The risk of clustering of deprivations in Spain: a tale of two crises

César García-Gómez (Departamento de Economía Aplicada, Universidad de Valladolid, Valladolid, Spain)
Ana Pérez (Departamento de Economía Aplicada, Universidad de Valladolid, Valladolid, Spain)
Mercedes Prieto-Alaiz (Departamento de Economía Aplicada, Universidad de Valladolid, Valladolid, Spain)

Applied Economic Analysis

ISSN: 2632-7627

Article publication date: 8 November 2024

173

Abstract

Purpose

The At Risk of Poverty or Social Exclusion (AROPE) rate is a key indicator for monitoring poverty in Europe. However, it is not sensitive to the degree to which individuals face multiple deprivations simultaneously. This paper aims to fill this gap by studying the relationship between the three dimensions of the AROPE rate at the lower tail of their joint distribution in Spain in the period 2009–2022.

Design/methodology/approach

To capture how the different dimensions of poverty are related at the lower tail of their joint distribution, this paper proposes a multivariate left tail concentration function based on copulas. This function quantifies lower tail dependence at a finite scale, which, for practical purposes, is more suitable than estimating asymptotic measures, and can be represented in a 2D graph, facilitating interpretation and temporal comparisons. This function also provides information on overall dependence, as it is closely related to the Blomqvist’s beta.

Findings

There is a considerable risk of clustering of deprivations in Spain, with low positions in one poverty dimension extending to others. This risk increased after the Great Recession but did not decrease with the economic recovery that followed. The crisis linked to COVID-19 did not have a significant impact on the risk of clustering of deprivations. Lower tail dependence provides new valuable insights on the dependence structure of poverty dimensions beyond the analyses based on overall dependence.

Originality/value

This paper provides new theoretical results and a pioneering application of multivariate lower tail dependence measures in welfare economics using non-parametric methods.

Keywords

Citation

García-Gómez, C., Pérez, A. and Prieto-Alaiz, M. (2024), "The risk of clustering of deprivations in Spain: a tale of two crises", Applied Economic Analysis, Vol. ahead-of-print No. ahead-of-print. https://doi.org/10.1108/AEA-03-2024-0113

Publisher

:

Emerald Publishing Limited

Copyright © 2024, César García-Gómez, Ana Pérez and Mercedes Prieto-Alaiz.

License

Published in Applied Economic Analysis. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence maybe seen at http://creativecommons.org/licences/by/4.0/legalcode


1. Introduction

Poverty is one of the most important social problems in Spain, being among the EU Member States with the highest poverty figures over the years. According to the At Risk of Poverty or Social Exclusion (AROPE) rate, over one in four people in Spain were at risk of poverty or social exclusion in 2022 (INE, 2023). This indicator, central for monitoring poverty in Europe in the framework of the 2030 Agenda for Sustainable Development, is constructed based on three dimensions – income, material needs and work intensity – and calculates the percentage of people who are below the imposed threshold in any of the three dimensions. Bringing these three indicators together in a way implies a multidimensional approach to identifying the poor and is an important step forward, as the target population is no longer identified on the basis of a single indicator, such as income. In spite of this advantage, the AROPE rate is not sensitive to the degree to which individuals face multiple forms of deprivation simultaneously [see, e.g. Nolan and Whelan (2011) and Bárcena-Martín et al. (2020)]. This limitation has significant implications, as it means that social policies target individuals experiencing deprivation in at least one dimension (the union criterion), without distinguishing between those facing multiple deprivations and those suffering from only one.

Understanding joint deprivations in poverty analysis is essential because deprivations can reinforce each other, creating a cycle of disadvantages. When households are deprived in one aspect of their lives, such as income, they could be more likely to accumulate further deprivation in other areas, such as material needs or work intensity. This reinforcing effect exacerbates poverty and makes it more difficult for people to escape poverty traps. Atkinson and Bourguignon (1982) already warned that:

Different forms of deprivation (such as low income, poor health, bad housing, etc.) tend to be associated, often drawing a contrast with what would be observed if they were independently distributed.

The influential work of Sen (1999) also remarks that coupling of disadvantages between different sources of deprivation is crucial to understand poverty and to design policies to tackle it. Similarly, Wolff and De-Shalit (2007) argue that the target group in anti-poverty polices is the group consisting of individuals who cluster disadvantages in all dimensions, being at the bottom of the joint distribution of poverty dimensions.

The main aim of this paper is to study how the three dimensions of the AROPE rate are related at the lower tail of their joint distribution in Spain using the data provided by the Encuesta de Condiciones de Vida (ECV) over the period 2009–2022. Focusing on this period allows studying the effect of the Great Recession and the COVID-19 crisis. In doing so, we complement the picture of poverty shown by the AROPE rate exploring overlapping forms of deprivation, a valuable information to design effective policies against poverty.

Now the problem is how to capture the relationship between the different dimensions of poverty, knowing that measuring the relationship between possible pairs is not sufficient and could hide important aspects (Durante et al., 2014). The well-known Pearson’s linear correlation coefficient is completely inappropriate because the distribution of the indicators that determine poverty hardly fits a Normal distribution. Furthermore, this coefficient does not allow quantifying multidimensional relationships nor does it detect relationships in the tails of the distribution. Therefore, alternative statistical techniques are required, such as stochastic dominance or the copula function. Stochastic dominance techniques allow the comparison of joint distribution functions without resorting to a given index; see Duclos et al. (2006). However, they do not allow us to quantify the level of dependence between variables and their application with more than two dimensions is almost intractable, as shown in García-Gómez et al. (2019).

Another approach to measure the dependence among poverty dimensions, which is the one we adopt in this paper, consists of transforming the original variables into positions (ranks) and focusing the analysis on the joint distribution function of these positions, which is, by definition, a copula; see Nelsen (2006), Joe (2014) and Durante and Sempi (2015) for an introduction to copulas. The copula-based methodology has certain advantages to achieve the objective of the paper. First, it allows for the definition of non-linear dependency concepts that have, in turn, a clear meaning in terms of poverty. It also allows the measurement of multivariate dependency relationships that could not be captured using only averages of bivariate measures across all possible pairs. Another advantage of copulas over, for example, stochastic dominance techniques is that they offer great flexibility in constructing and comparing multivariate distributions. These advantages come at a price, as the outcomes in the original variables are lost. This is not important for the purpose of this paper, but it could be in other cases.

