Copula‐based analysis of multivariate dependence patterns between dimensions of poverty in Europe

Copula-based analysis of multivariate dependence patterns between dimensions of poverty in Europe C´esar Garc´ıa-G´omeza, Ana P´ereza,b,∗and Mercedes Prieto-Alaiza aDpto. Econom´ıa Aplicada, Universidad de Valladolid. bIMUVA, Universidad de Valladolid. Abstract It is widely recognised that poverty is a multidimensional phenomenon involving not only income, but also other aspects such as education or health. In this multidimensional setting, analysing the dependence between dimensions becomes an important issue, since a high degree of dependence could exacerbate poverty. In this paper, we propose measuring the multivariate dependence between the dimensions of poverty in Europe using copulabased methods. This approach focuses on the positions of individuals across dimensions, allowing for other types of dependence beyond linear correlation. In particular, we analyse how orthant dependence between the dimensions of the AROPE rate has evolved in the EU-28 countries between 2008 and 2014 by applying non-parametric estimates of multivariate copula-based generalisations of Spearman’s rank correlation coefficient. We find a general increase in the dependence between dimensions, regardless of the coefficient used. ∗Corresponding Author. Dpto. Econom´ıa Aplicada. Universidad de Valladolid. C/ Avda. Valle Esgueva 6, 47011 Valladolid. Spain. Tel: 34 983423317, E-mail: [email protected] 1 "This is the peer reviewed version of the following article: García-Gómez, C., PÉREZ, A., PrietoAlaiz, M.: “Copula-based analysis of multivariate dependence patterns between dimensions of poverty in Europe”. Review of Income and Wealth, 2020 which has been published in final form at http://dx.doi.org/10.1111/roiw.12461 . This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions." Moreover, countries with higher AROPE rates also tend to experiment more dependence between its dimensions. JEL Classification: D63, I32, O52. Keywords: Spearman’s rho; Orthant dependence; Empirical copula; Poverty measurement; AROPE rate. 1 Introduction There is a widespread agreement that poverty is a multidimensional phenomenon involving not only low incomes, but also deprivations in other dimensions such as education, health or labour; see, for instance, Sen (1985, 1987). Because of that, attention has been increasingly focused on multidimensional approaches to the analysis of poverty, to the point where the European Union (EU), for example, has adopted a multidimensional poverty and social exclusion index as a tool to monitor and implement effective poverty-reduction policies in the framework of the Europe 2020 Strategy. The index at hand, namely the AROPE (At Risk Of Poverty or social Exclusion) rate, is based on three measures: relative income poverty, material deprivation and work intensity. Also, the United Nation Development Program (UNDP) adopted, in 2010, the Multidimensional Poverty Index (MPI), which is based on the Alkire and Foster (2011) proposal. This index also considers three dimensions: education, health and standard of living. Based on these indices (or any other multivariate indicator), several authors examined the incidence and intensity of multidimensional poverty in developed and non-developed countries; see, for instance, the contributions of Nolan and Whelan (2011), Whelan et al. (2014), Alkire and Apablaza (2016), White (2017) and Atkinson et al. (2017), in the European context. However, many of the multidimensional poverty indices, especially some of the most widely used, such as the AROPE rate and the MPI, are not sufficiently sensitive to the possible interrelation between the dimensions of poverty. Therefore, they could miss an important 2 part of the picture; see Duclos and Tiberti (2016). In this context, several authors argue that incorporating those relationships can be relevant, since higher dependence means higher concentration of deprivations and this could make overall poverty worse; see, for instance, Atkinson and Bourguignon (1982), Bourguignon and Chakravarty (2003), Duclos et al. (2006), Seth (2013) and Ferreira and Lugo (2013). In spite of its relevance, the problem of measuring the dependence between dimensions of poverty has been scarcely addressed in the literature and this is the scope of this paper. Noticeably, as we face a problem of studying dependence in a multivariate context, special care is required, since the step from two dimensions to three (or more) dimensions is not so obvious. Actually, as Durante et al. (2014) show, some bivariate dependence properties are not preserved in higher dimensions. In this framework, we propose complementing the analysis based on poverty indices by measuring the multivariate dependence among poverty dimensions using copula-based methods. The copula approach focuses on the positions of the individuals across dimensions, rather than on the specific values that those dimensions attain for such individuals. This approach has several advantages. First, it enables the decomposition of the joint distribution function of all dimensions into its univariate marginals and the dependence structure, which is captured by the copula. Nevertheless, as Genest and Neˇslehov´a (2007) point out, the copula alone does not characterize the dependence in the discrete case. Second, copulas allow building scaledfree measures of dependence that capture other types of dependence beyond linear correlation. Actually, the well-known Spearman’s rho and other related measures of bivariate association can be expressed in terms of copulas. Third, the copula approach facilitates the construction of multivariate generalisations of bivariate association coefficients, although the generalisation is not unique in some cases (see Section 2). Furthermore, dominance tests are also possible to establish copula-based orderings of dependence; see Decancq (2012) and the references therein. This would allow to rank pairs of multivariate distributions and perform full comparisons between two societies. However, as Decancq (2014) points out, this ordering could be ‘indecisive’ 3 in many cases, meaning that the societies cannot be ranked with respect to the dependence between the poverty dimensions considered. To overcome this drawback, one may prefer using copula-based dependence measures that can rank the distributions being compared. This is the approach we adopt in this paper. Applications of copula-based methods in welfare economics in a bivariate setting date back to Dardanoni and Lambert (2001), Quinn (2007) and Bø et al. (2012); see also the recent contribution of Aaberge et al. (2018). In a multidimensional framework, the first contribution employing copula-based methods in welfare economics is Decancq (2014). He