Measuring dependence between dimensions of poverty in Spain: An approach based on copulas

Transcription

1 6th Worl Congress of the nternational Fuzzy Systems Association (FSA) 9th Conference of the European Society for Fuzzy Logic an Technology (EUSFLAT) Measuring epenence between imensions of poverty in Spain: An approach base on copulas Ana Pérez, Mercees Prieto 2 Universia e Vallaoli 2 Universia e Vallaoli Abstract Welfare an close relate issues, like poverty an inequality, are multiimensional as they involve not only income, but also eucation, health or labour. This paper aims to measure the epenence among imensions using copula-base coefficients. This approach focuses on the positions of the iniviuals across imensions, rather than their specific values. We apply copula-base orerings of epenence to analyze how the concorance among imensions has evolve in Spain over the last years. We also compute multivariate extensions of the Spearman s rank coefficient. The variables consiere are those inclue in the AROPE rate: income, material nees an work intensity. Keywors: copulas, concorance, Spearman, orthant epenence, poverty.. Motivation There is a general agreement that welfare an close relate issues, like poverty an inequality, are multiimensional encompassing not only income, but also nonmonetary imensions such as eucation an health; see, for instance, Kolm (977), Atkinson an Bourbignon (982) an Sen (985). n this framework, much attention has been riven to the selection of the imensions involve (Ramos an Silber, 2005 an Anan et al., 2009) an the way of aggregating the information containe in such imensions (Maasumi, 986; Tsui, 995; Bourguignon an Chakravarty, 200; Atkinson, 200 an Alkire an Foster, 20). Moreover, the level of welfare, poverty an inequality epens on the egree of epenence among imensions (Bourbignon an Chakravarty, 200 an Duclos et al., 2006). However, the problem of measuring that epenence has been scarcely aresse in the literature an this is the main goal of this paper. The information about the multivariate epenence among imensions can be expresse either in terms of their joint istribution function or in terms of their implie copula. The copula approach focuses on the positions of the iniviuals across imensions, rather than the specific values that the corresponing variables attain for such iniviuals. The avantage of this approach for continuous variables is that it enables the ecomposition of the joint istribution function into its univariate marginals an the epenence structure capture by the copula. Moreover, copulas allow builing scale-free measures of epenence an orerings of multivariate epenence; see Nelsen (2006) an Joe (205) for a comprehensive review of copulas an relate concepts. n the bivariate case, there are several measures of epenence. The most well-known is Pearson s correlation coefficient, which is suitable for linear epenence an elliptical ata. However, empirical research shows that, in welfare economics, the ata selom belongs to this class. n this case, measures base on ranks, such as Spearman s rho or Kenall s tau, woul be more appropriate. These two coefficients measure a form of epenence known as concorance. Noticeably, they can be written in terms of copulas. n a multivariate framework, neither the concept of concorance nor the generalization of the bivariate coefficients of concorance is unique. For instance, in the trivariate case, there are more than eight copula-base generalizations of Spearman s rho; see García et al. (20) an the references therein. n welfare contexts, copula-base methos have been recently employe by Quinn (2007a, 2007b) an Bø et al. (202) in a two-imensional setting an by Decancq (20) in a multi-imensional framework. This author introuces a copula-base orering of epenence an characterizes two measures of multivariate epenence consistent with this orer: the multivariate Kenall s tau propose by Nelsen (996) an a multivariate version of Spearman s rho in Nelsen (2002). He illustrates the results with a Russian ata set incluing househol income, self-assesse health an years of schooling. n this paper, we exten the results in Decancq (20) by consiering: a) other two multiimensional versions of Spearman s rho, ue to Wolff (980), Joe (990) an Nelsen (996); b) the coefficients base on irectional epenence recently propose by Nelsen an Úbea- Flores (202) an García et al (20) for trivariate istributions. These measures are capable of revealing some forms of epenences that the coefficients analyse in Decancq (20) fail to etect. We also perform comparisons base on positive orthant epenent orerings using bootstrap methos. Our empirical application is evote to measuring how the epenence between the three imensions inclue in the AROPE (At Risk Of Poverty or social Exclusion) rate, i.e. income, material nees an work intensity, has evolve in Spain over the last years. We focus on this 205. The authors - Publishe by Atlantis Press 74

