains an small areas. A technical escription of the moels an estimators is inserte in Section 3. AMELI-WP2-D2.2

11 5 Section 4 contains a etaile escription of the specific estimators of the inicators an presents the results of Monte Carlo experiments. Section 5 is evote to a case stuy on a moel-assiste estimator of poverty rate; special attention is in the estimation of the variance of the estimator. Discussion is in Section Planne an unplanne omain structures Different omain structures can appear in practical applications of omain estimation (Lehtonen an Veijanen, 2009). Sampling esign may be base on nowlege of omain membership of units in population. If the sampling esign is stratifie, omains being the strata, the omains are calle planne (Singh, Gambino an Mantel, 1994). For planne omain structures, the population omains can be regare as separate subpopulations. Therefore, stanar population estimators are applicable as such. The omain size in every omain is often assume nown an the sample size n in omain sample s is fixe in avance. Stratifie sampling in connection to a suitable allocation scheme such as optimal (Neyman) or power (Banier) allocation is often use in practical applications, in orer to obtain control over omain sample sizes (e.g. Lehtonen an Pahinen, 2004). Singh, Gambino an Mantel (1994) escribe allocation strategies to attain reasonable accuracy for small omains, still retaining goo accuracy for large omains. Falorsi, Orsini an Righi (2006) propose sample balancing an coorination techniques for cases with a large number of ifferent stratification structures to be aresse in omain estimation. If the omain membership is not incorporate into the sampling esign, the sizes of omain samples will be ranom. The omains are then calle unplanne. Unplanne omain structures typically cut across esign strata. The property of ranom omain sample sizes introuces an increase in the variance of omain estimators. In aition, extremely small number (even zero) of sample elements in a omain can be realize, if the omain size in the population is small. Unplanne omain structures are commonly encountere in practice, because it is impossible to inclue all relevant omain structures into the sampling esign of a given survey. Unplanne omain structures are often assume in this report. n s

12 6 1.4 Direct an inirect estimators It is avisable to separate irect an inirect estimators for omains (Lehtonen an Veijanen, 2009). A irect estimator uses values of the variable of interest only from the time perio of interest an only from units in the omain of interest (Feeral Committee on Statistical Methoology, 1993). A Horvitz-Thompson (HT) type estimator provies a simple example of irect estimator. In moel-assiste estimation, irect estimators are constructe by using moels fitte separately in each omain. A irect omain estimator can still incorporate auxiliary ata outsie the omain of interest. This is relevant if accurate population ata about the auxiliary x-variables are only available at a higher aggregate level. An inirect omain estimator uses values of the variable of interest from a omain an/or time perio other than the omain an time perio of interest (Feeral Committee on Statistical Methoology, 1993). In general, inirect estimators are attempting to borrow strength from other omains an/or in a temporal imension. Inirect moel-assiste estimators for omains are iscusse in the literature (e.g. Estevao an Särnal, 1999, Lehtonen, Särnal an Veijanen, 2003, 2005, an Hiiroglou an Pata, 2004). Inirect estimators are use extensively in this report; this especially hols for omains whose sample size is small. Direct estimators are occasionally use in cases where the omain sample sizes are large. Direct estimators also serve as reference or benchmar estimators when investigating the bias an accuracy of the propose inirect estimators. 1.5 Estimation of poverty inicators The poverty (Laeen) inicators iscusse in this report can be ivie into two groups with respect to the selecte estimation approach. For the estimation of at-risof poverty rate base on poverty inicators, we use GREG an moel calibration type estimators (featuring esign-base moel assiste methos) an SYN an EBLUP or EBP type estimators (featuring moel-base methos). In all these estimators, logistic moels are use because the unerlying stuy variable is binary. Direct estimators, AMELI-WP2-D2.2