analysed the temporal evolution of well-being in Russia by means of a multivariate Kendall’s tau and a multivariate version of Spearman’s rho applied to the dimensions included in the Human Development Index (HDI). P´erez and Prieto (2015) extended Decancq’s results by considering other multivariate versions of Spearman’s rho to study how the dependence between the dimensions of the AROPE rate has evolved in Spain over the period 2009-2013. Also, P´erez and Prieto-Alaiz (2016a) analysed the multivariate dependence between the dimensions of the HDI using data from 187 countries and three copula-based measures of multivariate association: Spearman’s footrule, Gini’s gamma and Spearman’s rho. The contribution of this paper is twofold. First, we consider multivariate extensions of Spearman’s rho proposed by Nelsen (1996, 2002), which allow to capture some types of dependence which are essential in poverty analysis, namely those based on orthant dependence. Particularly useful is the coefficient based on lower orthant dependence, as it could measure the propensity of being simultaneously low-ranked in all dimensions of poverty. We also consider the generalizations of these coefficients to possibly non-continuous multivariate distributions proposed by Quessy (2009) and Mesfioui and Quessy (2010). Second, we apply these coefficients to perform cross-country and temporal comparisons of the multivariate dependence between the dimensions of the AROPE rate in the EU-28 countries over the period 2008-2014. As far as we know, this is the first time that these copula-based measures are applied in the European context. 4 The data we use comes from the EU-Statistics on Income and Living Conditions (EU-SILC) survey, which is the EU reference source for comparative statistics on income distribution and social inclusion at the European level. Our analysis complements the information on the incidence of multidimensional poverty, given by the AROPE rate, with information on the degree of multivariate dependence between its dimensions. In particular, we find that, in most EU countries, there has been an increase in the dependence between poverty dimensions over the period analysed. Noticeably, the highest increase corresponds to Spain, one of the countries most severely hit by the last economic crisis. Moreover, over all the years considered, the maximal dependence is generally found in the lower part of the joint distribution. These results imply that small values of income, no-material deprivation and work intensity tend to occur together, and this is more likely in 2014 than in 2008. We also detect strong dependence in the upper orthant of the joint distribution, suggesting that, after the crisis, most EU countries have become more polarised. Finally, we find that countries with higher AROPE rates also tend to experiment more dependence between its dimensions. The rest of the paper is organised as follows. Section 2 summarises the basic properties of copulas and describes orthant dependence concepts. It also introduces copula-based multivariate versions of Spearman’s rho coefficient and discusses how to estimate them non-parametrically using the empirical copula. New properties of the estimators considered are also included. Section 3 illustrates the use of these tools to measure how the dependence between the three indicators of the AROPE rate has evolved in the EU-28 countries over the period 2008-2014. Section 4 concludes the paper with a summary of the main results. 5 2 Methodology 2.1 Copulas and orthant dependence Copulas are joint distribution functions whose one-dimensional margins are uniform on I=[0,1]. More precisely, a d−dimensional copula Cis a multivariate distribution function C:Id→I defined for every u=(u1, . . . , ud)∈Idas C(u) = p(U≤u) = p(U1≤u1, . . . , Ud≤ud), where Uiis U(0,1), for i= 1, ..., d.1The importance of copulas in statistics relies on the Sklar’s theorem (Sklar, 1959). This theorem establishes that, if X= (X1, ..., Xd) is a d−dimensional random vector with joint distribution function F(x) = F(x1, ..., xd) = p(X1≤x1, ..., Xd≤xd) and univariate marginal distribution functions Fi(xi) = p(Xi≤xi), for i= 1, ..., d, then there exists a copula Csuch that, for all x= (x1, . . . , xd)∈Rd,Fcan be represented as F(x) = C(F1(x1), ..., Fd(xd)).(1) Hence, copulas are functions that join or “couple” multivariate distribution functions to their one-dimensional marginal distribution functions. If the margins F1, ..., Fdare all continuous, the copula Cin (1) is unique; otherwise Cis uniquely determined on RanF1×... ×RanFd. Conversely, if Cis a d−copula and F1, ..., Fdare univariate distribution functions, the function Fdefined in (1) is a joint distribution function with margins F1, ..., Fd. Throughout this section, we generally assume that F1, ..., Fdare all continuous, although some issues arising when dealing with possibly non-continuous variables will be duly pointed out. In a multidimensional poverty setting, the random vector Xrepresents the relevant ddimensions of poverty for a population and the transformed variables Ui=Fi(Xi), with i= 1, . . . , d, attach to each individual in the population its relative position in all dimensions. For instance, an individual with position vector (1,...,1) will be top-ranked in all dimensions. Each random 1An equivalent definition of a multivariate copula can be found in Nelsen (2006, p. 45) 6 variable Uiis U(0,1) and the joint distribution of the vector U= (U1, ..., Ud) is the copula C defined above.Therefore, for a given real vector u∈Id, the value C(u) represents the proportion of individuals in the population with positions outranked by u. For instance, C(0.25, ..., 0.25) will represent the probability that a randomly selected individual is simultaneously in the 1st quartile (“low ranked”) in all dimensions, i.e., in our setting, it will be the probability that he/she is simultaneously “poor” in all dimensions. Any copula Csatisfies the Fr´echet-Hoeffding bounds inequality W(u)≤C(u)≤M(u), for every u∈Id, where W(u) = max(u1+· · · +ud−d+ 1,0) and M(u) = min(u1, . . . , ud). M is always a copula and represents maximal dependence, i.e. the case when each of the random variables X1, ..., Xdis almost surely a strictly increasing function of any of the others (the outcomes in all dimensions are ordered in the same way). Wis only a copula if d= 2, in which case it represents perfect negative dependence. Another important copula is the independent copula, defined as Π(u) = u1× · · · × ud, which accounts for the case where the variables X1, . . . , Xdare independent. Finally, if U= (U1, ..., Ud) is a random vector of variables U(0,1) whose joint distribution function is the copula C, the survival function C:Id→Iis defined as: C(u) = p(U>u) = p(U1> u1, . . . , Ud> ud). In our setting, for instance, C(0.75, ..., 0.75) will represent the probability that a randomly selected individual is simultaneously in the 4th quartile (“top ranked”) in all dimensions, i.e., the probability that he/she is simultaneously “rich” in all dimensions. In general, Cis not a copula. Moreover, if U1, ..., Udare independent random variables, then its survival function is Π(u) = (1 −u1)× · · · × (1 −ud). For a comprehensive review of copulas, see Nelsen (2006). 