2 rate because it is the healine inicator to monitor an implement effective poverty-reuction policies in the framework of the EU 2020 Strategy. The rest of the paper is organize as follows. Section 2 introuces the copula an summarizes its basic properties. Section escribes the orthant epenence orerings an the way to implement them in practise. Section 4 introuces ten multivariate copula-base measures of epenence an iscusses its main properties. t also introuces its sample versions. Section 5 illustrates how these tools can be use to measure the evolution of the epenence between imensions of poverty in Spain over the last years. 2. Preliminaries an notation Accoring to Nelsen (2006), a -imensional copula C is a function C:, with =[0,], with the following properties: (i) For every u=(u,,u ) in, C(u) = 0 if at least one coorinate of u is 0, an C(u) = u j if all coorinates of u are except u j. (ii) C is -increasing, that is for every u =(u,,u ) an u 2 =(u 2,,u 2 ) in such that u j u j2, for all j=, 2 2 i i å å ( ) Cu ( i u ) i... -,..., ³ 0 i = i = From the statistical point of view, the importance of copulas comes up in the Sklar's theorem. This theorem establishes that, if X=(X,,X ) is a -imensional ranom variable with joint istribution function F an marginals F,,F, then, for all x=(x x ), F has a copula representation given by:,...,,..., F x F x x C F x F x () f F,,F are all continuous, then C is unique; otherwise C is uniquely etermine on RanF RanF. Conversely, if C is a -copula an F,,F are univariate istribution functions, then the function F efine in () is a -imensional istribution function with margins F,,F. Moreover, when F has continuous margins F,,F an copula C satisfies (), then, for any u in,,...,,..., C C u u F F u F u (2) u where F j, for j=,,, enotes the inverse of F j. The copula C itself, as efine in (2), is a multivariate istribution whose univariate marginals are all uniform on the interval (0,). Each copula function C is boune by its so-calle Fréchet-Hoeffing bouns, W(u) C(u) M(u), where W(u)=max(u + +u +, 0) an M(u)=min(u,,u ). M is always a copula but W is a copula only if =2. Taylor (2007) points out that M can be thought of as a state of maximal concorance, i.e. the state where each component X i of X is an almost surely increasing function of every other component X j. Unlike, if X is a vector of inepenent ranom variables, then its copula is the inepenent copula Π efine as Π(u)=u u. Another important function associate with a copula C is the survival function C efine as: C (u)=p(u>u)= p(u >u,,u >u ). n general, C is not a copula.. Copula-base orerings The concept of multivariate epenence can be efine in ifferent ways (see, for example, Mari an Kotz, 200). The one we hanle in this paper is positive epenence, that is, when large (small) values in one component ten to be associate with large (small) values in the other components. Following Nelsen (2006) an Decancq (20) we consier three copula-base orerings of epenence efine as follows. Let X an Y be two -imensional ranom variables with copula functions C X an C Y, respectively, an with X Y survival functions C an C, respectively. X is more positively lower-orthant epenent (PLOD) than Y if C X u u Y C for every u in () X is more positively upper-orthant epenent (PUOD) than Y if C X u Y u C for every u in (4) X is more positively orthant epenent (POD) than Y if both () an (4) hol. The three efinitions above were alreay introuce in Joe (997) in terms of cumulative istribution functions. As this author points out, the expressions () an (4) mean that the components of X are more likely simultaneously to have small an large values, respectively, compare with the components of Y. n the bivariate case, the three efinitions above are equivalent. n practise, the PLOD, PUOD an POD conitions may be checke once empirical versions of copulas an survival functions are known. To introuce these concepts, Note that this notion of POD is equivalent to the efinition of C X being more concorant than C Y in Dolati an Úbea-Flores (2006). 75