The use of copulas in poverty analysis has received increasing attention over the last years. To mention but a few, Decancq (2020) relies on the concepts of cumulative deprivation and derives a multivariate Spearman’s footrule-type index based on the diagonal section of the copula. These tools are further applied in Decancq (2023) using the Belgian MEQIN data set from year 2016, and also in Scarcilli (2024) using EU-SILC data from eight European countries over the period 2007–2019. García-Gómez et al. (2021, 2024) measure multivariate orthant dependence between the AROPE components for the EU-28 countries and for the Spanish regions, respectively, using three multivariate versions of the Spearman’s rho coefficient and covering several years from 2008 onwards. In a bivariate setting, other recent papers incorporating copula-based dependence measures to poverty analysis, with applications to Indonesia and European countries, respectively, are Hohberg et al. (2021) and Tkach and Gigliarano (2022).

In general, the aforementioned studies focus on measures of overall dependence based on averages across the full domain of the joint distribution. Hence, they may fail to discern changes in dependence across different segments of the distribution. By contrast, the paper at hand focuses on the relationship in the lower part of the joint distribution, i.e. on individuals suffering joint deprivations, and deal with the concept of lower tail dependence (LTD). Moreover, the dependence measures we propose are based on conditional probabilities rather than on averages. In this sense, our paper is closest to D’Agostino et al. (2023), who first introduced the concept of tail dependence in poverty analysis but confined their research to the bivariate case and two years (2009 and 2018). Like them, we consider the dimensions of the AROPE rate, but we incorporate, as a novelty, a multidimensional perspective and a new copula-based tool and we cover a longer period (2009–2022). Our paper also differs from theirs in the estimation procedure, as they use a semi-parametric method whereas we follow a fully non-parametric approach.

The multivariate proposals to be made in this paper, both methodological and applied, will be pioneering in poverty analysis and other fields, since, to the best of our knowledge, there is no empirical application of the multivariate LTD tools, even in finance or insurance, where bivariate tail coefficients are very popular. To capture the risk of clustering of deprivations we propose a multivariate extension of the bivariate left tail concentration function introduced by Venter (2001). This function has several advantages when it comes to analysing lower tail dependence. First, regardless of the number of dimensions considered, it represents, in a unit square, the degree of multivariate dependence in the lower tail of the joint distribution. Second, it quantifies lower tail dependence at a finite scale, which, for practical purposes, is more suitable than estimating asymptotic measures. Finally, it also provides information on overall dependence, as it is closely related to a multivariate version of Blomqvist’s beta.

The main findings of this paper are summarised as follows. First, there is a risk of clustering of deprivations in Spain, which means that deprivations in one dimension do not come alone but tend to extend to other dimensions. Second, the highest bivariate lower tail dependence occurs between income and either work intensity or material needs, whereas the lowest is between work intensity and material needs. However, despite the coupling of deprivations in these two dimensions having the lowest probability of occurring, when it does occur, it is the most corrosive one. Third, the risk of clustering of deprivations in Spain increased significantly after the Great Recession but did not decrease significantly with the economic recovery that followed. However, unlike the 2008 crisis, the recent crisis linked to COVID-19 did not have a significant impact on the risk of clustering of deprivations. As a result, this risk was still higher in 2022 than in 2009. Finally, overall dependence between poverty dimensions, as measured by Blomqvist’s beta, may decrease, while not doing so in the lower part of the joint distribution, as it occurred in Spain during the economic recovery period (2015–2020). Hence, lower tail dependence, which focuses on the bottom of the joint distribution, can provide insights beyond the analyses based on overall dependence.

The paper is organised as follows. In Section 2, we introduce different concepts of multivariate lower tail dependence and the so-called left tail concentration function and establish its main properties. In Section 3 we apply these tools to analyse the risk of clustering of deprivations in Spain over the period 2009–2022 using the AROPE components as poverty dimensions. Finally, Section 4 summarises the main conclusions of the paper. Proofs and examples are left to the Appendices in the supplementary material.

2. Methodology

The concept of multivariate tail dependence is related to the degree of dependence between extreme events, i.e. the amount of dependence in the lower-left or upper-right orthant of the joint distribution of dimensions. In particular, our main interest lies in tools for assessing lower tail dependence, which is the most relevant one in poverty analysis.

Now, the question arises on how to measure multivariate lower tail dependence. Among the different proposals in the literature, we focus on that based on conditional probabilities and the so-called left tail concentration function (LTCF, here onwards). This function is a copula-based tool that captures the probability of a subset of variables falling under a low quantile given that the others also do. Thus, in our setting, the LTCF will measure the probability that a household that is poor in some dimensions is also simultaneously poor in other dimensions. Moreover, the LTCF also allows detecting overall dependence, as it is closely related to a well-known measure of association, the Blomqvist’s beta. For an overview of copula-based measures of multivariate tail dependence, see Gijbels et al. (2020) and the references therein.

In this section, we summarise the basic concepts on copulas that we need for our purpose and introduce the LTCF, describing first the bivariate case and then moving to the more challenging multivariate framework. We also discuss its nonparametric estimation.

2.1 Copulas: basic concepts

The copula approach focuses on the positions of the households across the variables, rather than on the values that these variables attain for such households. To do so, let the random vector X = (X1, …, Xd) represent the relevant d dimensions of poverty and let Fi denote the marginal distribution of dimension i, with i = 1, …, d. Then, each original variable Xi is transformed by applying the so-called probability integral transformation, obtaining a transformed variable Ui = Fi(Xi), with i = 1, … d. These transformed variables attach to each household in the population its relative position in all dimensions. For instance, a household with position vector (1, …, 1) will be top-ranked in all dimensions, i.e. it will be the “richest” one in terms of income, working conditions, material well-being, etc. On the contrary, a household with position vector (0, …, 0) will be bottom-ranked in all dimensions, i.e. it will be the “poorest” one in terms of income, working conditions, material well-being, etc.

From probability theory, the transformed variables U1, …, Ud are standard uniform random variables U(0, 1), provided that the marginals F1, …, Fd are continuous, and the joint distribution of the vector U = (U1, …, Ud) turns out to be the copula function C. Therefore, the copula is a d-dimensional cumulative distribution function, C: Id → I, with I = [0,1], whose univariate marginals are U(0, 1). So, for a given real vector u = (u1, …, ud) ∈ Id, the value C(u) represents, in our setting, the proportion of households in the population with positions outranked by u, i.e.:

C(u)=p(Uu)=p(U1u1,,Udud).
For instance, C(0.2, …, 0.2) will represent the probability that a randomly selected household is simultaneously in the first quintile (“low-ranked”) in all dimensions.