7 In this paper, we use copulas to study measures of multivariate association derived from multivariate dependence concepts. The notions of dependence in the multivariate case can be defined in different ways. The one we handle in this paper is orthant dependence and it is defined as follows (Nelsen, 2006): •Xis positively lower orthant dependent (PLOD) if C(u)≥Π(u), for each u∈Id, that is, if the probability that the variables X1, ..., Xdare simultaneously small is at least as great as it would be were they independent. •Xis positively upper orthant dependent (PUOD) if C(u)≥Π(u), for each u∈Id, that is, if the probability that the variables X1, ..., Xdare simultaneously large is at least as great as it would be were they independent. •Xis positively orthant dependent (POD) if both inequalities hold. The corresponding negative concepts (NLOD, NUOD and NOD) are defined by reversing the sense of the inequalities above. For d= 2, PLOD and PUOD are the same and reduce to POD. Obviously, the same reduction occurs with the analogous negative concepts. For poverty analysis, lower orthant dependece will be the more relevant concept. In this framework, the differences [C(u)−Π(u)] and [C(u)−Π(u)] can be regarded as measures of “local” lower and upper orthant dependence, respectively; see Nelsen (1996). Accordingly, the copula-based measures of multivariate association to be introduced in next Section are based on these differences. 2.2 Copula-based multivariate extentions of Spearman’s rho One of the best-known measures of association between two random variables X1and X2is Spearman’s rank correlation coefficient, also known as Spearman’s rho (ρS). This measure, 8 which is the correlation coefficient of the transformed random variables F1(X1) and F2(X2), can be expressed in terms of their copula Cas follows (Nelsen, 1991): ρS= 12 ZI2 C(u1, u2)du1du2−3 = 12 ZI2 u1u2dC(u1, u2)−3.(2) When we move to a multivariate setting, several extensions of Spearman’s rho can be found in the literature. The first copula-based generalisation of bivariate Spearman’s rho, due to Wolff (1980) and Nelsen (1996), is a multivariate extension of the left-hand side expression in equation (2) and is defined as: ρ− d=2d(d+ 1) 2d−(d+ 1) ZId [C(u)−Π(u)]dΠ(u) = (d+ 1) 2d−(d+ 1) 2dZId C(u)dΠ(u)−1.(3) Following Nelsen (1996), ρ− dcan be regarded as a multivariate measure of average lower orthant dependence. In fact, ρ− dassesses, to some extent, the similarity between our multivariate data X(represented by its copula C) and the situation of independence (represented by copula Π) in the lower orthant. In a similar fashion, Nelsen (1996) defined a second generalisation of Spearman’s rho, derived from average upper orthant dependence. This measure, which is a multivariate extension of the right-hand side expression in equation (2), is given by: ρ+ d=2d(d+ 1) 2d−(d+ 1) ZId [C(u)−Π(u)]dΠ(u) = (d+ 1) 2d−(d+ 1) 2dZId Π(u)dC(u)−1.(4) From this expression, ρ+ dcould be thought of as the normalised average difference between C– representing the behaviour of our data in the upper orthant – and Π – representing independence in such orthant. The third copula-based multivariate version of Spearman’s rho, due to Nelsen (2002), is the 9 The three measures characterising the three dimensions of the AROPE rate are defined as follows. The measure of income is the equivalised disposable income, which is calculated as the total income of the household, after taxes and other deductions, divided by the equivalised household size.3The work intensity of a household is the ratio of the total number of months that all working-age household members have worked during the income reference year and the total number of months they could have theoretically worked during the same period.4Material deprivation is originally defined as the enforced lack in a number of essential items, namely: 1) the capacity of facing unexpected expenses; 2) one-week annual holiday away from home; 3) a meal involving meat, chicken or fish every second day; 4) an adequately warm dwelling; 5) a washing machine; 6) a colour television; 7) a telephone; 8) a car; 9) the capacity to pay their rent, mortgage or utility bills. For ease of interpretation we transform this variable into a variable that indicates the number of no-deprivations out of the nine possible, so that the new variable takes the following values: 0 (having all the 9 possible deprivations), 1 (having eight out of the nine aforementioned deprivations), . . . , 9 (having no deprivations). Thus, high values of the three variables considered (equivalised disposable income, work intensity, and number of no-deprivations) convey lower chance to be poor, while low values of each variable convey higher chance to be poor. The unit of analysis is the household. We only work with subsamples of households for which we have complete information for all the three variables. In particular, in these subsamples, households composed only of children, of students aged 18-24 and/or people aged 60 or more are excluded, due to their missing values in the work intensity variable.5In these subsamples, the sample sizes range from 2270 households (Cyprus, 2009) to 14773 households (Italy, 2008). 3The equivalised household size is defined according to the modified OECD scale, which gives a weight of 1 to the first adult, 0.5 to other household members aged 14 or over and 0.3 to household members aged less than 14. 4Eurostat considers that a working-age person is a person aged 18-59 years, excluding also the students aged 18-24 years. 