13 7 such as Horvitz-Thompson type estimators, are use as basic or reference estimators, sometimes also calle efault estimators in this report. In aition to the estimation of poverty rate for omains an small areas, we have examine methos for the ientification of omains that can be characterize as poor, i.e. omains whose estimate poverty rate falls below a given threshol. Raning of omains is part of so-calle triple-goal estimation, where the goal is to obtain goo rans, goo histogram an accurate omain estimates (Rao, 2003; Shen an Louis, 1998; Paoc et al., 2006). Juins an Liu (2000) present methos for improving the estimate range of omain estimators. The equivalize income constitutes the ey variable unerlying the poverty (monetary Laeen) inicators. Equivalise income is efine as the househol's total isposable income ivie by its "equivalent size", to tae account of the size an composition of the househol, an is attribute to each househol member (incluing chilren) (European Commission, 2006). Equivalization is mae on the basis of the OECD moifie scale, which assigns weight 1.0 for the first ault, 0.5 for every aitional person age 14 or over, an 0.3 for every chil uner 14. Relative meian at-ris-of poverty gap (poverty gap for short) an quintile share ratio (S20/S80 ratio) are examples of inicators that rely on meians or quantiles of the cumulative istribution function (CDF) of the unerlying continuous variable. For these inicators, HT type irect estimators, synthetic an composite estimators are evelope. A composite estimator is constructe as a linear combination of a esign-base irect estimator an a moel-base SYN estimator. In aition, for poverty gap we have stuie estimation of conitional expectations by simulation-base methos, resembling methos introuce in Molina an Rao (2010). In constructing the estimators, we use logarithmic transformation to correct for the sewness of the istribution of the stuy variable. In bac-transformation we first trie the RAST (Ratio Ajuste by Sample Total; Chambers an Dorfman, 2003, Fabrizi et al., 2007b) type transformation, an later evelope more elaborate transformations aime at improving the histogram of transforme preictions. The statistical properties (esign bias an accuracy) of the estimators of the selecte poverty inicators are examine with Monte Carlo simulation experiments. Real ata

14 8 taen from statistical registers of Statistics Finlan are use in constructing the frame populations. We have mae experiments also with the synthetic Amelia population (Alfons et al. 2011b). The populations contain a wie selection of socio-economic an emographic auxiliary variables. We have concentrate on esign-base simulation settings. Programs written in R language have been prouce for statistical computing of the selecte poverty inicators for omains an small areas. The R coes are escribe in a separate supplemental eliverable Veijanen an Lehtonen (2011). 1.6 The role of moels an auxiliary ata The role of moels Choice of statistical moel unerlying an estimator of a poverty inicator constitutes an important phase of the estimation proceure for omains an small areas. In constructing moel-assiste an moel-base estimators, we use selecte moels from the family of generalize linear mixe moels (GLMM, e.g. McCulloch an Searle, 2003). Linear an logistic fixe-effects an mixe moels are extensively use. Lehtonen, Särnal an Veijanen (2003, 2005) iscuss the choice of the moel in the context of GREG estimation. The rationale behin the choice of the assisting moel for GREG is the following. In GREG estimation for omains, various types of stuy variables can be use. For example, a linear moel formulation is appropriate for a continuous variable, an logistic moels are usually chosen for binary or polytomous variables. We call extene GREG family the GREG estimators that use GLMM s as assisting moels. In the parametrization of the assisting moel for an extene GREG family estimator, it is important for accurate omain estimation to account for the possible omain ifferences. Basically, omain ifferences can be accounte for either with a fixe- AMELI-WP2-D2.2

15 9 effects or a mixe moel specification. A fixe-effects moel is usually a efault in GREG estimation. Mixe moel specification offers a flexible approach for omain estimation (Lehtonen, Särnal an Veijanen, 2003, 2005) an is much use in our research. Because of this moel choice, the resulting estimators for omains are in most cases of inirect type The role of auxiliary information The availability of high-quality auxiliary information is crucial for reliable estimation for omains an small areas. Auxiliary information can be incorporate into the sampling esign (e.g. stratifie sampling, PPS sampling) or into the estimation proceure (or both). Stratifie sampling is often use to obtain sufficient sample size for the most important omains of interest (leaing to planne omains). In this report we concentrate on the use of auxiliary ata in the estimation proceure. Both equal probability an unequal probability sampling esign are iscusse, uner unplanne omain structures (referring to cases where the omains of interest are not efine as strata in the sampling esign). The reason for incorporating auxiliary ata in an estimation proceure is obvious: improve accuracy is attaine if strong auxiliary ata are available for omain estimation. Different types of auxiliary ata can be use in estimation for omains an small areas. The auxiliary ata can be aggregate at the population level or at the omain level, or at an intermeiate level. Aggregates are often taen from reliable auxiliary sources such as population census or other official statistics; this case is common in many European countries an North America. If the auxiliary ata are inclue in a sampling frame, as is the case in many European countries, notably in Scaninavia, the necessary auxiliary totals can be aggregate at the esire level from unit-level ata sources. A rapily progressing tren in official statistics prouction is the use of unit-level auxiliary ata for omain an small area estimation. These ata are incorporate in the estimation proceure by unit-level statistical moels. Uner this option, register ata (such as population census register, ifferent unit-level aministrative an statistical registers) can be available as frame populations an sources of auxiliary