3 further notation is neee. Let X i =(X i,,x i ), i=,...,n, be a ranom sample of size n from the -imensional continuous ranom variable X. ts marginal istribution functions are estimate by their empirical counterparts: n Fˆ j ( x ) = å, for j= an { X x} x ÎÂ ij n i= where { A } enotes the inicator function on a set A. Let R ij be the rank of X ij among {X j,...,x nj }, with i =,...,n an j =,...,, an efine Rij n R.The copula C is estimate by n Cˆ ( u) { Uˆ ij ui } = å, for u=(u,,u ) in n i= j= where A enotes the inicator function of a set A an R ˆ ˆ ij Uij = Fj ( Xij ) =. The empirical survival function is n efine in a similar way. Fermanian et al. (2004) establish regularity conitions for Ĉ to converge towars a Gaussian process. As the asymptotic istribution is unfeasible in most cases, bootstrap methos are commonly use to approximate it. Conitions () an (4) can then be checke out by comparing the values of both the empirical copula an the empirical survival function on the two -imensional istributions X an Y being compare. For this comparison to be feasible, a gri of points on the hypercube shoul be efine. Let 0<p < <p k < be k points in, the k points in that make up the gri are p,..., i p i, with i j =,,k an j=,,. At each of these points, the values of the empirical copula of both X an Y are compute an it is checke if one of the copulas outranks the other at all points in the gri. The same proceure is carrie out with the empirical survival function. Obviously, this comparison is very restrictive as it is merely escriptive. However, the proceure can be significantly improve by bootstrapping. n oing so, we will be able to formally test the hypothesis X Y H o : C ³ C (PLOD conition). Similarly, the equivalent hypothesis for the survival function, i.e., the PUOD conition, coul also be teste. 4. Copula-base multivariate measures of epenence The concorance orering efine in the preceing section prouces an incomplete ranking. n contrast, a epenence measure allows a complete ranking of all the istributions being compare. n this section, we first iscuss three epenence measures which can be regare as multivariate generalizations of the wellknown Spearman s rho. These measures are relate to the three multivariate epenence orering conitions ij introuce in Section. A complete escription of these an other copula-base measures of multivariate epenence can be foun in Wolff (980), Joe (990), Nelsen (996, 2002), Dolati an Úbea-Flores (2006) an Schmi an Schmit (2007). The problem of estimating such measures is aresse in Joe (990), Schmi an Schmit (2007) an Pérez an Prieto (205). The first copula-base multivariate extension of Spearman s rho that we consier, r -, is ue to Wolff (980) an Nelsen (996) an it is efine as follows: C( u) u- P( u) u r - = M ( u) u- P( u) u Since the enominator of this expression represents the maximum value of its own numerator, i.e. its value at the maximal copula C = M, it is guarantee that the maximum value of r - is. By working out some of the integrals in (5), the following alternative expression turns out: ( ) é ( ) ( ) ù é 2 2 ( ) ( ) ù r - = + C u u- = + P u C u - 2 -( + ) 2 -( + ) ê ë úû êë úû where P ( ) = ( -u j ) j= (5) u. Following Nelsen (996), r - can be regare as a multivariate measure of average lower orthant epenence. The secon multivariate version of Spearman s rho consiere in this paper, r +, was originally propose by Nelsen (996) as a multivariate measure of average upper orthant epenence an it is efine as follows: P( u) C( u) - P( u) u r + = P ( u) M ( u) - P ( u) u Again, the enominator of this expression resembles its own numerator evaluate at the maximal copula C = M an so the maximum value of r + is. The expression (6) can be alternatively written as: ( ) é ù r + = + 2 P( ) C( - 2 -( + ) u u) ê ë úû When the copula of X is the upper boun M, both r - r + an attain their maximum value,, an they become zero when the components of X are inepenent, i.e. when C = Π. A lower boun for both r - an r + is é 2 ( )! ù {! é2 ( ) ù êë - + úû êë - + úû} ; see Nelsen (996). (6) 76