5The representation of each country in the whole cross-country sample does not change when going from the full sample to the restricted one. 16 As we explained in Section 2, copula-based methods requires ranking the households in each dimension. In doing so, ties could arise in one or multiple variables. In our case, for example, the work intensity and material deprivation variables are of non-continuous nature, thus leading to a considerable number of ties. The problem of having ties in a copula-based framework was already mentioned in Section 2, where it was remarked that, in the presence of ties, the copula in (1) is no longer unique. Therefore, the values of the copula-based multivariate extensions of Spearman’s rho can vary widely even based on the same joint distribution. Different alternatives to deal with ties can be found in the literature; see, for example, Quessy (2009), Mesfioui and Quessy (2010), Genest et al. (2013) and Decancq (2014). In this paper, we focus on two of these alternatives in order to analyse how robust our results are to the method used. On one hand, we compute the tie-corrected estimators of the multivariate extensions of Spearman’s rho defined in (18)-(20), as proposed by Genest et al. (2013). On the other hand, following Decancq (2014) we break the ties using additional information from other secondary variables so that we eventually get, for each variable, unique ranks, {1,2, . . . , n}, and hence the coefficients defined in (12)-(14) can be directly applied to these ranks; see below. We are aware that it is unclear the effect of using additional secondary variables on the concordance properties of the original variables. In spite of that, we will see later that both approaches lead to very similar conclusions regarding the evolution of the dependence between poverty dimensions in Europe. To start with, we will explain in detail how we use additional information to break the ties. Firstly, when a tie occurs in work intensity, households are ranked according to two secondary ranking variables measuring the intensity in both education and health of the household. The intensity of education is the sum of the highest ISCED (International Standard Classification of Education) level attained by all members of the household that are not currently in education divided by the highest possible value of this sum. The health intensity indicator is constructed in a similar way as the sum of the values of the self-assessed health indicator of all members of the household divided by the highest possible value of this sum. The choice of these secondary 17 variables is not arbitrary. Both the relationships between educational and labour market outcomes and between health and labor market attainments are well documented in the literature; see, for example, Nickell (1979), Mincer (1991), Wolbers (2000), Farber (2004) and Riddell and Song (2011), regarding the former and Chirikos (1993), Ettner et al. (1997), Currie and Madrian (1999), Pelkowski and Berger (2004) and Garc´ıa G´omez and L´opez Nicol´as (2006), regarding the latter. As secondary ranking variable for material deprivation, we use the burden of the housing cost. An overburden of the housing cost can be seen as an indicator of financial stress (Whelan and Maˆıtre, 2012; Deidda, 2015) and as an indicator of vulnerability (Brandolini et al., 2010). We use both a dummy variable taking the value 1 if the housing cost is a burden for the household and the value of the housing cost itself. Thus, households for which the housing cost is a burden are assigned worse positions than those for which it is not. If a tie still exists for those households for which the housing cost is a burden they are ranked using the value of the housing cost. That is, the higher is the housing cost the worse is the position of the household. Both in the case of work intensity and material deprivation, if ties still exist after ranking households according to the secondary variables, the ties are broken at random. Thus, after this procedure, households are eventually assigned unique ranks, {1,2, . . . , n}, for each variable and the estimators ˆρ− d, ˆρ+ dand ˆρdin (12)-(14) can be computed using these ranks. 3.2 A primer look at the transformed data In this section we show some examples of multivariate association in our data. To illustrate cross-country comparisons, Figure 1 represents the unique ranks described above, rescaled to [0, 1] as defined in (11), for the three dimensions of the AROPE rate in Bulgaria and Romania in 2008. As we can see, the points are not uniformly distributed over the unit cube, indicating departure from independence. Actually, in both countries we observe a positive association, as the points tend to concentrate around the main diagonal of the cube, that is, the three variables tend to be jointly large or small together. Moreover, both plots are denser around 18 the vertexes (0, 0, 0) and (1, 1, 1), but in Bulgaria the concentration is higher around the former than around the latter, suggesting that dependence in the lower orthant is higher than in the upper orthant. The contrary occurs in Romania, where there is a higher concentration of observations around the vertex (1, 1, 1), suggesting that upper orthant dependence is higher than lower orthant dependence. As a matter of fact, these patterns are properly captured by the coefficients ˆρ− 3and ˆρ+ 3, which in the case of Bulgaria will fulfil the condition ˆρ− 3>ˆρ+ 3, while they will behave the other way round in Romania. <Insert Figure 1 here > To illustrate temporal comparisons, Figure 2 displays two scatter plots representing the scaled ranks for Spain in 2008 and 2014. As we can see, there has been an increase in the multivariate dependence between dimensions of the AROPE rate in Spain over this period, as the concentration of the observations around the main diagonal is higher in 2014 than in 2008. Moreover, in both years, the concentration of points in the lower orthant seems to be higher than in the upper orthant. Hence, we would expect ˆρ− 3>ˆρ+ 3, being both coefficients higher in 2014 than in 2008. <Insert Figure 2 here > To complement this graphical analysis, we have split the unit cube [0,1]3in 64 boxes of the same size and we have computed (see Table 1) the observed relative frequencies in the four boxes along the main diagonal for the same countries and time periods represented in Figures 1 and 2. The four boxes are denoted as {u≤0.25, 0.25 <u≤0.5, 0.5<u≤0.75, u>0.75}, where u≤0.25 denotes the component-wise inequality, i.e. ui≤0.25 for i= 1,2,3,and so this first box records the share of households being simultaneously in the 1st quartile (low-ranked) in all dimensions. The other three boxes are defined similarly. <Insert Table 1 here > 19 If the three variables were independent, the proportion of points in each box would be the same and equal to 1.56%. However, in all the examples in Table 1, there is a larger proportion of points concentrated around the main diagonal implying departure from independence. Furthermore, in all cases, the frequencies are higher in the extreme boxes, suggesting positive orthant dependence, in agreement with the patterns displayed in Figures 1 and 2. 