TY - JOUR
AU - Zelli, Roberto
AB - Summary Multilateral comparison of outcomes drawn from multiple groups pervade the social sciences and measurement of their variability, usually involving functions of respective group location and scale parameters, is of intrinsic interest. However, such approaches frequently mask more fundamental differences that more comprehensive examination of relative group distributional structures reveal. Indeed, in categorical data contexts, location- and scale-based techniques are no longer feasible without artificial and questionable cardinalisation of categories. Here, Gini’s transvariation measure is extended and employed in providing quantitative and visual multilateral comparison tools in discrete, continuous, categorical, univariate, or multivariate settings which are particularly useful in paradigms where cardinal measure is absent. Two applications, one analysing Eurozone cohesion in terms of the convergence or divergence of constituent nations income distributions, the other, drawn from a study of ageing, health, and income inequality in China, exemplify their use in a continuous and categorical data environment. 1. INTRODUCTION Following early concerns about the measurement of aggregate differences (Dalton, 1920; Gini, 1921) and the path-breaking work of Fisher (1932, 1935), multilateral comparisons of grouped outcomes have become ubiquitous in the empirical sciences1 rendering unit free measurement of their collective variation of intrinsic interest. Unit free measures are preferred because they are comparable across different entities. Generally, studies of relative variation within and between groups employ standard summary statistics of location (means and medians) and dispersion (variances and ranges) in various combinations in three basic approaches: range measures, average distance from a central value measures, and average distance between all possible pairs measures. Range (largest less the smallest number) or interquartile range measures divided by a location parameter are examples of the first unit-free approach, the coefficient of variation or Theil’s entropic measures (Theil, 1967; Maasoumi, 1986; Maasoumi et al., 2007) exemplify average difference from the average approaches, and the Gini coefficient (Gini, 1921; Yitzhaki, 1983; Chakravarty, 1988; Lambert and Aronson 1993) is an example of the third approach. Each has its pros and cons. Range measures are easily computed and capture the potential span of differences but fail to reflect the extent of bilateral differences between groups within the interior of the collection—they are not subgroup decomposable so that a subgroups impact on overall variability cannot always be established. The second group, in accounting for the difference from the average of each element, reflects the totality of differences much better and, like the ANOVA technique, they are usually subgroup decomposable. However, Sen (1995) and Yitzhaki (2003) argue that measures of average absolute differences, such as the Gini, capture more of the totality of differences than difference from mean-based measures. Unfortunately, when analysing subgroup impacts, Gini-type measures are not subgroup decomposable (Bourguignon, 1979) except in exceptional circumstances (Mookherjee and Shorrocks, 1982). A common problem with these approaches, highlighted in the contexts of treatment effects and growth and convergence models (Carneiro et al., 2002, 2003; Durlauf and Quah, 2002), is that, in confining analyses to subsets of conditional moments, important information concerning differences in moments beyond those subsets is ignored and can thus be misleading. Somewhat trivially, in a collection of distributions with identical means, difference in means tests have zero power against more general distributional differences such as differences in variances. In essence, employing just means and variances creates a ‘veil of ignorance’ that is only countervailed by comparing subgroup distributions in their entirety across their complete range. Moreover, such analyses are not feasible in ordinal environments encountered, for example, in subjective well-being measurement literature without arbitrary assignment of cardinal scales to ordinal categories. Unfortunately, arbitrary scale assignment is not a solution because of the scale dependency problem (Schroder and Yitzhaki, 