4 Nelsen (2002) proposes another multivariate version of Spearman s rho, r, efine as the average of r - an r +, i.e.: - + r ( + r + ) ì é ù ü - r = = ï2 C( u) ( u) ( u) C( u) í P + P - ï ý 2 2 -( + ) ê ïî ë úû ïþ (7) This measure is further iscusse in Dolati an Úbea- Flores (2006), where it is shown that it is a measure of average orthant epenence. t has also been use in Decancq (20). n the biimensional case (=2) the three coefficients efine above reuce to bivariate Spearman s rho, r S, that is, r - 2 = r + 2 = r2 = rs. n the trivariate case (=), the three coefficients efine above become: r - = 8 C( u, u, u ) u u u - (8a), 2 2 r + = 8 C( u, u, u ) u u u - (8b), r + r r2 + r + r2 r = = (8c), 2 2 where ( r 2, r, r 2 ) are the three possible pairwise Spearman s rho coefficients. The avantage of r - an r + is that they are capable of revealing some forms of epenences that r fails to etect. Nelsen (996) illustrates this feature with an example where r = 0, presumably inicating no epenence at all, while r + is positive an r - is negative, inicating some egree of positive upper orthant epenence an negative orthant epenence, that is misse by r. n spite of this avantage, there are still some forms of multivariate epenence that the coefficients r - an may fail to etect when they take values near 0. To r + overcome this rawback, a general class of irectional -coefficients has recently been propose by Nelsen an Úbea-Flores (202). Let (, 2, ), with i ={,- }, enote the eight vertexes of the cube etermining the eight irections in which we coul measure epenence in the -imensional case. For each irection (, 2, ), a irectional -coefficient is efine as: aar aar aar r + r aa a r = + aa 2a (9) 2 Hence, each of these eight irectional coefficients is a simple linear combination of the pairwise measures an the two measures r - an r + of trivariate association. (,,) Noticeably, r = r + (-,-,-) - an r = r. Finally, we consier the new inex of maximal epenence r propose by García et al. (20) as the larg- max est of the eight irectional -coefficients efine in (9). These authors prove that their inex can also be calculate using the three pairwise Spearman s rho coefficients an the three common -imensional versions of Spearman s rho in (8) as follows: max 2 max { 2,, 2, } min - {, + r = r r r r - r r } (0) The non-parametric estimator of the coefficient r + in (6) was alreay given in Joe (990). For =, this estimator becomes: n 8 n + r ˆ + = RRR 2 i i2 i- nn ( + )( n - å ) i= n - () Pérez an Prieto (205) propose a non-parametric estimator of r -, which reuces to the following expression for =: 8 n + n rˆ - = RiRi2Ri- 2 nn + - å i= n - (2) ( )( n ) Note that Schmi an Schmit (2007) propose other nonparametric estimators of r - an r + but, as Pérez an Prieto (205) show up, these are unfeasible, since they can take values out of the parameter space. Note also that there is a typo in García et al. (205), since their expressions for the estimators - an + are the other way roun, being their equation (2) the correct expression of + an their equation () the proper expression of -. Finally, the parameter ρ in (8c) can be estimate as: rˆ rˆ + rˆ rˆ + rˆ + rˆ = = For the biimensional case (=2), all expressions above collapse to the well-known sample bivariate Spearman s rank correlation. To estimate the irectional coefficients in (9) an the inex of maximal epenence in (0), García et al. (20) suggest using plug-in estimators, that is, replacing each pairwise correlation by its well known sample counterpart an replacing r - an r + by their nonparametric estimators in (2) an (), respectively. 77

5 5. Empirical application Several recent works (Atkinson an Morelli, 20; Foessa, 20 an Prieto et al. 205) point out that poverty in Spain has increase significantly over the last years of economic crisis. As we sai before, poverty epens on the egree of interepenence among the poverty imensions. n this section, we apply the copula-base methos escribe in Section an Section 4 to measure the evolution of the epenence among the poverty imensions in Spain over the perio This analysis will give us a general picture of how the egree of poverty has evolve in Spain in the years after the crisis. 5.. Data an variables The imensions of poverty we have selecte for our empirical application are those inclue in the AROPE rate 2, namely income, material nees an work intensity. We select these variables because the AROPE rate is the healine inicator to monitor an implement effective poverty-reuction policies in the framework of the Europe 2020 Strategy. The income of a househol is the total income of the househol, after tax an other euctions ivie by the equivalent househol size. The material eprivation is originally efine as the enforce inability to: ) pay unexpecte expenses; 2) affor a one-week annual holiay away from home; ) have a meal involving meat, chicken or fish every secon ay: 4) heat aequately a welling; 5) have a washing machine; 6) have color television; 7) have telephone; 8) have car; 9) manage with payment arrears (mortgage or rent, utility bills, hire purchase instalments or other loan payments). For the sake of simplicity an easy of interpretation, we consier a transformation of this variable that takes its complementary values, namely: 0 (having all the 9 possible eprivations), (having eight out of the nine aforementione eprivations),, 9 (having no eprivations). Hence, this new variable inicates the number of no-privations out of the nine possible. The work intensity of a househol is the ratio of the total number of months that all working-age 4 househol members have worke uring the year an the total number of months the same househol members theoretically coul have worke in the same perio. 