16 10 ata. Moreover, the registers often contain unique ientification eys that can be use in merging at micro level ifferent register sources an ata from registers an sample surveys. Known omain membership for all population elements is often assume. Many countries, both in Europe an in the European Union, are progressing in the evelopment of reliable population registers that can be accesse for statistical purposes. Goo examples are Austria, Estonia, Finlan an Slovenia, which have representation in the AMELI project. Obviously, access to micro-merge register an survey ata provies great flexibility for the evelopment of methos for omain estimation an in the omain estimation practice. All estimator types (except HT an relate irect estimators) examine in this report aim at using information about auxiliary variables in the population. We have first assume access to unit-level auxiliary information. The reason is that this option offers much flexibility for estimator construction. Uner this option, a moel is fitte to the sample ata, preictions are calculate for all population elements using the estimate moel parameters an the nown values of the auxiliary variables, an the preictions in the population contribute to the estimation of the inicators of interest, such as poverty rate in the given omains an small areas. Because the option of the use of unit level auxiliary ata for statistical purposes is not (yet) commonly available in statistical infrastructures within the EU, we exten the methoology to cases where only aggregate-level auxiliary ata are available. In the metho we only assume that the population totals of continuous auxiliary variables, or population frequencies of classes of iscrete variables, are nown. A calibration metho is introuce to calculate the necessary preicte values. We have not applie Bayesian methos (e.g., Fabrizi et al., 2005) or moels involving spatial or temporal correlations (Chanra et al., 2007). SAE methos that borrow strength in spatial or temporal imension were evelope an investigate to some extent in the context of the EU s FP5 project EURAREA Estimation uner outlier contamination AMELI-WP2-D2.2

17 11 In eveloping estimators that are robust against outlier contamination we iscuss the contamination mechanisms an moels propose in the WP4 woring ocument by Hulliger an Schoch (2010). Outlying mechanisms consiere are OCAR (outlying completely at ranom) an OAR (outlying at ranom), an the contamination moels are CCAR (contaminate completely at ranom), CAR (contaminate at ranom), an NCAR (not contaminate at ranom). The efinitions of these concepts are given in the woring ocument referre above. 2 Basic properties of omain estimators Known esign-base properties relate to bias an accuracy of esign-base moelassiste estimators an moel-epenent estimators for omains an small areas are summarize in Table 1 (Lehtonen an Veijanen, 2009). Moel-assiste estimators such as GREG an calibration are esign consistent or nearly esign unbiase by efinition, but their variance can become large in omains where the sample size is small. Moel-epenent estimators such as synthetic an EBLUP estimators are esign-biase: the bias can be large for omains where the moel oes not fit well. The variance of a moel-epenent estimator can be small even for small omains, but the accuracy can be poor if the square bias ominates the mean square error (MSE), as shown for example by Lehtonen, Särnal an Veijanen (2003, 2005). For a moel-epenent estimator, the ominance of the bias component together with a small variance can cause poor coverage rates an invali esign-base confience intervals. For esign-base estimators, on the other han, vali confience intervals can be constructe. Typically, moel-assiste estimators are use for major or not-sosmall omains an moel-epenent estimators are use for minor or small omains where moel-assiste estimators can fail. Table 1 inicates that small omains present problems in the esign-base approach. Purcell an Kish (1980) call omain a mini omain when its share of population is smaller than 1%. In so small omains, especially irect estimators can have large variance. Small omains are the main reason to prefer inirect moel-base estimators to irect esign-base estimators (Rao, 2003)