2017; Liddell and Kruschke, 2018; Bond and Lang, 2019) and, since objects like the range, coefficient of variation, and Gini coefficients are monotone scale dependent, this issue carries over to inequality measurement. In answer to these concerns, measures of distributional differentness or inequality are introduced here. These measures compare a collection of distributions across their complete range of variation. Specifically, Gini’s bilateral distributional transvariation (Gini, 1916; 1959) is extended to multilateral environments in generating three new general measures, together with their respective asymptotically normal standard errors, which are distributional analogues of the aforementioned three basic measures of variation in collections of numbers. The measures, which focus on relative distributional differences in collections of discrete, continuous, categorical, and potentially multivariate distributions, are in the respective forms of a multilateral transvariation (MGT) statistic, a distributional coefficient of variation (DCV), and a distributional Gini (DisGini) coefficient. All come in population weighted and unweighted forms. MGT is a generalisation to K distributions of Gini’s bilateral transvariation, originally introduced in Anderson et al. (2017), here its sampling distribution is provided. Like its range counterpart, it is simple to compute and provides a measure of the extremes of variation of the collection of distributions but gives no sense of the extent of bilateral distributional differences or aggregate differences from the ‘average’ distribution. DCV, like its coefficient of variation counterpart, measures aggregated differences of distributions from an average distribution and is particularly useful for studying convergence and divergence issues in collections of distributions. Its development prompted a new concept of universal convergence/divergence whereby all groups in the collection are converging/diverging in concert and can be usefully visualised in a radar chart and tests which are provided. DisGini measures the totality of bilateral similarities or differences in a collection of distributions and is the most comprehensive measure of distributional differences. Conceptually, it is based on an extension of the between-group means component of the three-fold subgroup decomposition of the Gini (Bourguignon, 1979). While the Gini coefficient between-group component captures between-group inequalities in terms of differences between subgroup means, DisGini captures between-group dissimilarities in terms of the totality of subgroup distributional differences. Section 2 introduces the three new instruments for assessing multilateral distributional differences. Estimators for these measures along with their distributional properties are derived in Section 3. Section 4 reports a convergence application, addressing the question of increasing commonality in the income distributions of the Eurozone’s constituent nations. A second study, exemplifying the efficacy of the techniques in ordered categorical data contexts, examines the progress of health–income inequalities over the ageing process and is provided in the appendix. Some conclusions are drawn in Section 5. 2. MULTILATERAL TRANSVARIATION 2.1. MGT: generalising Gini’s transvariation measure In Gini's original bilateral transvariation measure (GT), Gini (1916, 1959) provided a measure of the difference between two distributions2 which, for two distributions |$f_i\left(x\right)\!, \, f_j\left(x\right)$| whose support3 is confined to |$\mathbb {R}^{+}$|⁠, can be defined, following Anderson et al. (2017), as follows: $$\begin{eqnarray} \text{GT}_{ij}&=&\frac{1}{2}\int _0^{{\infty }}\left|f_i\left(x\right)-f_j \left(x\right)\right|\mathit {dx} \\ &=&\frac{1}{2}\int _0^{{\infty }}[\text{max}\left(f_i\left(x\right)\!,f_j\left(x \right)\right)-\text{min}\left(f_i\left(x\right)\!,f_j\left(x\right)\right)] \mathit {dx}. \end{eqnarray}$$(2.1) Since |$0 \le \int _0^{{\infty }}|f_i\left(x\right)-f_j\left(x\right)|\mathit { dx} \le 2$|⁠, pre-multiplying by 0.5 yields a statistic that will be 0 when the two distributions are identical and 1 when they have mutually exclusive support.4 Note that, by definition, |$\text{GT}_{ij}=\text{GT}_{ji}$|⁠; furthermore it has a one to one relationship with distributional overlap |$\text{OV}_{ij}$| measuring the extent of