From an statistical point of view, the most important result of the theory of copulas is the Sklar’s theorem (Sklar, 1959). This theorem ensures that, given a d-dimensional random vector X = (X1, …, Xd) with joint distribution function F and univariate marginal distribution functions Fi, for i = 1, …, d, there exists a copula C: Id → I such that, for all (x1, …, xd) ∈ Rd:

(1) F(x1,,xd)=C(F1(x1),,Fd(xd)).
Conversely, if C is a d-copula and F1, …, Fd are univariate distribution functions, the function F defined in (1) is a joint distribution function with margins F1, …, Fd. Thus, copulas link joint distribution functions to their univariate marginals. If F1, …, Fd are all continuous, the copula C in (1) is unique. Otherwise, C is uniquely determined on Range F1 × … × Range Fd. Over the rest of this section we will assume that the marginal distributions F1, …, Fd are all continuous, although some issues arising when dealing with possibly non-continuous variables will be duly pointed out in Section 2.4. For a detailed discussion on the pitfalls related to non-continuity of the marginal distributions, see Genest and Nešlehová (2007) and the references therein.

Any copula C satisfies the Fréchet–Hoeffding bounds inequality:

(2) W(u)C(u)M(u),
for every u = (u1,…, ud) ∈ Id, where W(u) = max(u1 + ⋯ + udd + 1, 0) and M(u) = min(u1, …, ud). M is always a copula and represents perfect positive dependence, i.e. the case when each of the random variables X1, …, Xd is almost surely a strictly increasing function of any of the others (the outcomes in all dimensions are ordered in the same way). W is only a copula if d = 2, in which case it represents perfect negative dependence, that is, the case in which one variable is almost surely a decreasing function of the other. Another important copula is Π(u) = u1 × ⋯ × ud, which accounts for the case where the variables F1, …, Fd are independent.

2.2 Bivariate lower tail dependence

To analyse tail dependence in a multivariate setting one could consider the set of all possible bivariate relationships or define proper multivariate dependence measures. In this section, we address the first issue and describe some tools to capture bivariate tail dependence in a multivariate distribution, or more precisely, in a trivariate distribution, which is the most relevant case for our purposes. The problem of building up proper multivariate tail dependence coefficients is left for next section.

To measure bivariate lower tail dependence, we use the LTCF introduced by Venter (2001). This function allows to quantify dependence near the bottom corner [0,0] of the copula domain by means of the conditional probability that one variable takes on small values given that another variable also takes small values. In the trivariate case, one has three random variables, (X1, X2, X3), and so three possible bivariate relationships come up. Hence, if C is the three-dimensional copula associated to the joint distribution of (X1, X2, X3), whose marginal cumulative distributions are F1, F2 and F3, respectively, then, for each pair of variables (Xi, Xj) and any u ∈ (0,1), the LTCF is defined as the conditional probability that Xi is below its uth quantile given that Xj is below its uth quantile, that is:

(3) λLi|j(u)=p(XiFi1(u)|XjFj1(u))=p(Uiu|Uju)=Cij(u,u)u,
for all 1 ≤ i < j ≤ 3, where C12(u, u) = C(u, u, 1), C13(u, u) = C(u, 1, u) and C23(u, u) = C(1, u, u) are the bidimensional marginals of the 3-copula C, and Fi1 stands for the inverse of Fi [1] [2].

In poverty analysis, the special interest resides in the lower part of this function, which gives the probability that an individual is low-ranked in one dimension given that he/she is low-ranked in another dimension. For instance, considering the three dimensions of the AROPE rate, say income (1), work intensity (2) and material needs (3), the functions λL1|2(u), λL2|3(u) and λL3|1(u) will capture the probability of being poor in income, work intensity or material needs, given that one is already poor in work intensity, material needs or income, respectively, that is, the probability of being poor in two dimensions, given that one is one-dimensionally poor. Note, however, that the bivariate LTCF does not impose a hierarchy between the variables Xi and Xj, since λLi|j(u)=λLj|i(u), and therefore one variable cannot be identified as the source of deprivation in the other.

The LTCF in (3) is bounded between 0 and 1 and it approaches unity for u near 1, so it does not distinguish much between copulas there. However, its behaviour on the left tail varies significantly across the copulas (Venter, 2001; Durante et al., 2015) and its limit, as u approaches 0, is the well-known lower tail dependence coefficient λLi|j=limu0+λLi|j(u). We will not further discuss this asymptotic coefficient λLi|j since our purpose is understanding tail dependence behaviour at the quantile level u as given by the LTCF [3]. To ease the interpretation of the LTCF, it is informative to determine its value for some specific cases. For example, for independent variables, λLi|j(u)=u; for perfect positive dependence, λLi|j(u)=1; and for perfect negative dependence, λLi|j(u)=max(21/u,0).

It is noteworthy that the LTCF also provides information on the degree of overall dependence, as it is closely related to the well-known measure of association Blomqvist’s beta. This measure, also known as medial correlation coefficient, compares the propensity for two variables to take both values either above or below their respective medians, with the propensity for the occurrence of the contrary event. For each pair of random variables (Xi, Xj) with medians x˜i=Fi1(0.5) and x˜j=Fj1(0.5), respectively, and associated copula Cij, the bivariate Blomqvist’s beta is defined as:

(4) β2ij=p[(Xix˜i)(Xjx˜j)>0]p[(Xix˜i)(Xjx˜j)<0]=4Cij(0.5,0.5)1.
Notice that from (3) and (4), the following relationship holds (Durante et al., 2015):
(5) λLi|j(0.5)=(1+β2ij)/2.
Therefore, the value of the LTCF at u = 0.5 provides information on overall dependence, that is, the higher this value, the larger the propensity of Xi and Xj to take larger or smaller values than their corresponding medians simultaneously.

So far we have focused our attention on the study of bivariate tail dependence between the components of (X1, X2, X3) through the functions λLi|j(u), for all 1 ≤ i < j ≤ 3. However, these functions may fail to detect multivariate association among the three variables involved. To overcome this drawback, different proposals of proper multivariate tail dependence appear in the literature. In the next section we will discuss some of them.

2.3 Multivariate lower tail dependence

The literature on multivariate tail dependence is scarcer than in the bivariate case and is mainly focused on theoretical results for some parametric families of copulas. In fact, to our knowledge, there is no empirical application of multivariate tail dependence using non-parametric methods. One of the reasons for this is that, in the multivariate case, tail dependence concepts and properties are more complicated. To start with, the presence of more than two variables brings additional difficulties to the very definition of tail dependence. For instance, the bivariate tail dependence coefficients discussed in previous section cannot be generalised to the multivariate case (d > 2) in a straightforward and unique way, as we will see next.