18 12 Table 1. Design-base properties of moel-assiste an moel-epenent estimators for omains an small areas Bias Precision (Variance) Accuracy (Mean Square Error, MSE) Confience intervals Design-base moel-assiste methos GREG an calibration estimators Design unbiase (approximately) by the construction principle Variance may be large for small omains Variance tens to ecrease with increasing omain sample size MSE = Variance (or nearly so) Vali esign-base intervals can be constructe Moel-base an moel-epenent methos Synthetic an EBLUP estimators Design biase Bias oes not necessarily approach zero with increasing omain sample size Variance can be small even for small omains Variance tens to ecrease with increasing omain sample size MSE = Variance + square Bias Accuracy can be poor if the bias is substantial Vali esign-base intervals not necessarily obtaine In practice, there are two main approaches to esign-base estimation for omains: irect estimators that are usually applie for planne omain structures (such as strata whose sample sizes n are fixe in the sampling esign) an inirect estimators whose natural applications are for unplanne omains (whose omain sample sizes are ranom). In moel-base or moel-epenent SAE, inirect estimators that aim at borrowing strength are often use. AMELI-WP2-D2.2

19 13 3 Moels an estimators The fixe an finite population of interest is enote U = {1, 2,...,,..., N}, where refers to the label of population element. A omain is a subset of population U such as a regional population in NUTS3 or NUTS4 region or a emographic subivision within the regional areas. Poverty rate estimates, for example, are require not only for regions but also for classes efine by age an gener. Consier a region r an a class c. They efine a omain : in population U, a subset U = U r U c contains people belonging to class c ( U ) in region r ( U ). The number of units in the omain c in population is enote by N. In sample s, corresponing subsets are efine as s = sr sc with n observations. Naturally, regions are special cases of omains. A small area is a omain whose realize sample size is small (even zero). r Many poverty inicators are compose of omain totals, frequencies an meians. The omain total of the stuy variable y (equivalize incomes) is efine as t = y, (1) U where y enotes the value of the stuy variable for element. The frequency f of a class C, such as the frequency of persons with income smaller than a threshol, is written as a sum of class inicators v = I{ y C} : f = v. (2) U For a binary inicator, (1) an (2) obviously coincie

20 Moels an auxiliary ata Auxiliary information is use in moel-assiste an moel-base methos. The available auxiliary information consists of an auxiliary x-vector an a omain membership specification I = 1 if U, I = 0 otherwise, = 1,..., D, for every unit U. Letting x enote the value of the auxiliary vector for unit, we thus assume that both x an omain membership I is nown for every U. Moels are incorporate in moel-assiste (GREG, moel calibration) an moelbase (synthetic, EBLUP, EBP) methos. Consier a generalize linear fixe-effects moel, E ( Y ) = f( x ; β ), for a given function f (; β ), where β requires estimation, an m E m refers to the expectation uner the moel (Lehtonen an Veijanen, 2009). Examples of f (; β ) are a linear functional form an a logistic function. The moel fit to the sample ata {( y, ); s} x yiels the estimate ˆβ of β. Using the estimate parameter values, the vector value x an the omain membership of, we compute the preicte value assumptions. yˆ = f ( x ; βˆ ) for every U, which is possible uner our A similar reasoning applies to a generalize linear mixe moel involving ranom effects in aition to the fixe effects. The moel specification is E ( Y u ) = f ( x ( β+ u )), where u is a vector of ranom effects efine at the m omain level. Using the estimate parameters, preicte values are compute for all U. yˆ = f ( x ( βˆ + uˆ )) Let us iscuss linear moels in more etail. For a linear fixe-effects moel Y = x β + ε, U we erive two special cases, a common moel formulation an a moel formulation involving omain-specific intercepts. AMELI-WP2-D2.2

21 15 Uner the common moel formulation, we have x = (1, x1,..., x J ), nown for every U, an β = ( β0, β1,..., β J ) where β j are fixe effects common for all omains, j = 0,..., J. Uner the moel formulation with omain-specific intercepts, we have x = ( I1,..., ID, x1,..., x J ), I = 1 if U, I = 0 otherwise, = 1,..., D, an β = ( β01,..., β0d, β1,..., β J), where β 0 are omain-specific intercepts an β j are common slopes, j = 1,..., J. In both special cases, preicte values ŷ ˆ = x β are calculate for every U. The rationale behin the two special cases is the following. If a single (common) fixe-effects moel is assume to hol in all omains, possible ifferences between omains are not necessarily capture by the estimator, although in GREG the weighte sum of resiuals corrects for esign bias cause by the possible moel misspecification. For fixe effects moel, there is some theoretical support for using omain-specific intercepts, or at least regional inicators, to account for possible ifferences between regions. Then the beta parameters, or slopes, associate with explanatory x-variables are often specifie common to all omains. The two special cases of moels result in an inirect omain estimator. A irect estimator is obtaine by using separate slopes for every omain in aition to the separate intercepts, that is, a moel Y = x β + ε, U. This moel woul probably result in too unstable omain estimates, in particular if the omain sample size is small. On the other han, a omain-specific moel might be realistic for omains with a large sample size. In orer to account for possible ifferences between regions, a linear mixe moel incorporates omain-specific ranom effects u N σ for omain U, or 2 ~ (0, u) regional ranom effects 2 r ~ (0, u) u N σ for region U r, where U Ur. For omainspecific ranom intercepts, a linear mixe moel is given by 2 Y = x β + u + ε, U, ε ~ N(0, σ ),