commonality between the two distributions (Anderson et al., 2012) which is given by: $$\begin{eqnarray} \text{OV}_{ij}=\int _0^{{\infty }}\text{min}(f_i\left(x\right)\!,f_j\left(x \right))\mathit {dx}. \end{eqnarray}$$(2.2) Essentially GT = 1−OV. Generalising equation (2.1) to K distributions indexed |$k=1,\ldots ,K$|⁠, suggests contemplating a multilateral Gini transvariation measure (MGT), defined as follows: $$\begin{eqnarray} \text{MGT}&=&\frac{1}{K}\int _{0}^{{\infty }}[\text{max}\left( f_{1}\left( x\right)\! ,f_{2}\left( x\right)\! ,\ldots ,f_{K}\left( x\right) \right) \\ && - \ \text{min}\left( f_{1}\left( x\right)\! ,f_{2}\left( x\right)\! ,\ldots ,f_{K}\left( x\right) \right) ]\mathit {dx}. \end{eqnarray}$$(2.3) As in the bilateral comparison, when the distributions have mutually exclusive support then MGT = 1, when the distributions are identical then MGT = 0. A weighted version of MGT, MGT-W is also possible, and has the form ) $$\begin{eqnarray} \text{MGT-W} &=& \int _{0}^{{\infty }}[\text{max}\left( w_{1}f_{1}\left( x\right)\! ,w_{2}f_{2}\left( x\right)\! ,...,w_{K}f_{K}\left( x\right) \right) \\ && - \ \text{min}\left( w_{1}f_{1}\left( x\right)\! ,w_{2}f_{2}\left( x\right)\! ,...,w_{K}f_{K}\left( x\right) \right) ]\mathit {dx,} \end{eqnarray}$$(2.4) where |$w_{k}$| are possible weights associated to the distributions |$f_{k}$|⁠, |$k=1,\ldots ,K.$| When the K distributions are regarded as subgroups of an overall distribution, |$w_{k}$| are the proportions associated with each density function. One problem with the multilateral transvariation measure is its maximum–minimum nature. Like the range statistic for a collection of numbers which does not reflect differences in objects in the mid range, the MGT does not reflect the many bilateral functional differences and similarities camouflaged by just considering extreme density values. Indeed, it is in essence the distributional analogue of the relative range measure of a collection of numbers wherein the relative locations of interior and low weight members have little or no impact on its value. An alternative which is in effect an aggregation of all distributional differences from the average distribution or what will be referred to as the DCV is introduced in the next section. 2.2. DCV: a distributional coefficient of variation The collection of K subgroups indexed |$k=1,\ldots ,K$| with respective distributions |$f_{k}(x)$| may be considered in the context of individual distributions being components within a mixture |$f(x)$| representing the overall population distribution: $$\begin{eqnarray} f(x)=\sum _{k=1}^Kw_kf_k(x), \sum _{k=1}^K w_k=1 \ \text{and} \ w_k{\,\ge\, }0 \ \text{for all} \ k , \end{eqnarray}$$(2.5) where |$w_k$| are weights reflecting the importance of the component within the population. So, for example, |$f(x)$| may refer to a societal income distribution with |$f_k(x)$| being the income distribution of the k-th constituency and |$w_k$| its relative population size. Alternatively, from a representative agent or treatment effect perspective, the distributions describing outcomes of particular groups could be compared directly, without reference to their relative importance in the collection, in which case |$w_k$| would be set to |$1/K$| for all k. Indeed, from a policy application perspective, there is no reason why |$f(x)$| should not be defined for policy purposes as some ‘target’ distribution that constituencies should aspire to so that DCV provides a measure of the extent to which the policy has not been achieved. |$\text{OV}_{ko}$|⁠, the distributional overlap between the k’th subgroup distribution and the overall mixture is such that: $$\begin{eqnarray} \text{OV}_{ko} = \int _{0}^ {\infty } \text{min}\left(f_{k} (x), f(x) \right) dx. \end{eqnarray}$$(2.6) The corresponding subgroup/overall transvariation is related to the overlap measure as follows: GT|$_{ko}=1-$|OV|$_{ko}.