The first generalisation of the bivariate LTCF in (3) to a random vector (X1, …, Xd) with d-copula C, was proposed by Sweeting and Fotiou (2013) and is given by:

(6) λL,d(u)=p(U1u,,Ud1u|Udu)=C(u,,u)u,
for any u ∈ (0,1). By definition, the function λL,d(u) in (6) is bounded between 0 and 1 and is invariant with respect to the conditioning variable, since all the variables Ui, with i = 1, …, d, share the same marginal distribution U(0, 1). Therefore, this function provides, for each u, the conditional probability that a group of d − 1 variables take simultaneously values under its u quantile, given that the remaining variable also does. In a poverty setting, this function will capture the risk that a household that is unidimensionally poor is also multidimensionally poor, although it cannot identify which single dimension is more likely to trigger disadvantages in others.

Here onwards, we will refer to λL,d(u) as the multivariate left tail concentration function (shortly, multivariate LTCF). In particular, for d = 3, we get the following trivariate LTCF:

(7) λL,3(u)=C(u,u,u)u,
for any u ∈ (0,1) For instance, considering that income, material needs and work are the three dimensions of poverty, λL,3(0.2) will measure the probability that a household that is in its second decile in one dimension, say income, is also simultaneously in the second decile in both work intensity and material needs, i.e. the risk that a household that is unidimensionally poor is also tridimensionally poor.

As in the bivariate case, the multivariate LTCF in (6) approaches 1 for u near 1, and its limit, as u approaches 0, is the multivariate lower tail dependence coefficient, λL,d=limu0+λL,d(u),defined in Hua and Joe (2011) and Joe (2014). But its main advantage is that, by varying u in (0, 1), a set of lower tail dependence coefficients at a finite scale come up and one may trace out how the intensity of dependence behaves at the different parts of the joint distribution. Hence, analysing the function λL,d(u) is more informative than looking only at its limiting value λL,d. Another advantage of the function λL,d(u) is that, despite its multivariate nature, it can be represented in a two-dimensional graph allowing the visualisation of multivariate tail dependence and making temporal and cross-section comparisons easier, as we will see later.

The following result, whose proof is in Appendix 1 in the supplementary material, establishes the main properties of the multivariate LTCF.

Proposition 1: The multivariate LTCF in (6) has the following properties:

(a) In the case of independence, it becomes λL,dΠ(u)=ud1, whereas in the case of perfect positive dependence, it becomes λL,dM(u)=1.

(b) If C and C′ are two d-copulas such that C′(u) ≤ C(u), ∀uId, then λL,dC(u)λL,dC(u).

(c) ∀u ∈ (0,1), max[d − (d − 1)/u,0] ≤ λL,d(u) ≤ 1, and λL,d(1) = 1.

(d) For any d ≥ 3, it is non-increasing in d, i.e. λL,d(u) ≤ λL,d−1(u).

(e) It is closely related to a multivariate version of Blomqvist’s beta βd proposed by Nelsen (2002), through the following equation [4]:

(8) λL,d(0.5)=(2d11)βd+12d1.
Interestingly, property (d) means that the more dimensions we have, the smaller the lower tail dependence between them, that is, if one household is income-poor, it is less likely for it to be also poor in both work intensity and material needs simultaneously, than to be also poor in another single dimension, say work intensity. Notice also that property (e) implies that, for fixed d, the higher λL,d(0.5), the higher the global multivariate dependence, as measured by Blomqvist’s βd, and vice versa. Thus, the value of the multivariate LTCF at u = 0.5 provides information on the overall multivariate dependence.

To better appreciate the usefulness of the multivariate LTCF, Figure 1 displays its trivariate version in (7) for three different parametric copulas (Clayton, Frank and Gumbel) sharing the same Blomqvist’s beta: β3=0.2 (left panel); β3=0.5 (central panel); and β3=0.8 (right panel) [5]. To ease the interpretation, in all panels, the trivariate LTCF in the case of independence is depicted in blue.

Several conclusions emerge from this figure. First, we can see that, for fixed β3, the three copula models display very different shapes of the LTCF, but they all cross at u = 0.5, where its value is λL,3(0.5)=(3β3+1)/4. Second, the main differences arise in the lower tail, around the bottom corner (0,0), as expected, whereas the upper corner (1, 1) is not informative, as limu1λL,3(u)=1 in all cases. In particular, over the range 0 < u < 0.5, the Clayton copula clearly shows more lower tail dependence than both the Gumbel and Frank copulas, for which limu0+λL,d(u)=0, and the differences become larger for higher values of β3. Third, Figure 1 shows that, for the three models, as the degree of overall dependence, measured by β3, increases, the LTCF moves upwards, indicating a higher degree of lower tail dependence, whereas as β3 decreases, the LTCF approaches its value at independence. Therefore, the position of the LTCF with respect to this benchmark gives important information on both the degree of lower tail dependence and the degree of overall dependence in the multivariate distribution, making the LTFC a potential useful tool for model selection, as pointed out by Venter (2001).

So far, we have discussed multivariate lower tail dependence measures that generalise the bivariate LTCF in (3) by still conditioning on one single variable, as defined in (6). However, other possible generalisations from bivariate to multivariate tail dependence can be worked out by conditioning on different subsets of variables. In doing so, several possibilities come up. For instance, with four dimensions (d = 4), one could define the probability that one variable keeps below its uth quantile, given that the other three variables also do, but it could also be possible to define the probability that two variables keep below their uth quantiles, given that the remaining two variables also do, and so on. Obviously, as the number of dimensions increases, the more complicated the problem becomes and the more cumbersome the notation is; see, for instance, Li (2009) and Fernández-Sánchez et al. (2016). Consequently, we just consider here the trivariate case, which is the relevant one for our purposes [6].

By conditioning on all possible subsets of two variables, in the trivariate case (d = 3) we can define three different trivariate versions of the bivariate LTCF in (3), which we will refer to as tri-bivariate LTCF, namely:

(9) λLk|ij(u)=p(Uku|{Uiu,Uju})=C(u,u,u)Cij(u,u),
for any u ∈ (0,1), where k ∈{1,2,3}, k i and 1 ≤ i < j ≤ 3, and Cij is the corresponding bidimensional marginal of the 3-copula C. The study of trivariate lower tail dependence based on the new LTCF in (9) corresponds, for small values of u, to measure how likely one variable is in (0,u] given that the other two variables are also in (0,u]. When the limit exists, the asymptotic tri-bivariate lower tail dependence coefficients are given by λLk|ij=limu0+λLk|ij(u); see Li (2009) and Fernández-Sánchez et al. (2016) for further results on these and other asymptotic multivariate tail dependence coefficients.