22 16 or, more generally, y = Xβ + Zu +ε 2 2 for a matrix Z. The parameters β, σ u an σ are first estimate from the ata, an the values of the ranom effects are then preicte. An example of a generalize linear mixe moel formulation is a binomial logistic mixe moel for a binary y-variable. We want to estimate the totals all omains U. The logistic mixe moel is of the form t = U y for exp( x ( β+ u)) Em( y u) = Py { = 1 u} = 1 + exp( x ( β+ u ) for U, = 1,..., D, where x is a nown vector value for every U, β is a vector of fixe effects common for all omains, an u is a vector of omain-specific ranom effects. Here again, preictions y ˆ = exp( x ( βˆ + uˆ )) /(1 + exp( x ( βˆ + u ˆ )) are calculate for every U. Lehtonen, Särnal an Veijanen (2005) give several special cases of the moel. An inirect estimator for omains is obtaine with mixe moel specification. We have fitte most of the mixe moels with R function nlme. By efault it uses the maximum lielihoo metho. In nlme, the esign weights o not contribute to estimation. Design weights can be inclue in moel fitting with R function glmer (pacage lme4). When fitting the fixe effects moels, we have use esign weights. AMELI-WP2-D2.2

23 Design-base estimators Horvitz-Thompson estimator Horvitz-Thompson (HT) estimator of omain total (1) is a weighte sum of values in the sample: tˆ = ay, (3) s where the esign weights a are inverses of inclusion probabilities π ( a = 1/ π ). An HT estimator is a irect estimator. It oes not incorporate any moel. The estimator is esign unbiase but it can have large variance, especially for small omains. HT estimator is often use uner planne omain structures, where the omain sample sizes are sufficiently large Generalize regression estimators Generalize regression (GREG) estimators (Särnal et al., 1992; Lehtonen an Veijanen, 2009) are assiste by a moel fitte to the sample. By choosing ifferent moels we obtain a family of GREG estimators with same form but ifferent preicte values (Lehtonen et al., 2003, 2005, 2007). Orinary GREG estimator (4) tˆ = yˆ + a ( y yˆ ) ; GREG U s incorporating a linear regression moel is use to estimate omain totals (1) of a continuous stuy variable. For a binary or polytomous response variable, a linear moel formulation will not necessarily fit the ata well. A logistic moel formulation might be a more realistic choice. LGREG (logistic GREG; Lehtonen an Veijanen, 1998) estimates the frequency f of a class C in each omain. A logistic regression

24 18 moel is fitte to the inicators v = I{ y C}, s, using the esign weights. The fitte moel yiels estimate probabilities pˆ = Pv { = 1; x, β ˆ}. The LGREG estimator of the class frequency in U is. (5) fˆ = pˆ + a ( v pˆ ) ; LGREG U s Here ˆ is the sum of preicte values in the population. Thus it is necessary p U to have access to unit level population information about the persons auxiliary variables. The last component of (5), i.e. an HT estimator of the resiual total, aims at correcting the possible bias of the first (synthetic) part. It is obvious that for certain moel choices, notably for a omain-specific moel formulation, the last component vanishes. A so-calle omain size correction (Lehtonen an Veijanen, 2009) is incorporate into an estimator efine as ˆ N f = pˆ + a ( v pˆ ); Nˆ = a. (6) ; LGREG(2) ˆ U N s s In the MLGREG estimator (Lehtonen an Veijanen, 1999, Lehtonen, Särnal an Veijanen, 2005, Torabi an Rao, 2008), an alternative logistic mixe moel involving fitte values pˆ = Pv { = 1; x, βu ˆ, ˆ } is use instea of a fixe-effects logistic moel. The ranom effects are associate with omains or with regions. This moel formulation may be a realistic option for many situations in practice Moel calibration Calibration is typically use to construct an estimator as weighte sample sum with weights chosen so that the weighte sample sums of auxiliary variables are ientical with nown population totals (Estevao an Särnal, 2004; Kott, 2009). In moel calibration introuce by Wu an Sitter (2001) an Wu (2003), preictions are use instea of auxiliary variables. We have generalize moel calibration for omain AMELI-WP2-D2.2