3.3 Estimation results In this section, we analyse the evolution of the multivariate dependence between poverty dimensions in the EU-28 countries over the period 2008-2014 using both the non-parametric estimators in (12)-(14) applied to the unique ranks as explained in Section 3.1, and the tiecorrected estimators in (18)-(20). As we pointed out in Section 2.3, the asymptotic variances of these estimators are complex. Therefore, we rely on a nonparametric bootstrap method to compute the bootstrap standard errors as the sample standard deviation of 1000 bootstrapped point estimates of the coefficients. Figure 3 displays, for the EU-28 countries and over the whole period analysed, the evolution of the values of ˆρ− 3(in Panel A) and ˆρ+ 3(in Panel B) together with the 95% standard confidence intervals using the bootstrap standard errors.6Figure 4 displays similar results for the tiecorrected estimators bρ−z 3(in Panel A) and bρ+z 3(in Panel B). <Insert Figure 3 here > <Insert Figure 4 here > Several conclusions emerge from these figures. First, the patterns of the evolution of dependence over the period analysed are very similar whether we use the continuous (Figure 3) or tiecorrected (Figure 4) versions of the coefficients, although the former seem to have slightly 6We have also computed the 95% bootstrap percentile confidence intervals obtaining very similar results not displayed here to save space. The results are available upon request. 20 larger values than the latter. Second, all the coefficients are always positive, indicating a positive multivariate association between poverty dimensions both in the lower and in the upper orthant. This means that low (high) values of income tend to occur with low (high) values of the other two poverty dimensions. Third, Figure 3 shows that, regardless of the year and the country, the value of ˆρ− 3(Panel A) is greater than that of ˆρ+ 3(Panel B), except for the case of Romania, and the same result holds for the tie-corrected versions of the coefficients (Figure 4). This means that average lower orthant dependence tends to be higher than average upper orthant dependence, that is, the probability of being simultaneously low-ranked in all poverty dimensions tends to be higher than the probability of being simultaneously high-ranked in all dimensions. Fourth, there are different cross-country profiles in the evolution of multivariate dependence. For instance, in Spain there is a clear increasing trend in the multivariate dependence between dimensions of poverty in both the lower and the upper orthant over the period analysed. An increasing trend is also found in other countries such as Cyprus, Denmark, Italy or The Netherlands. However, no decreasing trend shows up in any country. On the other hand, in some countries such as Greece and the UK, there is not a clear trend, but the dependence in 2014 is clearly higher than in 2008, since the corresponding confidence intervals do not overlap. However, in countries like Austria, Germany or Sweden, there is a considerable overlap in the confidence intervals for these two years and thus we cannot give meaningful conclusions on the variation of the dependence coefficients. To get a better insight regarding the change in multivariate dependence between 2008 and 2014, Table 2 reports point estimates (with standard errors) for these two years and for ˆρ− 3, ˆρ+ 3and ˆρ3. In columns 3, 6 and 9, we also display the results of a two-independent sample t-test with unequal variances, calculated using bootstrap standard errors. In particular, we perform a onesided test to determine if the increases or decreases in the value of the coefficients between 2008 and 2014 are statistically significant. The corresponding p-value (in parentheses) is computed assuming asymptotic normality of the t-statistic. Table 3 displays the same results for bρ−z 3, 21 bρ+z 3and bρz 3. Interestingly, in most EU-28, we find a significant increase in all the coefficients over the period analysed. Thus, we can say that there has been a general increase in the multivariate orthant dependence between dimensions of poverty in the EU over the period 2008-2014. Moreover, this increase is found both in the lower and in the upper orthant, which means that, over the period analysed, there has been a general increase in both the probability of being simultaneously low-ranked and the probability of being simultaneously high-ranked in all dimensions of poverty. Noticeably, the highest increase in both the lower and upper orthant dependence is found in Spain, one of the countries most hardly hit by the economic crisis. Another country severely affected by the crisis, namely Greece, also experienced a substantial increase in these two types of dependence. <Insert Table 2 here > <Insert Table 3 here > To complement the analysis of three-dimensional dependence, we have also analysed all possible pairwise relationships between the dimensions of the AROPE rate. The results are displayed in Tables 4 and 5. The first feature that is worth pointing out is that the bivariate coefficients share many of the properties of the trivariate coefficients. In particular, in all the countries and for both years, all of them are positive and, in most of the countries, they are larger in 2014 than in 2008, with the differences being statistically significant at 5% in most cases. Additionally, these tables reveal that, in general, the dependence tends to be higher between income and the other two dimensions than between work intensity and no-material deprivation. <Insert Table 4 here > <Insert Table 5 here > 22 Finally, as we said in the Introduction, quantifying the dependence between the dimensions of the AROPE rate provides a useful complement to the information given by this indicator. In this context, we wonder whether those countries with higher AROPE rates are also countries with high levels of dependence between its dimensions. To address this issue, Panel A of Figure 5 depicts two scatter plots showing the relationship between the AROPE rate and the coefficient ˆρ− 3for the EU-28 countries in the years 2008 and 2014.7Panel B of the same figure displays the same results for the coefficient bρ−z 3. In all graphs, the horizontal and vertical reference lines represent the corresponding values for the whole EU-28. We focus on ˆρ− 3and bρ−z 3 because they measure lower orthant dependence, which is the key point in poverty analysis. The main features from these figures are the following: a) there is a positive relationship between the AROPE rate and lower orthant dependence, that is, countries with high incidence of multidimensional poverty tend to experience also a high degree of multivariate dependence between its dimensions in the lower orthant; b) those countries with either very low or very high values of both ˆρ− 3and bρ−z 3in 2008 have converged, over the period analysed, to the situation of the majority of the EU-28 countries; c) in the EU-28 as a whole (see the reference lines), there has been an increase in the AROPE rate accompanied with an increase in the multivariate dependence between its dimensions. <Insert Figure 5 here > 4 Conclusions This paper proposes to measure the dependence between dimensions of poverty using copulabased multivariate generalisations of Spearman’s rho. Two of these coefficients, namely ρ− dfor continuous data and ρ−z dfor possibly non-continuous data, turn out to be essential in poverty analysis as they enable to measure the dependence between the poverty dimensions in the 7The AROPE rate is calculated here as the proportion of households in our sample that are poor in at least one of the three dimensions considered. 