In our poverty framework, the trivariate function in (7) measures how likely it is that suffering disadvantage in one dimension triggers further disadvantages in others. Wolff and De-Shalit (2007) call this sort of disadvantages corrosive disadvantages. However, this function cannot distinguish which single deprivation is more likely to spread its “badness” to the others. In turn, the coefficients in (9) assess the risk of being tridimensional poor given that one is bidimensional poor and allow identifying which coupling of deprivations is more corrosive in leading to multivariate poverty.

To close this section, we notice that there is a close relationship between the coefficients defined in (3), (7) and (9). In particular:

(10) λL,3(u)=λLi|j(u)λLk|ij(u).
This implies that, in a trivariate setting, it is enough to compute λL,3(u) and the three possible bivariate coefficients, {λL2|3(u),λL1|3(u),λL1|2(u)}, since the others immediately follow. Moreover, from equation (10), one can deduce the next inequalities:
λL,3(u)min{λL2|3(u),λL1|3(u),λL1|2(u)},λL,3(u)min{λL1|23(u),λL2|13(u),λL3|12(u)}.
In our example, this means, first, that a household that is unidimensionally poor has a larger risk to be also bidimensionally poor than to be tridimensionally poor, and second, that the risk for a household to be tridimensionally poor is larger if it is bidimensionally poor than if it is only poor in one dimension.

2.4 Non-parametric estimation

Now, the question arises on how to estimate the LTCF in practice, since the copula C is unknown and must be estimated from the data. To do that, let {(X1j, …, Xdj)}j = 1,…,n be a sample of n serially independent random vectors from the d-dimensional vector X = (X1, …, Xd) with associated copula C. Then, it is possible to estimate non-parametrically the copula C by the corresponding empirical copula, namely:

(11) C^n(u)=1nj=1ni=1d1{U˜ijui}, for u=(u1,,ud)Id,
where 1A denotes the indicator function on a set A and U˜ij are the transformed data to [0,1] by scaling ranks, i.e:
(12) U˜ij=Rij/n,
where Rij denotes the rank of Xij among {Xi1, …, Xin}, with i = 1, …, d and j = 1, …, n.

Following the work of Patton (2013) and Durante et al. (2015) in the bivariate setting, we propose to estimate the multivariate LTCF in (6) by replacing the copula function by its empirical counterpart in (11). In doing so, the empirical version of the multivariate LTCF is given, for any 0 < t < 1, by:

λ^L,d(t)=C^n(t,,t)t.
In a similar fashion, the bivariate LTCF will be estimated non-parametrically by replacing the bivariate copula function in (3) by their corresponding empirical counterpart, which is just a particular case of (11) with d = 2. Finally, we propose to estimate the coefficients λLk|ij(u), through a plug-in estimator that consists of replacing in (10) both the bivariate and the trivariate LTCF by their empirical versions previously defined, to get:
λ^Lk|ij(t)=λ^L,3(t)λ^Li|j(t).
Similarly, the Blomqvist’s beta will be estimated through a plug-in estimator based on equation (8).

So far, we have assumed that the marginal distributions are continuous. However, some of the variables usually encountered in poverty analyses are not continuous, as it will be the case in our empirical application. In this setting, the copula C in (1) is not unique and the values of the measures of tail dependence previously discussed can vary widely even based on the same joint distribution. To overcome this problem, different solutions are available in the literature.

One solution is to perform a “data jittering” of the original discrete variables by adding a continuous random noise; see D’Agostino et al. (2023) for an application in poverty analysis. The resulting continuous random vector has a unique copula, which turns to be the checkerboard copula. This copula retains some of the dependence structure of the original variables and can be consistently estimated in two ways; either by applying the empirical copula Cn to the jittered data or through the so-called empirical checkerboard copula, which is a multilinear extension of the empirical copula Cn, applied to the original data. Genest et al. (2017) show that both procedures provide consistent estimators of the checkerboard copula but the latter is more precise than the former. Given this, in our empirical application we estimate the LTCF using the empirical checkerboard copula. For further solutions to deal with binary data, see Kobus and Kurek (2018) and the references there.

3. The risk of clustering of deprivations in Spain

In this section, we analyse lower tail dependence between poverty dimensions in Spain to address the following questions. Is there a risk of clustering of deprivations in Spain? If this is so, which are the deprivations and couples of deprivations that are more corrosive? How did the risk of clustering of deprivations in Spain respond to the Great Recession and the crisis derived from the COVID-19 pandemic? What is the value-added of tail dependence analyses compared to the overall dependence analyses used so far in the multidimensional poverty framework?

3.1 Data and variables

The dimensions of poverty we consider are income, work intensity and material needs. These are the dimensions included in the AROPE rate. The three measures characterising these dimensions are defined as follows:

  1. 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 [7]. The income is collected for the calendar year prior to the interview. For example, if the survey is conducted in 2022, the income data corresponds to 2021.

  2. The 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 [8].

  3. Material 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, which measures material needs in the household, 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 data we use comes from the cross-sectional ECV surveys for years in the period 2009–2022 [9]. The unit of analysis is the household and we only work with subsamples of households for which we have complete information for all the three variables. In these subsamples, the sample sizes range from 9,475 observations in 2013 to 18,498 observations in 2022.

3.2 A first look at the risk of clustering of deprivations

In this section, we get a first overview of the risk of clustering of deprivations in Spain, i.e. the tendency of bad positions in some poverty dimensions to extend to other dimensions. In particular, our aim is to determine if this risk exists and, if it does, which are the most corrosive deprivations.

To this aim, we will use the different measures of lower tail dependence discussed in Section 2. In our framework, the trivariate LTCF λL,3(u) in (7) allows to study the probability that a household that is badly positioned in one of the AROPE dimensions is also badly positioned simultaneously in the other two dimensions. Moreover, using equation (10), we can decompose this probability into:

  1. the probability that a household is simultaneously low-ranked in two dimensions given that it is low-ranked in one of them, captured by the bivariate LTCF, λLi|j(u) in (3); and

  2. the probability that a household is simultaneously low-ranked in the three dimensions given that it is low-ranked in two of them, captured by the tri-bivariate LTCF, λLk|ij(u) in (9).