2 The AROPE rate is the percentage of people who are either at risk of monetary poverty, or severely materially eprive or living in a househol with a very low work intensity. We use the OECD equivalence scale which assigns a weight.0 to the first ault; 0.5 to the secon an each subsequent person age 4 an over; 0. to each chil age uner 4. 4 A working-age person is a person age 6-65 years, with the exclusion of stuents in the age group between 6 an 24 years. With our transformation of the variable measuring material nees we ensure that the three variables consiere keep the same relationship with phenomenon being measure, poverty. That is, high values of each variable convey lower chance to be poor, while low values of each variable convey higher chance to be poor. The ata comes from the EU-Statistics on ncome an Living Conitions (EU-SLC) survey, which is the EU reference source for comparative statistics on income istribution an social inclusion at the European level. To face our goal, we base our analysis on ata from surveys launche in 2009 an 20. The unit of analysis is the househol. The samples sizes of the EU-SLC survey are 60 an 29 in 2009 an 20, respectively. However, we only work with subsamples of 086 househols in 2009, an 9695 househols in 20, for which we have complete information for all the three ranking variables. n particular, in these subsamples, househols compose only of chilren, of stuents age less than 25 an/or people age 65 or more are exclue, ue to their missing values in the variable work intensity. Although the full samples are constructe to be representative for the entire Spain, this restricte subsamples might not be. Nevertheless, the ifference between some statistics obtaine with the entire sample an those obtaine with the subsamples are tiny. For example, the means househol income obtaine with the entire sample are 6945 an 6087 in 2009 an 20 respectively, while both means with the reuce sample are 7842 an n the copula-base framework we eal with how the iniviuals are ranke in the three imensions, rather than with the exact outcome levels of the iniviuals in these imensions. Hence, in each imension, househols are ranke accoring to the variable efining such imension. f a tie occurs, the househols tie are ranke ranomly. As we will illustrate later, this ranom assignment o not noticeably change the results Results We first test the hypothesis of increase epenence in Spain over the perio by performing copula base-orerings comparisons, as escribe in Section. n orer to o that, for each year, we compute the empirical copula function over a gri of 27 points in efine by (0.25, 0.5, 0.75) an then, for each point in the gri, we compare the values of the empirical copula for both years as escribe in section. Then, we approximate the sampling istribution of each empirical copula by resampling with replacement repeately (000 subsamples) from the original sample. Afterwars, on the basis of the bootstrap istributions obtaine, we test for the ifference between the two years consiere to be significant. The results are isplaye in Table (see the Appenix). Several conclusions emerge from this table. First, the empirical copula in year 20 outranks the empirical 78

6 copula in year 2009 at all the gri points. Secon, the ifferences between 20 an 2009 are significantly non-negative at 5% in all cases. Hence, the PLOD conition hols at all gri points at 5% significant level. This means that the three poverty imensions (income, no material nees an work intensity) are more likely to have simultaneously small values in 20 than in 2009; that is, being simultaneously poor in all the three imensions is more likely to occur in 20 than it was in Therefore, the situation for the poors has become even worse in 20 than it was in We next illustrate the multivariate epenence among income, no material nees an work intensity through the multivariate versions of Spearman s rho introuce in Section 4. n orer