23 lower orthant of the joint distribution. Hence, they capture the propensity of a household to be simultaneously low-ranked in all dimensions. Our empirical application provides a more comprehensive picture on how multidimensional poverty has evolved in the EU-28 countries over the period 2008-2014, by complementing the information about the incidence of poverty with measures of the multivariate dependence between its dimensions. In particular, we use multivariate generalisations of Spearman’s rho to assess multivariate dependence and we consider, as variables characterising poverty, those included in the AROPE rate: income, material needs and work intensity. The nature of the last two variables entails the presence of ties when ranking the households according to such variables. To address this problem, we adopt two different approaches, namely the use of estimators for the continuous case after breaking the ties using additional information and the use of tie-corrected estimators for possibly discontinuous data. Interestingly, the results obtained keep robust to the approach used. Our first conclusion is that, for all the EU-28 countries and all the years considered, there is a positive multivariate association between poverty dimensions, regardless of the coefficient used. Moreover, this dependence has noticeably increased in Europe between 2008 and 2014 and for most of the countries this increase is statistically significant and it is especially remarkable in those countries most hardly hit by the economic crisis like Spain and Greece. Another important conclusion is that, in the vast majority of European countries, the maximal dependence is found in the lower orthant. Therefore, small values of the three poverty dimensions tend to occur together and this simultaneous concentration of small values of income, no-material deprivations and work intensity is more likely to occur in 2014 than in 2008. Finally, we detect a positive relationship between the incidence of multidimensional poverty, measured by the AROPE rate, and the dependence between its dimensions. This means that countries with a high poverty incidence tend to experiment also a higher degree of dependence between the dimensions of poverty. 24 Acknowledgments We thank the referees for their helpful comments and suggestions that have improved the paper. The financial support from Spanish Ministry of Economy and Competitiveness (Project ECO2016-77900-P) and ERDF is acknowledged. The second author also acknowledges financial support from Junta de Castilla y Le´on-Consejer´ıa de Educaci´on (Project VA148G18). The third author also acknowledges financial support from Autonomous Community of Madrid and European Commission (Project S2015/HUM-3416-DEPOPOR-CM). The usual disclaimers apply. 25 Table 1. Share of households in the main diagonal of the unit cube [0,1]3 u≤0.25 0.25 <u≤0.5 0.5 <u≤0.75 u>0.75 Total Bulgaria (2008) 11.06% 3.43% 3.30% 7.28% 25.07% Romania (2008) 5.80% 2.34% 2.57% 9.10% 19.81% Spain (2008) 7.23% 2.28% 2.35% 3.29% 15.15% Spain (2014) 8.14% 2.77% 2.32% 5.27% 18.5% 32 Table 2. Coefficients of trivariate dependence between the dimensions of the AROPE rate ˆρ− 3ˆρ+ 3ˆρ3 2008 2014 t-test 2008 2014 t-test 2008 2014 t-test Austria 0.373 0.412 2.458 0.338 0.372 2.262 0.355 0.392 2.442 (0.011) (0.011) (0.007) (0.011) (0.010) (0.012) (0.011) (0.010) (0.007) Belgium 0.501 0.558 4.221 0.436 0.484 3.549 0.468 0.521 4.012 (0.010) (0.010) (0.000) (0.009) (0.010) (0.000) (0.009) (0.009) (0.000) Bulgaria 0.572 0.553 -1.167 0.528 0.529 0.046 0.550 0.541 -0.575 (0.011) (0.011) (0.122) (0.012) (0.011) (0.482) (0.011) (0.011) (0.283) Cyprus 0.383 0.446 3.535 0.379 0.437 3.326 0.381 0.441 3.559 (0.014) (0.011) (0.000) (0.013) (0.011) (0.000) (0.013) (0.011) (0.000) Czech Republic 0.394 0.421 2.097 0.370 0.383 1.065 0.382 0.402 1.653 (0.008) (0.010) (0.018) (0.008) (0.009) (0.143) (0.007) (0.009) (0.049) Germany 0.437 0.447 0.928 0.397 0.401 0.397 0.417 0.424 0.692 (0.008) (0.008) (0.177) (0.007) (0.008) (0.346) (0.007) (0.007) (0.245) Denmark 0.276 0.374 5.911 0.229 0.325 6.381 0.252 0.349 6.340 (0.012) (0.012) (0.000) (0.010) (0.011) (0.000) (0.011) (0.011) (0.000) Estonia 0.425 0.441 0.966 0.392 0.410 1.175 0.408 0.425 1.105 (0.012) (0.011) (0.167) (0.011) (0.010) (0.120) (0.011) (0.010) (0.134) Greece 0.412 0.520 8.573 0.404 0.508 8.134 0.408 0.514 8.646 (0.010) (0.008) (0.000) (0.010) (0.008) (0.000) (0.009) (0.008) (0.000) Spain 0.342 0.499 15.792 0.314 0.465 15.909 0.328 0.482 16.372 (0.007) (0.007) (0.000) (0.007) (0.007) (0.000) (0.007) (0.007) (0.000) Finland 0.378 0.410 2.769 0.330 0.353 2.144 0.354 0.381 2.557 (0.008) (0.008) (0.003) (0.008) (0.008) (0.016) (0.007) (0.008) (0.005) France 0.412 0.441 2.596 0.373 0.395 2.014 0.393 0.418 2.387 (0.008) (0.008) (0.005) (0.008) (0.008) (0.022) (0.008) (0.008) (0.008) Croatia NA 0.502 NA NA 0.497 NA NA 0.500 NA NA (0.011) NA NA (0.011) NA NA (0.011) NA Hungary 0.449 0.525 6.980 0.434 0.509 6.429 0.442 0.517 6.966 (0.008) (0.007) (0.000) (0.009) (0.008) (0.000) (0.008) (0.007) (0.000) Ireland 0.546 0.562 1.144 0.491 0.545 3.748 0.519 0.554 2.579 (0.010) (0.009) (0.126) (0.011) (0.010) (0.000) (0.010) (0.009) (0.005) Italy 0.384 0.443 7.168 0.355 0.407 6.621 0.369 0.425 7.136 (0.006) (0.006) (0.000) (0.005) (0.006) (0.000) (0.006) (0.006) (0.000) Lithuania 0.444 0.506 3.821 0.402 0.483 5.239 0.423 0.494 4.673 (0.012) (0.011) (0.000) (0.011) (0.011) (0.000) (0.011) (0.011) (0.000) Luxembourg 0.382 0.377 -0.277 0.339 0.329 -0.525 0.360 0.353 -0.414 (0.013) (0.013) (0.391) (0.013) (0.013) (0.300) (0.012) (0.012) (0.339) Latvia 0.475 0.477 0.102 0.445 0.460 0.976 0.460 0.469 0.552 (0.012) (0.011) (0.459) (0.011) (0.010) (0.165) (0.011) (0.010) (0.290) Malta 0.491 0.481 -0.590 0.486 0.453 -1.874 0.488 0.467 -1.268 (0.013) (0.012) (0.277) (0.013) (0.012) (0.030) (0.013) (0.011) (0.102) Netherlands 0.272 0.387 9.497 0.232 0.331 8.949 0.252 0.359 9.575 (0.009) (0.008) (0.000) (0.008) (0.008) (0.000) (0.008) (0.008) (0.000) Poland 0.438 0.472 3.868 0.434 0.454 2.082 0.436 0.463 3.067 (0.006) (0.007) (0.000) (0.006) (0.007) (0.019) (0.006) (0.006) (0.001) Portugal 0.426 0.486 3.830 0.417 0.469 3.237 0.421 0.477 3.652 (0.012) (0.010) (0.000) (0.013) (0.010) (0.001) (0.012) (0.009) (0.000) Romania 0.439 0.419 -1.481 0.478 0.448 -2.246 0.458 0.433 -1.931 (0.009) (0.010) (0.069) (0.009) (0.010) (0.012) (0.009) (0.009) (0.027) Sweden 0.357 0.375 1.166 0.315 0.316 0.066 0.336 0.345 0.660 (0.010) (0.012) (0.122) (0.009) (0.011) (0.474) (0.009) (0.011) (0.255) Slovenia 0.410 0.463 4.922 0.395 0.437 4.019 0.402 0.450 4.664 (0.008) (0.008) (0.000) (0.007) (0.007) (0.000) (0.007) (0.007) (0.000) Slovak Republic 0.408 0.451 2.802 0.387 0.412 1.745 0.397 0.432 2.368 (0.011) (0.011) (0.003) (0.011) (0.010) (0.040) (0.010) (0.010) (0.009) United Kingdom 0.423 0.522 8.377 0.377 0.482 8.835 0.400 0.502 8.914 (0.009) (0.007) (0.000) (0.009) (0.008) (0.000) (0.009) (0.007) (0.000) Note: Standard errors for the coefficients and p-values for the one-side t-test are displayed in parentheses. 