25 19 estimation (Lehtonen et al., 2009). A moel is first fitte to the sample. We iscuss only a logistic regression moel, although any moel coul be applie. The estimator of the total frequency is a weighte sum of inicators over the whole sample, region or the omain. The weights are chosen so that the weighte sum of estimate probabilities over a subset of sample equals the sum of preicte probabilities over a corresponing subset of population. The sum of weights over the sample subset must equal the size of the population subset. Moreover, the weights shoul be close to the esign weights. The proceure of fining such weights is calle calibration (e.g. Särnal, 2007). In population level calibration (Wu an Sitter, 2001), the weights must satisfy calibration equation wz, ˆ i i = zi = N pi, (7) i s i U i U where z i = (1, pˆ ). Using the technique of Lagrange multiplier ( λ ), we minimize i ( ) 2 w a λ wz i i zi s a i s i U uner the conitions (7). The first part of the equation is the istance between the weights w an the nown esign weights constraints (7). The equation is minimize by weights a. The latter part correspons to the w = ag; g = 1+ λ z, where λ = zi az i i azz i i i i U i s i s

26 20 The omain estimator is efine as a omain sum fˆ = wv. (8) ; pop s In our experiments, this estimator has not performe well. The first choice for omain level calibration is equation w, ˆ izi = zi = N pi, (9) i s i U i U where the weights sizes must be nown. We minimize w i are specific to the omain. From (9) we see that the omain 2 ( w a ) λ wizi zi s a i s i U uner the calibration equations (9). The solution is w = ag ; g = 1+ λ z, where λ = zi az i i azz i i i i U i s i s 1. The frequency in the omain is estimate by a weighte sum of inicators over the omain: fˆ = w v. (10) ; s s AMELI-WP2-D2.2

27 21 We call this estimator semi-irect, referring to the fact that the sum contains only observations from the omain. It is not irect, however, as the weights are etermine by a fitte moel that incorporates all sample values. Next we introuce some semiinirect estimators incorporating observations outsie the omain. The first semi-inirect omain level calibration estimator is a sum over the whole sample with omain-specific weights w that are close to weights a in the omain an close to zero outsie the omain. In other wors, the weights shoul be close to I { s} a = I a ( I = I { s} ). The calibration equation is w z = z. (11) i s i i i i U We minimize 2 ( w Ia ) λ wizi zi s a. i s i U The solution is w = I a + λ az; λ = zi Iiaz i i azz i i i i U i s i s 1. The estimator is efine as a weighte sum over the whole sample: f = w v. (12) ˆ ; s s Alternatively, the summation extens only over the omain. We have also consiere a similar estimator efine as a regional sum:

28 22 fˆ = w ν, (13) ; s sr where the subset s r of sample contains all the people in the same region r as the omain. The calibration equation is w z = z. i sr i i i i U We minimize 2 ( w Ia ) λ wizi zi s a r i sr i U obtaining w = I a + λ az; λ = zi Iiaz i i azz i i i i U i sr i sr 1. This estimator apparently borrows strength from other omains in same region. Estevao an Särnal (2004) have shown that borrowing strength is not always a goo iea, but they consier a ifferent class of calibration estimators. In contrast with their estimators, our estimator is a sum over a set larger than the omain, an the weights are close to zero outsie the omain. AMELI-WP2-D2.2

29 Moel-base estimators Synthetic estimator Synthetic (SYN) estimator is typically a sum of preicte values over the population elements in a omain. In the case of a logistic moel, synthetic estimator is the sum of preicte probabilities: fˆ ; LSYN = pˆ. (14) U For logistic SYN (LSYN) estimator using a logistic fixe-effects moel, the preictions are pˆ = Pv { = 1; x, β ˆ}, an pˆ = Pv { = 1; x, βu ˆ, ˆ } for a MLSYN estimator using a logistic mixe moel. Obviously, LSYN estimator (14) constitutes the first component of the LGREG estimator (5) EBLUP an EBP estimators The EBLUP estimator (empirical best linear unbiase estimator, e.g. Rao, 2003, p. 95) is use in the context of a linear mixe moel Y = x β + u + ε, U, or, more generally, y = Xβ + Zu +ε for a matrix Z. Uner the first mixe moel the omain total s conitional expectation given the ranom effects u is E Y u = x β + Nu. U U