Figure 2 displays, for years 2009, 2015, 2020 and 2022, the estimated trivariate (left panel), bivariate (central panel) and tri-bivariate (right panel) LTCFs. In all cases, the functions are calculated for t ∈ [0.05, 0.95] over 100 points and, as a benchmark, the theoretical LTCF of independence is also displayed in red. Here onwards, in all figures and tables the superindices (1), (2) and (3) denote income, work intensity and material needs, respectively.

The first conclusion that can be drawn from Figure 2 is that in Spain there is a risk of clustering of deprivations since, for the four years considered, the estimated LTCFs are above the theoretical function of the independence case. This means that, in Spain, bad positions in some poverty dimensions tend to extend to other dimensions, creating a vicious circle that exacerbates the poverty conditions of individuals. This finding highlights the need to complement the AROPE rate with an analysis of the risk of clustering of disadvantages, a phenomenon not detected by the AROPE rate alone. Moreover, looking at the bivariate LTCFs (central panel) we observe that, for all t, the highest bivariate lower tail dependence is between income and either work intensity or material needs. In contrast, the lowest bivariate lower tail dependence is between work intensity and material deprivation. This feature confirms the previous findings of D´Agostino et al. (2023) at an European level. In turn, focusing on the tri-bivariate LTCFs (right panel) we observe that, conditioning on two dimensions, the highest lower tail dependence occurs precisely when conditioning on the pair material needs and work intensity. This means that, although the bivariate coupling of deprivations in work intensity and material needs is the least likely to happen, when it does it is the most corrosive one.

3.3 Trends in the risk of clustering of deprivations

In the previous section, we identified a risk of clustering of deprivations in Spain, where disadvantages in one dimension of poverty do not come alone, but often trigger disadvantages in others. The objective of this section is to analyse how this risk evolved from 2009 to 2022, paying special attention to the impacts of both the Great Recession and the recent COVID-19 pandemic crisis.

To that aim, Figure 3 displays the estimated LTCFs for years 2009 (black line), 2015 (blue line), 2020 (red line) and 2022 (green line). The 95% standard bootstrap confidence intervals, generated from 1000 bootstrap replicates, are also displayed. Panel (a) presents the trivariate LTCF, panel (b) depicts pairwise LTCFs and panel (c) shows the tri-bivariate LTCFs. As a benchmark, the theoretical LTCF of independence is displayed in the dashed black line.

Several conclusions emerge from Figure 3. First, in general, the curves for 2015 are above those for 2009, i.e. the Great Recession led to an increase in the risk of clustering of deprivations in Spain. This holds for the trivariate LTCF, for the pairwise LTCFs (except for λL1|2 at the lower end of the distribution, where the conclusions are unclear), and for the tri-bivariate LTCFs (except for λL2|13 and λL1|23 at the lower end of the distribution). Second, when comparing 2015 with 2020, we cannot establish general conclusions, as the curves often overlap or intersect. However, if we focus on the lower end of the joint distribution (for t ≤ 0.2), we observe that the curves for 2020 are not below those for 2015 (except for λL3|12), indicating that the risk of clustering of deprivations did not decrease during the economic recovery period (2015–2020). Finally, we observe that the curves for 2020 and 2022 are very close together, indicating that the crisis linked to the COVID-19 pandemic did not have a significant impact on the risk of clustering of deprivations.

The LTCFs enable the analysis of lower tail dependence at any quantile. However, as noted by Venter (2001), this function does not offer much information for high quantiles, as it tends towards 1 as u approaches 1. In addition, when examining multidimensional poverty, the focus lies primarily on assessing dependence in the lower part of the joint distribution. Therefore, for the remainder of this section, we will concentrate on the value of LTCFs for u = 0.2 (similar results for u = 0.1 can be found in Appendix 3 in the supplementary material). Notice that λL,3(0.2) captures the probability that a household in the first quintile in one of the dimensions of AROPE is also simultaneously in the first quintile in the other two dimensions. Conversely, λLk|ij(0.2) captures the probability that a household that is simultaneously in the first quintile in two dimensions i and j is also in the first quintile in the remaining one. Finally, λLi|j(0.2) captures the probability that a household in the first quintile in one dimension j is also in the first quintile in another dimension i. By analysing the evolution of these coefficients, we aim to shed more light on the evolution of the risk of clustering of deprivations in Spain between 2009 and 2022.

Figure 4 shows the evolution of the estimated lower tail dependence measures at t = 0.2 throughout the period 2009–2022: Panel (a) depicts the evolution of λ^L,3(0.2); Panel (b) displays the evolution of λ^Li|j(0.2); Panel (c) shows the evolution of λ^Lk|ij(0.2). In all cases, 95% confidence intervals are provided, calculated using bootstrap with 1,000 replicates. To complement this information Table 1 presents the results of two-independent sample one-side t-tests with unequal variances, calculated using bootstrap standard errors, to test the significance of the changes in the lower tail dependence measures over the subperiods 2009–2015, 2015–2020, 2020–2022, and over the full period 2009–2022. The p-values of the tests (in parentheses) are computed assuming asymptotic normality of the t-statistics. The estimated values of all coefficients for each year in the period analysed, with bootstrap standard errors, can be found in Table 5 of Appendix 3 in the supplementary material.

Looking at Figure 4 and Table 1 we observe that the vast majority of the coefficients show significant increases between 2009 and 2022. Moreover, we observe that this is due to the fact that they significantly increased after the Great Recession (2009–2015), but did not significantly decrease during the period of economic recovery (2015–2020). On the contrary, some of the coefficients even increased significantly during this period. Finally, we find that the vast majority of the measures show no significant changes between 2020 and 2022, with them increasing immediately after the COVID-19 crisis but rapidly declining afterwards. An interesting result is that this evolution is slightly different from that of the AROPE rate, according to the data publicly published by the Spanish National Statistical Institute (INE) [10]. Both the AROPE rate and the risk of clustering deprivations exhibited similar trends during the crises: they increased significantly during 2009–2015 and remained almost constant in the period 2020–2022. However, the economic recovery from 2015 to 2020 resulted in a decrease in the AROPE rate (although it did not return to 2009 levels), while the risk of clustering deprivations remained almost unchanged. Once again, this highlights the need to complement the AROPE rate with measures of dependence among its indicators to ensure an accurate measurement of multidimensional poverty.