33 Table 3. Tie-corrected coefficients of trivariate dependence between the dimensions of the AROPE rate bρ−z 3bρ+z 3bρz 3 2008 2014 t-test 2008 2014 t-test 2008 2014 t-test Austria 0.346 0.374 1.836 0.302 0.325 1.750 0.324 0.349 1.842 (0.011) (0.010) (0.033) (0.010) (0.009) (0.040) (0.010) (0.009) (0.033) Belgium 0.484 0.535 3.899 0.408 0.458 4.022 0.446 0.497 4.049 (0.010) (0.009) (0.000) (0.009) (0.009) (0.000) (0.009) (0.009) (0.000) Bulgaria 0.545 0.518 -1.688 0.488 0.474 -0.927 0.517 0.496 -1.355 (0.011) (0.011) (0.046) (0.011) (0.011) (0.177) (0.011) (0.011) (0.088) Cyprus 0.368 0.434 3.766 0.360 0.418 3.550 0.364 0.426 3.781 (0.014) (0.011) (0.000) (0.012) (0.011) (0.000) (0.013) (0.010) (0.000) Czech Republic 0.355 0.374 1.547 0.321 0.321 -0.029 0.338 0.347 0.841 (0.008) (0.009) (0.061) (0.007) (0.008) (0.488) (0.007) (0.008) (0.200) Germany 0.426 0.424 -0.185 0.373 0.367 -0.681 0.399 0.395 -0.427 (0.007) (0.008) (0.427) (0.006) (0.007) (0.248) (0.007) (0.007) (0.335) Denmark 0.243 0.337 6.143 0.198 0.285 7.132 0.220 0.311 6.735 (0.011) (0.011) (0.000) (0.008) (0.009) (0.000) (0.009) (0.010) (0.000) Estonia 0.398 0.419 1.307 0.346 0.373 1.938 0.372 0.396 1.647 (0.012) (0.011) (0.096) (0.010) (0.010) (0.026) (0.011) (0.010) (0.050) Greece 0.392 0.510 9.514 0.379 0.496 9.755 0.386 0.503 9.946 (0.010) (0.008) (0.000) (0.009) (0.008) (0.000) (0.009) (0.008) (0.000) Spain 0.357 0.508 15.976 0.331 0.472 16.020 0.344 0.490 16.454 (0.007) (0.006) (0.000) (0.006) (0.006) (0.000) (0.006) (0.006) (0.000) Finland 0.357 0.384 2.468 0.305 0.322 1.807 0.331 0.353 2.221 (0.008) (0.008) (0.007) (0.006) (0.006) (0.035) (0.007) (0.007) (0.013) France 0.372 0.395 2.085 0.322 0.338 1.674 0.347 0.367 1.941 (0.008) (0.008) (0.019) (0.007) (0.007) (0.047) (0.007) (0.007) (0.026) Croatia NA 0.499 NA NA 0.479 NA NA 0.500 NA NA (0.010) NA NA (0.010) NA NA (0.010) NA Hungary 0.420 0.485 5.931 0.388 0.447 5.387 0.404 0.466 5.855 (0.008) (0.007) (0.000) (0.008) (0.007) (0.000) (0.008) (0.007) (0.000) Ireland 0.538 0.549 0.824 0.477 0.525 3.456 0.508 0.537 2.222 (0.010) (0.009) (0.205) (0.010) (0.009) (0.000) (0.010) (0.009) (0.013) Italy 0.389 0.439 6.261 0.355 0.399 6.057 0.372 0.419 6.349 (0.005) (0.006) (0.000) (0.005) (0.005) (0.000) (0.005) (0.005) (0.000) Lithuania 0.408 0.481 4.519 0.355 0.437 5.769 0.381 0.459 5.254 (0.011) (0.011) (0.000) (0.010) (0.010) (0.000) (0.010) (0.011) (0.000) Luxembourg 0.341 0.335 -0.399 0.305 0.296 -0.612 0.323 0.315 -0.507 (0.012) (0.012) (0.345) (0.010) (0.010) (0.270) (0.011) (0.011) (0.306) Latvia 0.433 0.458 1.606 0.382 0.421 2.731 0.407 0.439 2.204 (0.012) (0.011) (0.054) (0.010) (0.010) (0.003) (0.011) (0.010) (0.014) Malta 0.485 0.470 -0.858 0.464 0.430 -2.069 0.474 0.450 -1.491 (0.013) (0.012) (0.195) (0.012) (0.011) (0.019) (0.012) (0.011) (0.068) Netherlands 0.269 0.341 9.085 0.230 0.314 9.353 0.250 0.343 9.415 (0.008) (0.008) (0.000) (0.006) (0.007) (0.000) (0.007) (0.007) (0.000) Poland 0.413 0.451 4.237 0.394 0.419 2.902 0.404 0.435 3.697 (0.006) (0.007) (0.000) (0.006) (0.006) (0.002) (0.006) (0.006) (0.000) Portugal 0.392 0.458 4.343 0.358 0.424 4.471 0.375 0.441 4.548 (0.012) (0.010) (0.000) (0.012) (0.009) (0.000) (0.011) (0.009) (0.000) Romania 0.379 0.361 -1.436 0.397 0.367 -2.432 0.388 0.364 -1.978 (0.009) (0.009) (0.075) (0.008) (0.009) (0.008) (0.008) (0.009) (0.024) Sweden 0.305 0.332 1.927 0.259 0.274 1.351 0.282 0.303 1.708 (0.009) (0.011) (0.027) (0.007) (0.009) (0.088) (0.008) (0.009) (0.044) Slovenia 0.352 0.427 7.051 0.316 0.380 6.853 0.334 0.404 7.196 (0.008) (0.007) (0.000) (0.007) (0.007) (0.000) (0.007) (0.007) (0.000) Slovak Republic 0.364 0.405 2.655 0.327 0.355 2.133 0.346 0.380 2.482 (0.011) (0.011) (0.004) (0.009) (0.009) (0.016) (0.010) (0.010) (0.007) United Kingdom 0.401 0.495 8.122 0.346 0.444 9.378 0.373 0.469 8.955 (0.009) (0.007) (0.000) (0.008) (0.007) (0.000) (0.008) (0.007) (0.000) Note: Standard errors for the coefficients and p-values for the one-side t-test are displayed in parentheses. 34 Table 4. Coefficients of pairwise dependence between the dimensions of the AROPE rate ˆρincome,work ˆρincome,no−deprivation ˆρwork,no−deprivation 2008 2014 t-test 2008 2014 t-test 2008 2014 t-test Austria 0.429 0.467 1.990 0.388 0.419 1.571 0.249 0.289 1.906 (0.014) (0.013) (0.023) (0.014) (0.014) (0.058) (0.015) (0.014) (0.028) Belgium 0.570 0.618 3.127 0.506 0.568 3.877 0.329 0.378 2.492 (0.011) (0.010) (0.001) (0.011) (0.011) (0.000) (0.014) (0.014) (0.006) Bulgaria 0.571 0.552 -0.975 0.643 0.568 -4.237 0.435 0.502 3.180 (0.014) (0.014) (0.165) (0.012) (0.013) (0.000) (0.015) (0.014) (0.001) Cyprus 0.465 0.489 1.052 0.469 0.513 2.164 0.209 0.322 4.412 (0.018) (0.014) (0.146) (0.016) (0.013) (0.015) (0.020) (0.017) (0.000) Czech Republic 0.431 0.459 1.776 0.427 0.439 0.789 0.290 0.307 1.077 (0.010) (0.012) (0.038) (0.010) (0.012) (0.215) (0.010) (0.013) (0.141) Germany 0.487 0.468 -1.502 0.467 0.504 2.948 0.297 0.301 0.297 (0.009) (0.009) (0.067) (0.009) (0.009) (0.002) (0.010) (0.011) (0.383) Denmark 0.327 0.427 4.838 0.284 0.397 5.501 0.145 0.224 3.607 (0.015) (0.014) (0.000) (0.014) (0.015) (0.000) (0.015) (0.016) (0.000) Estonia 0.471 0.493 1.124 0.439 0.411 -1.379 0.315 0.372 2.700 (0.014) (0.013) (0.131) (0.015) (0.014) (0.084) (0.016) (0.014) (0.003) Greece 0.462 0.538 4.708 0.505 0.647 10.481 0.257 0.358 5.390 (0.013) (0.010) (0.000) (0.011) (0.008) (0.000) (0.014) (0.013) (0.000) Spain 0.510 0.560 4.314 0.294 0.503 16.637 0.179 0.382 15.018 (0.008) (0.008) (0.000) (0.010) (0.008) (0.000) (0.010) (0.009) (0.000) Finland 0.459 0.506 3.535 0.386 0.378 -0.602 0.217 0.260 2.730 (0.010) (0.009) (0.000) (0.010) (0.010) (0.274) (0.011) (0.011) (0.003) France 0.403 0.419 1.122 0.498 0.515 1.330 0.277 0.320 2.858 (0.011) (0.010) (0.131) (0.009) (0.009) (0.092) (0.011) (0.010) (0.002) Croatia NA 0.638 NA NA 0.496 NA NA 0.366 NA NA (0.011) NA NA (0.014) NA NA (0.015) NA Hungary 0.514 0.559 3.357 0.474 0.578 7.602 0.337 0.414 4.931 (0.010) (0.009) (0.000) (0.011) (0.009) (0.000) (0.012) (0.010) (0.000) Ireland 0.610 