30 24 This woul be an optimal preictor of the omain total in the sense of minimizing MSE. Its best linear unbiase preictor (BLUP) is tˆ ˆ x β ˆ, BLUP = ( σu, σ ) + Nu ( σu, σ ) U where the optimal estimators of β an u epen on unnown variance components 2 2 σ u an σ as follows: For 2 ( ; ) 2 R = Cov εσ, G = Cov( u ; σ u ) an V = R + ZGZ, ˆ(, ) ( ) σu σ ( = ) β XV X XV y an ˆ u = uˆ ( σ, σ ) GZ V ( y Xβ ). In EBLUP (empirical BLUP), the variances are estimate an plugge into the BLUP equation: ˆ tˆ ( ˆ, ˆ ) ˆ ( ˆ, ˆ EBLUP = x β σu σ + Nu σu σ ). U Another in of EBLUP, here calle EBLUP(Y) (Saei an Chambers, 2004), contains the conitional expectation of only that part of sum which is not observe in sample, E Y u = + ( N n ) u U s U s x β. The sample observations are inclue in the EBLUP(Y) estimator tˆ ˆ ˆ ˆ N n uˆ ˆ ˆ y EBLUP( Y ) = x β ( σu, σ ) + ( ) ( σu, σ ) + U s s AMELI-WP2-D2.2

31 25 EBLUP an EBLUP(Y) shoul have smaller MSE than GREG estimators, but they may have consierable esign bias, especially if the esign weights vary substantially. The EBLUP estimators can be written using the preicte values yˆ = x β ˆ + uˆ in forms resembling the synthetic estimator: t ˆ = y ˆ ˆ σ ˆ σ 2 2 ; EBLUP ( u, ) U For a logistic mixe moel the EBP (empirical best preictor, e.g. Jiang an Lahiri, 2006) estimators are of the form fˆ = pˆ ˆ σ ˆ σ (15) 2 2 ; EBP ( u, ) U s, (16) fˆ = pˆ ˆ σ ˆ σ + v 2 2 ; EBP( Y ) ( u, ) U s s where preictions are pˆ = exp( x βˆ + uˆ ) /(1 + exp( xβ ˆ + uˆ ). 3.4 Transformations of preictions The synthetic estimator of a poverty inicator constructe from preictions is usually biase, in part ue to the transformation of observations. As the income y is approximately istribute as lognormal, a moel is fitte to z = log( y + 1), an the fitte values z ˆ are bac-transforme to yˆ = exp( zˆ ) 1. This shoul be followe by a bias correction. A RAST bias correction term c RAST, (Ratio Ajuste by Sample

32 26 Total; Chambers an Dorfman, 2003, Fabrizi et al., 2007b) woul be chosen in each omain so that the weighte sample sum of c ˆ, y over the omain equals the RAST weighte omain sample sum of the original incomes y. However, RAST correction merely corrects the mean of preictions without affecting significantly their sprea. It ignores the fact that the tails of the istribution of incomes usually contribute significantly to a poverty inicator. For example, the quintile share incorporates the first an last quintiles. Unfortunately, the istribution of preictions is concentrate aroun the average an the income istribution erive from the preictions is unrealistically even. Therefore, synthetic estimates of Gini coefficient an poverty gap ten to be too small an quintile share estimate from preictions is often too large. Moreover, the ifferences between synthetic omain estimates are too small. We introuce linear an non-linear transformations as generalizations of the RAST correction. We transform preictions so that they have similar histogram as the observe values. The transformation incorporates esign weights even when they cannot be use in fitting the moel, as is the case in many current R pacages. This may reuce the esign bias. Consier preictions y ˆ for units in population omain ( U ). We compare the istributions of preictions an sample values by ifferences of percentiles. The percentiles of the y ˆ ( U ) are enote by p ˆ c, 1 c 99. The corresponing percentiles of the sample values y ( s ), enote by p c, are obtaine from the HT estimate of the cumulative istribution function. Thus esign weights contribute to the proceure. Our goal is to fin a linear transformation efine by parameters a an b so that the percentiles of expane preictions y = a + byˆ are close to * corresponing percentiles * y, U p c of observations. Let. We minimize the ifferences between the percentiles * p c enote the cth percentile of * p c an p c : AMELI-WP2-D2.2