To sum up, the analysis of lower tail dependence reveals a long-lasting effect of the Great Recession on the risk of clustering of deprivations, as this risk increased during the crisis and did not decrease with the subsequent economic recovery. This suggests an asymmetric effect of the Great Recession on lower tail dependence between poverty dimensions; see Ayala and Cantó (2022) for a discussion on the asymmetric effect of the economic cycle on poverty and inequality in Spain. Conversely, the recent economic crisis linked to the COVID-19 pandemic does not appear to have had significant effects on the risk of clustering of deprivations. This suggests that both crises had different impacts on the dependency structure of poverty dimensions. Although further research is needed, a potential explanation for these different behaviours may be found not only in the distinct nature of the two crises but also in the different reactions of governments in terms of welfare policies. As noted by Moreira and Hick (2021), European governments responded promptly to the economic consequences of COVID-19 by adapting welfare policies and implementing emergency measures that were, in general, more extensive than those adopted during the Great Recession. In the Spanish case, Boscá et al. (2021) show that the set of policies implemented during the COVID-19 crisis had a significant impact on mitigating the decline in economic activity. Moreover, Cantó et al. (2022) show that the Spanish government’s fiscal response, with a big role of earnings compensation schemes, helped to lessen the shock and protect incomes, especially at the bottom of the distribution. In any case, the risk of clustering of deprivations was still significantly higher in 2022 than in 2009. This fact suggests the need to rethink public policies and steer them towards the declustering of deprivations. Otherwise, multidimensional poverty runs the risk of becoming chronic.

3.4 Tail dependence versus overall dependence

To close the paper, we wonder if the tail dependence analysis performed in the previous sections provides a value-added when comparing to the analysis based on overall dependence. To address this question, in this section we will study the evolution of overall dependence between the three AROPE dimensions using Blomqvist’s beta and we will compare the results with those obtained in the previous section. Recall that this measure is closely related to the LTCF; see equations (5) and (8). Figure 5 shows, in Panel (a) the evolution of the trivariate Blomqvist’s beta in Spain over the period 2008–2022, and in Panel (b) the evolution of the three pairwise coefficients. To complement this figure, Table 2 displays the results of a two-independent sample one-side t-test with unequal variances, calculated using bootstrap standard errors, to test the significance of the change in the coefficients in the periods 2009–2015, 2015–2020, 2020–2022 and 2009–2022, with the p-values of the tests in parenthesis.

The results in Figure 5 and Table 2 show that the evolution of overall dependence over the period analysed was substantially different to that observed for lower tail dependence in Section 3.3. In particular, we find that the level of overall dependence between the AROPE dimensions was not significantly different in 2022 than in 2009. Moreover, this is due to the fact that overall dependence significantly increased after the Great Recession but, contrary to what happened with lower tail dependence, significantly decreased during the economic recovery period. Hence, we do not find, for overall dependence, the permanent effect of the Great Recession that we found when analysing lower tail dependence. On the other hand, as it also occurred with lower tail dependence, the COVID-19 crisis did not have a significant negative effect on the level of overall dependence between poverty dimensions. In fact, the coefficients slightly increased immediately after the outbreak of the crisis, but quickly decreased afterwards. This suggests, again, that the two economic crises experienced over the last two decades had very different impacts on the structure of dependence between poverty dimensions.

The fact that the analyses based on overall and lower tail dependence yield different results is of great importance, not only from a research standpoint but also from a public policy perspective. As we have seen in the Spanish cases, measures of overall dependence such as Blomqvist’s beta may fail to detect the permanent impact of the Great Recession on the dependence at the lower tail of the joint distribution of poverty dimensions. The concept of lower tail dependence overcomes this drawback by capturing the risk that low positions in some dimensions extend to the others. Therefore, this concept is particularly relevant in the multidimensional poverty framework, as it can provide relevant information when designing, implementing and monitoring public policies aimed at declustering deprivations.

4. Conclusions

The AROPE rate is the main indicator of poverty and social exclusion in the European Union and therefore in Spain. One of the main criticisms of this measure is that it does not capture the extent to which individuals accumulate deprivations. This paper proposes complementing the AROPE rate with measures of lower tail dependence, which allow for assessing the risk of clustering deprivations in society. This entails calculating the probability that a household ranked low in one or several dimensions of poverty is also ranked low in other dimensions. Specifically, we introduce and apply various measures of lower tail dependence to analyse the risk of clustering deprivations in Spain and its evolution over the past years, based on the three dimensions of the AROPE rate. We also compare the tail dependence behaviour with overall dependence. This analysis yields several relevant results.

Firstly, there is a risk of clustering of deprivations in Spain. That is, low positions in some of the dimensions of the AROPE rate tend to extend to the rest of the dimensions, which exacerbates the poverty conditions of individuals. Secondly, the highest bivariate lower tail dependence occurs between income and either work intensity or material needs, whereas the lowest is between work intensity and material needs. However, despite the coupling of deprivations in these two dimensions having the lowest probability of occurring, when it does occur, it is the most corrosive one. Thirdly, the risk of clustering of deprivations in Spain responded differently to the last two crises. It increased significantly after the Great Recession but did not decrease significantly with the economic recovery that followed, suggesting an asymmetric response to the economic cycle, with a long-lasting effect of the Great Recession. By contrast, the crisis linked to COVID-19 did not have a significant impact on the risk of clustering of deprivations. As a result, this risk was still higher in 2022 than in 2009. Although more research is needed, the different welfare policies implemented in the two crises may play a role in explaining these patterns. Finally, we find that overall dependence between poverty dimensions, as measured by Blomqvist’s beta, may decrease, while not doing so in the lower part of the joint distribution, as it occurred in Spain during the economic recovery period (2015–2020).

A significant implication of our findings is the need to complement the AROPE rate with measures that capture the risk of clustering deprivations to ensure a more accurate measurement of multidimensional poverty, acknowledging the interdependence among poverty dimensions. In terms of policy recommendations, it is essential to rethink public policies and steer them towards reducing the clustering of deprivations. Addressing only individual dimensions of poverty is insufficient; policies must be designed to tackle the interrelated nature of these dimensions to prevent poverty from becoming chronic. The different responses to the Great Recession and the COVID-19 crisis underline the impact of welfare policies on the clustering of deprivations. A more comprehensive and well-coordinated approach could provide a roadmap for future interventions aimed at mitigating multidimensional poverty.

Figures

Trivariate LTCFs of Clayton (solid), Frank (dashed) and Gumbel (dotted) copulas with 

β′3=0.2 (left), 

β′3=0.5 (center) and 

β′3=0.8 (right)

Figure 1.