to o that, we compute the estimators of the ten -imensional inexes introuce in Section 4 for the two years consiere. Again, we approximate the sampling istribution of these estimators by bootstrap methos, using the resampling scheme explaine above. The results are isplaye in Table 2 in the Appenix. This table also inclues the three possible pairwise Spearman s rho correlation coefficients, namely, the correlation between: income an material nees ( 2 ), income an work intensity ( ) an material nees an work intensity ( 2 ). As expecte, most coefficients have increase its magnitue in 20 as compare to Actually, the ifferences between 20 an 2009 are all significantly non-negative at 5% level, except for the irectional coefficients - (,, ) (,,), - (,, ), - -. Hence, the epenence among imensions of poverty has increase significantly over the perio Moreover, in both years, the largest coefficient is -, inicating that the irection of maximal epenence is (-,-,-). This means that small values of the three poverty imensions (income, no material nees an work intensity) ten to occur together, an this simultaneous occurrence is more likely in 20 than in Noticeably, the secon largest coefficient of multivariate epenence is +, which measures epenence in irection (,,). This means that high values of the three poverty imensions ten to occur together, i.e., househols with high income are more likely to have simultaneously high work intensity an few material nees. Moreover, this simultaneous occurrence of goo rankings in the three imensions is more likely in 20 than in To summarize, our results show that not only the epenence among imensions of poverty has increase significantly from 2009 to 20, but also the society has become more polarize, with poverty imensions (income, no material nees an work intensity) being more likely to have simultaneously small (large) values in 20 than in Conclusions n this paper we analyse how the epenence among the imensions of poverty has evolve in Spain over the perio The imensions consiere are those inclue in the AROPE rate: income, material nees an work intensity. n orer to o that, we have applie copula-base orerings of epenence over a gri of 27 points to test whether such epenence has increase over the perio consiere. We have also compute several multivariate extensions of the Spearman s rank coefficient for the two years consiere an we have teste, using bootstrap methos, whether the ifferences between both years are significant. Our main conclusion is that the egree of epenence among income, material nees an work intensity has noticeably increase over the perio consiere. Moreover, in both years, the maximal epenence among the three imensions occur in irection (-,-,-) an the irection of the secon largest epenence is (,,). This means that small (high) values of the three poverty imensions ten to occur together, an this simultaneous concentration of small (large) values of income, no material privations an work intensity, is more likely to occur in 20 than in Therefore, after the crisis, the Spanish society has become more polarize, with the poor being more likely to keep being poor an the rich being more likely to keep being rich. 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8 Appenix: Tables Table. Estimate copula (stanar errors in parenthesis) for the three AROPE imensions evaluate at 27 gri points u=(u,u 2,u ) for the two years 2009 an 20, together with t-statistics to test for significant ifferences between the two years consiere u=(u,u 2,u ) t-statistic (0.25, 0.25, 0.25) (0.50, 0.25, 0.25) (0.75, 0.25, 0.25) (0.25, 0.50, 0.25) (0.50, 0.50, 0.25) (0.75, 0.50, 0.25) (0.25, 0.75, 0.25) (0.50, 0.75, 0.25) (0.75, 0.75, 0.25) (0.25, 0.25, 0.50) (0.50, 0.25, 0.50) (0.75, 0.25, 0.50) (0.25, 0.50, 0.50) (0.50, 0.50, 0.50) (0.75, 0.50, 0.50) (0.25, 0.75, 0.50) (0.50, 0.75, 0.50) (0.75, 0.75, 0.50) (0.25, 0.25, 0.75) (0.50, 0.25, 0.75) (0.75, 0.25, 0.75) (0.25, 0.50, 0.75) (0.50, 0.50, 0.75) (0.75, 0.50, 0.75) (0.25, 0.75, 0.75) (0.50, 0.75, 0.75) (0.75, 0.75, 0.75) (0.00) (0.00) (0.00) (0.00) 0.80 (0.00) (0.00) (0.00) (0.00) (0.00) 0.7 (0.00) (0.00) (0.00) 0.52 (0.00) Table 2. Estimate parameters (stanar errors in parenthesis) of the trivariate correlation coefficients between the three AROPE imensions for the two years 2009 an 20, together with t-statistics to test for significant ifferences between the two years consiere Estimator t-statistic (0.008) (0.009) 0.29 (0.009) (0.006) (0.006) (,,) (,,) (,, -) (,,) (,, ) (,, ) max (0.008) (0.009) 0.40 (0.009) (0.006)