33 27 c * ( ) 2 c c S = p p. By noting that p = a + bpˆ we obtain * c c ( ˆ ) 2. c c S= a + bp p c Obviously, S is minimize for parameters a an linear regression moel with preictions are pˆ c b by OLS corresponing to a x= an y = pc. The transforme omain y = a + byˆ. (17) * ˆ ˆ Wea auxiliary information may lea to negative transforme preictions (17). Here we outline a proceure for avoiing negative values. We erive non-linearly transforme preictions corresponing percentiles of log ( ) y with percentiles of log ( ) y, U, close to y, s. As the percentiles of log-transforme vectors are logarithms of the original percentiles (although this oes not always hol for the meian), we minimize c ( a log ( ˆ ) log ( )) 2 + b pc pc. The parameters a an b are again foun by OLS. Expane preictions y are then efine by log ( y ) = aˆ + bˆ log ( yˆ ), that is, ( ˆ ˆ log ( ˆ )) y = exp a + b y. (18)

34 28 These expane preictions are never negative. The log-transformation appears more natural for log-normally istribute observations than (17). For practical purposes, function log( x + 1) was applie instea of log( x ). However, the proportion of negative or zero incomes shoul not excee 1%, to avoi unefine logarithms. In a small omain, there is not enough ata for reliable estimation of the percentiles of observations, an consequently the estimate parameters in the transformation (18) are inaccurate. With the Finnish ata set we ecie to calculate the p c from the whole sample instea of each omain, but such a proceure may result in bias. With the Amelia ata, we obtaine better results by minimizing the following sum over omains : c ( a log ( ˆ ) log ( )) 2 + b pc pc This amounts to fitting a linear fixe-effects moel with omain-specific intercepts a an common slope b. The expansion transformation is then ( ˆ ˆlog ( ˆ )) y = exp a + b y. In the Amelia ata, about 1.5% of the people ha zero equivalize income (variable EDI2), an negative incomes i not occur. In orer to tae the zeroes into account, we incorporate zero preictions into the transformation as follows. Let p 0 enote the proportional frequency of zero among the equivalize incomes in the sample. In a sorte vector of N omain preictions, roughly 0 N p smallest elements are replace by zero. Then the percentiles p ˆ c are calculate from the positive preictions an the p c are calculate from positive sample values. Transformation (18) is applie only to the positive preictions, an zero preictions are inclue in the estimator. AMELI-WP2-D2.2

35 29 To account for negative income values, we propose that the log-transformation in (18) is performe by function log( x+ c+ 1), where c is the absolute value of the minimum over all observations an preictions, if negative observations or preictions occur an c = 0 otherwise. Zero observations are then not treate separately, an all observations an preictions contribute to the percentiles. Instea of function exp, we woul apply f( x) = exp( x) ( c+ 1) in (18). This approach is aopte in the R algorithms, but it was not necessary in the simulation experiments, as negative incomes i not appear. The range of percentage points may have large impact on the estimator. The percentiles are calculate at c=1, 2,,99 for quintile share an Gini coefficient. For poverty gap, we use c=1, 2,,50 in Table 12 an with Amelia ata, but in tables we use percentiles up to the poverty line. If the ata are suspecte of containing a lot of outliers, their effect is probably reuce by excluing some of the largest percentiles. If the moel incorporates few auxiliary variables, the number of istinct preictions is small, an the histogram of expane preictions will consist of few bars, representing a poor approximation of the true istribution. When some of the auxiliary variables also efine the omains, this problem is pronounce. For example, if the omains are efine by country, gener an age class, then with x-variables gener, age class an urbanisation, preictions in each omain have only three istinct values corresponing to the classes of urbanisation. Then the preictor involving expane preictions may not yiel goo results. 3.5 Frequency-calibrate preictors calculate using nown omain marginal totals of auxiliary variables We evelop here a new metho that may be feasible in situations where only aggregate-level auxiliary ata are available. Suppose that only the totals of auxiliary variables are nown in a omain of population. In the case of qualitative x-variables, this means that the omain sizes an omain frequencies of classes are nown in the