Trivariate LTCFs of Clayton (solid), Frank (dashed) and Gumbel (dotted) copulas with β3=0.2 (left), β3=0.5 (center) and β3=0.8 (right)

Trivariate (left), bivariate (center) and tri-bivariate (right) LTCFs in Spain for years 2009, 2015, 2020 and 2022

Figure 2.

Trivariate (left), bivariate (center) and tri-bivariate (right) LTCFs in Spain for years 2009, 2015, 2020 and 2022

Trivariate [Panel (a)], bivariate [Panel (b)] and tri-bivariate [Panel (c)] LTCFs for Spain in years 2009 (black), 2015 (blue), 2020 (red) and 2022 (green) with 95% bootstrap confidence intervals

Figure 3.

Trivariate [Panel (a)], bivariate [Panel (b)] and tri-bivariate [Panel (c)] LTCFs for Spain in years 2009 (black), 2015 (blue), 2020 (red) and 2022 (green) with 95% bootstrap confidence intervals

In Panel (a), evolution of 

λ^L,3(0.2). In Panel (b), evolution of 

λ^L1|2(0.2) in solid, 

λ^L1|3(0.2) in dashed and 

λ^L2|3(0.2) in dotted. In Panel (c), evolution 

λ^L3|12(0.2) in solid, 

λ^L2|13(0.2) in dashed and 

λ^L1|23(0.2) in dotted

Figure 4.

In Panel (a), evolution of λ^L,3(0.2). In Panel (b), evolution of λ^L1|2(0.2) in solid, λ^L1|3(0.2) in dashed and λ^L2|3(0.2) in dotted. In Panel (c), evolution λ^L3|12(0.2) in solid, λ^L2|13(0.2) in dashed and λ^L1|23(0.2) in dotted

Evolution of trivariate [Panel (a)] and bivariate [Panel (b)] Blomqvist’s beta. In Panel (b), 

β212 in solid, 

β213 in dashed and 

β223 in dotted

Figure 5.

Evolution of trivariate [Panel (a)] and bivariate [Panel (b)] Blomqvist’s beta. In Panel (b), β212 in solid, β213 in dashed and β223 in dotted

t-Tests for the significance of the variation in λL,3(0.2),λLi|j(0.2) and λLk|ij(0.2) in Spain over the subperiods 2009–2015, 2015–2020, 2020–2022 and the full period 2009–2022

Coefficient2009–20152015–20202020–20222009–2022
λ^L,3(0.2) 4.573*** (0.000) 1.320 (0.093) −0.629 (0.265) 6.498*** (0.000)
λ^L1|2(0.2) 1.880** (0.030) 2.687*** (0.004) 0.668 (0.252) 6.268*** (0.000)
λ^L1|3(0.2) 3.782*** (0.000) 0.522 (0.301) −3.212*** (0.001) 1.845** (0.033)
λ^L2|3(0.2) 7.315*** (0.000) −0.191 (0.424) −0.298 (0.383) 8.211*** (0.000)
λ^L3|12(0.2) 4.338*** (0.000) −0.582 (0.280) −1.320 (0.093) 3.319*** (0.000)
λ^L2|13(0.2) 3.062*** (0.001) 1.245 (0.107) 1.590 (0.056) 6.547*** (0.000)
λ^L1|23(0.2) −0.780 (0.782) 2.092** (0.018) −0.596 (0.276) 0.791 (0.214)
Notes:

** and ***indicate significance at 5 and 1%, respectively.

p-values in parentheses

Source: Authors’ own creation

t-Tests for the significance of the variation of trivariate and bivariate Blomqvist’s beta in Spain over the subperiods 2009–2015, 2015–2020, 2020–2022 and the full period 2009–2022

Coefficient2009–20152015–20202020–20222009–2022
β^3 8.132*** (0.000) −6.463*** (0.000) −2.059** (0.020) 0.291 (0.386)
β^212 5.261*** (0.000) −4.412*** (0.000) −1.681** (0.046) −0.595 (0.276)
β^213 5.184*** (0.000) −5.990*** (0.000) −0.285 (0.388) −0.785 (0.216)
β^223 8.176*** (0.000) −4.890*** (0.000) −2.511*** (0.006) 1.553 (0.060)
Notes:

** and ***indicate significance at 5 and 1%, respectively.

p-values in parentheses

Source: Authors’ own creation

Notes

1.

The values of the LTCF are called “finite tail dependence coefficients” by Sweeting and Fotiou (2013) and “penultimate tail dependence coefficients” by Manner and Segers (2011).

2.

If Fi is not strictly increasing, the quasi-inverse Fi(1)(u)=inf{x/F(x)u} should be used.

3.

Interested readers in coefficient λLi|j can refer to Nelsen (2006) and Joe (2014) for more theoretical issues and Oh and Patton (2018), Supper et al. (2020) and Panagiotou and Stavrakoudis (2023) for applications.

4.

We follow Úbeda-Flores (2005) and Ferreira and Ferreira (2020) and denote this coefficient as βd to distinguish it from other existing multivariate versions of Blomqvist’s beta.

5.

See Appendix 2 in the supplementary material for a brief description of these three copulas and the corresponding formulae of the multivariate Blomqvist’s beta.

6.

For higher dimensions, the problem is greatly simplified when all possible marginals of C coincide, as it happens with Archimedean copulas, for instance; the interested reader in the multivariate Archimedean copulas tail behaviour is referred to Charpentier and Segers (2009) and Fernández-Sánchez et al. (2016).

7.

The 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.

8.

Eurostat considers that a working-age person is a person aged 18–59 years, excluding also the students aged 18–24 years.

9.

In our tables and graphics, the year indicated refers to the survey year, not the income year. Recall that the reference period for income and work intensity data is the year prior to the survey, while material deprivation is contemporaneous. This introduces a lag in the income and work intensity variables with respect to the material deprivation dimension. We acknowledge that this lag may affect the direct comparability of income and work intensity with the material deprivation dimension; however, this is consistent with the methodology used in the AROPE rate.

Supplementary material

The supplementary material for this article can be found online.

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Acknowledgements

The authors thank the referees for their helpful comments and suggestions that have improved the paper. The first author gratefully acknowledges the financial support provided by the 2022 I + D+i National Projects (Generación de Conocimiento) by the Spanish Ministry of Sciences and Innovation (Ref: PID2022-143254OB-I00).

Corresponding author

César García-Gómez can be contacted at: cesar.garciag@uva.es

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