0.663 3.348 0.528 0.529 0.079 0.418 0.468 2.520 (0.012) (0.010) (0.000) (0.013) (0.013) (0.468) (0.015) (0.013) (0.006) Italy 0.505 0.506 0.159 0.356 0.432 7.524 0.248 0.337 7.862 (0.007) (0.007) (0.437) (0.007) (0.007) (0.000) (0.008) (0.008) (0.000) Lithuania 0.530 0.578 2.602 0.404 0.490 4.116 0.335 0.415 3.660 (0.014) (0.013) (0.005) (0.015) (0.014) (0.000) (0.016) (0.015) (0.000) Luxembourg 0.448 0.454 0.254 0.490 0.460 -1.449 0.144 0.146 0.096 (0.016) (0.016) (0.400) (0.014) (0.015) (0.074) (0.019) (0.019) (0.462) Latvia 0.524 0.571 2.564 0.494 0.487 -0.365 0.364 0.348 -0.729 (0.014) (0.012) (0.005) (0.013) (0.012) (0.358) (0.015) (0.015) (0.233) Malta 0.651 0.646 -0.302 0.475 0.429 -2.021 0.340 0.326 -0.537 (0.013) (0.011) (0.381) (0.017) (0.015) (0.022) (0.019) (0.017) (0.296) Netherlands 0.374 0.478 7.154 0.283 0.371 5.747 0.099 0.227 7.888 (0.011) (0.010) (0.000) (0.011) (0.011) (0.000) (0.011) (0.011) (0.000) Poland 0.449 0.527 7.085 0.494 0.490 -0.346 0.365 0.372 0.537 (0.007) (0.008) (0.000) (0.007) (0.008) (0.365) (0.009) (0.009) (0.295) Portugal 0.478 0.546 3.514 0.497 0.521 1.233 0.289 0.365 3.446 (0.015) (0.012) (0.000) (0.015) (0.012) (0.109) (0.018) (0.013) (0.000) Romania 0.518 0.498 -1.231 0.515 0.459 -3.392 0.342 0.343 0.079 (0.011) (0.012) (0.109) (0.011) (0.012) (0.000) (0.013) (0.014) (0.468) Sweden 0.402 0.444 2.244 0.361 0.356 -0.268 0.245 0.237 -0.426 (0.012) (0.014) (0.012) (0.012) (0.015) (0.394) (0.013) (0.015) (0.335) Slovenia 0.495 0.559 5.004 0.410 0.449 2.749 0.302 0.342 2.738 (0.009) (0.009) (0.000) (0.010) (0.010) (0.003) (0.010) (0.011) (0.003) Slovak Republic 0.420 0.498 4.130 0.451 0.443 -0.461 0.320 0.354 1.694 (0.013) (0.013) (0.000) (0.013) (0.013) (0.322) (0.014) (0.014) (0.045) United Kingdom 0.506 0.571 4.720 0.420 0.529 7.390 0.274 0.405 7.782 (0.010) (0.009) (0.000) (0.011) (0.009) (0.000) (0.013) (0.011) (0.000) Note: Standard errors for the coefficients and p-values for the one-side t-test are displayed in parentheses. 35 Table 5. Tie-corrected coefficients of pairwise dependence between the dimensions of the AROPE rate ˆρz income,work ˆρz income,no−deprivation ˆρz work,no−deprivation 2008 2014 t-test 2008 2014 t-test 2008 2014 t-test Austria 0.382 0.426 2.282 0.386 0.394 0.473 0.205 0.229 1.296 (0.014) (0.013) (0.011) (0.012) (0.011) (0.318) (0.014) (0.013) (0.097) Belgium 0.528 0.576 3.014 0.486 0.539 3.848 0.323 0.374 2.927 (0.012) (0.011) (0.001) (0.010) (0.010) (0.000) (0.012) (0.012) (0.002) Bulgaria 0.510 0.474 -1.836 0.656 0.585 -4.191 0.383 0.428 2.135 (0.014) (0.014) (0.033) (0.012) (0.012) (0.000) (0.015) (0.014) (0.016) Cyprus 0.381 0.424 1.839 0.520 0.554 1.847 0.191 0.300 4.490 (0.018) (0.015) (0.033) (0.014) (0.012) (0.032) (0.019) (0.016) (0.000) Czech Republic 0.347 0.356 0.538 0.449 0.467 1.313 0.218 0.219 0.067 (0.010) (0.012) (0.295) (0.009) (0.011) (0.095) (0.009) (0.011) (0.473) Germany 0.446 0.420 -1.974 0.499 0.515 1.523 0.252 0.250 -0.186 (0.009) (0.009) (0.024) (0.007) (0.008) (0.064) (0.010) (0.010) (0.426) Denmark 0.245 0.352 5.448 0.272 0.376 6.338 0.144 0.205 3.524 (0.014) (0.014) (0.000) (0.011) (0.012) (0.000) (0.012) (0.013) (0.000) Estonia 0.396 0.424 1.435 0.458 0.444 -0.774 0.261 0.320 2.958 (0.014) (0.013) (0.076) (0.013) (0.012) (0.220) (0.015) (0.013) (0.002) Greece 0.374 0.494 7.325 0.556 0.673 9.682 0.227 0.343 6.524 (0.013) (0.010) (0.000) (0.010) (0.007) (0.000) (0.013) (0.012) (0.000) Spain 0.444 0.518 6.188 0.377 0.550 16.084 0.210 0.402 15.733 (0.009) (0.008) (0.000) (0.008) (0.007) (0.000) (0.009) (0.008) (0.000) Finland 0.390 0.440 3.738 0.392 0.377 -1.277 0.211 0.240 2.244 (0.009) (0.009) (0.000) (0.008) (0.008) (0.101) (0.009) (0.009) (0.012) France 0.342 0.352 0.665 0.475 0.485 0.831 0.223 0.263 2.866 (0.011) (0.010) (0.253) (0.008) (0.008) (0.203) (0.010) (0.010) (0.002) Croatia NA 0.595 NA NA 0.525 NA NA 0.347 NA NA (0.011) NA NA (0.013) NA NA (0.015) NA Hungary 0.439 0.468 2.160 0.489 0.592 7.887 0.285 0.339 3.489 (0.010) (0.009) (0.015) (0.010) (0.008) (0.000) (0.011) (0.010) (0.000) Ireland 0.574 0.640 4.055 0.533 0.521 -0.661 0.417 0.449 1.747 (0.013) (0.010) (0.000) (0.011) (0.012) (0.254) (0.013) (0.013) (0.040) Italy 0.456 0.462 0.599 0.422 0.484 7.058 0.238 0.310 6.942 (0.007) (0.007) (0.275) (0.006) (0.006) (0.000) (0.007) (0.008) (0.000) Lithuania 0.423 0.498 3.931 0.435 0.515 4.145 0.285 0.365 3.823 (0.014) (0.013) (0.000) (0.014) (0.013) (0.000) (0.015) (0.015) (0.000) Luxembourg 0.348 0.364 0.724 0.468 0.412 -3.246 0.154 0.170 0.803 (0.016) (0.016) (0.234) (0.012) (0.012) (0.001) (0.014) (0.015) (0.211) Latvia 0.425 0.500 3.964 0.518 0.519 0.033 0.278 0.299 1.033 (0.014) (0.012) (0.000) (0.012) (0.012) (0.487) (0.015) (0.014) (0.151) Malta 0.612 0.582 -1.705 0.493 0.458 -1.626 0.318 0.309 -0.359 (0.013) (0.012) (0.044) (0.016) (0.014) (0.052) (0.018) (0.016) (0.360) Netherlands 0.355 0.464 7.408 0.281 0.344 5.543 0.112 0.220 8.217 (0.011) (0.010) (0.000) (0.008) (0.008) (0.000) (0.009) (0.009) (0.000) Poland 0.385 0.451 5.868 0.514 0.525 1.053 0.312 0.329 1.393 (0.008) (0.008) (0.000) (0.007) (0.008) (0.146) (0.008) (0.009) (0.082) Portugal 0.374 0.455 4.125 0.523 0.549 1.519 0.229 0.319 4.334 (0.016) (0.012) (0.000) (0.014) (0.011) (0.064) (0.016) (0.013) (0.000) Romania 0.377 0.350 -1.675 0.553 0.497 -3.575 0.234 0.246 0.650 (0.011) (0.012) (0.047) (0.010) (0.012) (0.000) (0.012) (0.013) (0.258) Sweden 0.350 0.383 1.870 0.307 0.316 0.637 0.189 0.210 1.303 (0.012) (0.014) (0.031) (0.010) (0.011) (0.262) (0.011) (0.012) (0.096) Slovenia 0.369 0.463 7.173 0.420 0.465 3.561 0.213 0.282 4.985 (0.010) (0.009) (0.000) (0.009) (0.009) (0.000) (0.010) (0.010) (0.000) Slovak Republic 0.337 0.429 4.949 0.454 0.441 -0.702 0.246 0.270 1.314 (0.013) (0.013) (0.000) (0.013) (0.013) (0.241) (0.013) (0.014) (0.094) United Kingdom 0.456 0.517 4.417 0.411 0.519 8.394 0.253 0.372 7.726 (0.011) (0.009) (0.000) (0.010) (0.009) (0.000) (0.011) (0.010) (0.000) Note: Standard errors for the coefficients and p-values for the one-side t-test are displayed in parentheses. 36 Figure 1: Scatter plots of scaled ranks for Bulgaria (2008) and Romania (2008) 37 Figure 2: Scatter plots of scaled ranks for Spain (2008 and 2014) 38 Panel A: lower orthant dependence Panel B: upper orthant dependence Figure 3: Evolution of ˆρ− 3(Panel A) and ˆρ+ 3(Panel B) and their bootstrap standard 95% confidence intervals for EU-28 over the period 2008-2014. 39 Panel A: lower orthant dependence Panel B: upper orthant dependence Figure 4: Evolution of bρ−z 3(Panel A) and bρ+z 3(Panel B) and their bootstrap standard 95% confidence intervals for EU-28 over the period 2008-2014 40 Panel A Panel B Figure 5: Relationship between AROPE rate and ˆρ− 3(Panel A) and between AROPE rate and bρ−z 3(Panel B) for EU-28 and years 2008 (left) and 2014 (right). 41