$| Then DCV, the weighted average of subgroup–overall distribution transvariations, may then be written as: $$\begin{eqnarray} \text{DCV}=\frac{1}{(1-\sum _{k=1}^{K}w_{k}^{2})}\sum _{k=1}^{K}w_{k}GT_{ko}= \frac{1}{(1-\sum _{k=1}^{K}w_{k}^{2})}\sum _{k=1}^{K}w_{k}(1-\text{OV}_{ko}). \end{eqnarray}$$(2.7) Note that when subgroup distributions are identical they will be identical to their weighted sum so that |$\text{GT}_{ko}=0$| for all k and DCV = 0. When the subgroups have mutually exclusive support |$\text{GT}_{ko}=1-w_{k}$| so that DCV=1. As with the Sen (1995) and Yitzhaki (2003) critiques of mean deviation measures, DCV still does not reflect the full panoply of distributional differences between groups. However, a ‘distributional’ Gini coefficient will. 2.3. DisGini: the ‘distributional’ Gini coefficient To fully explore distributional differences consider instead: $$\begin{eqnarray} \text{DisGini}=\frac{1}{\varphi }\sum _{i=1}^K \sum _{j=1}^K 0.5\int _0^{\infty } w_i w_j |f_i(x)-f_j(x)| \mathit {dx} =\frac{1}{\varphi }\sum _{i=1}^K \sum _{j=1}^K w_i w_j \text{GT}_{ij}, \end{eqnarray}$$(2.8) where |$\varphi$| is a scaling parameter. Note the term |$\int _0^{\infty } w_i w_j |f_i(x)-f_j(x)|\mathit {dx}$| may be written as ‘|$w_i w_j 2 \text{GT} _{i,j}$|’ which is twice Gini’s transvariation of subdistributions |$f_{i}(x)$| and |$f_{j}(x)$|⁠, multiplied by the product of the respective population shares. Given the relationship (2.2) between GT and the overlap measure, OV, (2.8) may be written as: $$\begin{eqnarray} \text{DisGini}=\frac{1}{\varphi }\sum _{i=1}^K \sum _{j=1}^K w_i w_j(1-\text{OV}_{ij}). \end{eqnarray}$$ Which, letting c be a K element column vector of ones, may be written in matrix form: $$\begin{eqnarray} \frac{1}{\varphi }c^{\prime }\left[ {\begin{array}{*{10}c}{\begin{array}{*{10}c}0 & w_1w_2(1-\mathit {OV}_{12}) \\ w_2w_1(1-\mathit {OV}_{21}) & 0 \end{array}} & {\begin{array}{*{10}c}\dots & w_1w_K(1-\mathit {OV}_{1K}) \\ {\dots } & w_2w_K(1-\mathit {OV}_{2K}) \end{array}} \\ {\begin{array}{*{10}c}\vdots & {\vdots } \\ w_Kw_1(1-\mathit {OV}_{\mathit {K1}}) & w_Kw_2(1-\mathit {OV}_{\mathit {K2}}) \end{array}} & {\begin{array}{*{10}c}\ddots & {\vdots } \\ {\dots } & 0 \end{array}} \end{array}} \right] c. \end{eqnarray}$$(2.9) Consider a typical element |$w_i w_j(1-\text{OV}_{ij})$|⁠, when |$i=j$| the element will be zero, also when |$f_{i}(x)=f_{j}(x)$| for all x (i.e., subgroups i and j have identical distributions), the term will be 0. It follows that when all subgroups have identical distributions, expression (2.9) will be 0 since all of the elements are nonnegative this will constitute a lower bound for DisGini. Now consider the situation where all of the respective subgroup income distributions have mutually exclusive support, i.e., the subgroups are completely segmented so that for all |$i\ne j$| and a given x, |$f_{i}(x)\ge 0 \Rightarrow f_{j}(x) = 0$| and |$f_{j}(x)\ge 0 \Rightarrow f_{i}(x) = 0$|⁠. This corresponds to the mixture distribution situation where there is no distributional overlap between any constituency pairing, thus Gini’s transvariation would be at a maximum value of 1. In this case (2.8) may be written: $$\begin{eqnarray} \frac{1}{\varphi }c^{\prime }\left[ {\begin{array}{*{10}c}{\begin{array}{*{10}c}0 & w_1w_2 \\ w_2w_1 & 0 \end{array}} & {\begin{array}{*{10}c}\dots & w_1w_K \\ {\dots } & w_2w_K \end{array}} \\ {\begin{array}{*{10}c}\vdots & {\vdots } \\ w_Kw_1 & w_Kw_2 \end{array}} & {\begin{array}{*{10}c}\ddots & {\vdots } \\ {\dots } & 0 \end{array}} \end{array}} \right] c =\frac{1}{\varphi } \sum _{k=1}^K w_k(1-w_k) =\frac{1-\sum _{k=1}^K w_k^2}{\varphi }. \end{eqnarray}$$ If the scaling parameter |$\varphi$| is set to |$(1-\sum _{k=1}^K w_k^2)$| then DisGini will always fall in the interval [0,1] and be equal to 1 when there is complete distributional inequality in terms of complete segmentation of the constituency distributions. It follows that DisGini may finally be written as: $$\begin{eqnarray} \text{DisGini}&=&\frac{1}{(1-\sum _{k=1}^K w_k^2)}\sum _{i=1}^K \sum _{j=1}^K w_i w_j(1-\text{OV}_{ij}) \\ &=& \frac{1}{(1-\sum _{k=1}^K w_k^2)}\sum _{i=1}^K \sum _{j=1}^{K} w_i w_j \text{GT}_{ij}. \end{eqnarray}$$(2.10) If comparison of the distributions without subgroup weighting is desired, as in the aforementioned representative agent type scenarios, simply set |$w_{i}= \frac{1}{K}$| for all |$i=1,\ldots ,K$|⁠. By noting that the transvariation and overlap of two multivariate distributions is given by: $$\begin{eqnarray} \int \sum |f(x,y) - g(x,y)|\mathit {dx} \, \, \text{and}\, \, \int \sum \text{min}(f(x,y) - g(x,y))\mathit {dx} , \end{eqnarray}$$ respectively, where integration is over all continuous variables x and summation is over all discrete variables y, the foregoing formulae are readily extended to multivariate situations. Furthermore, by replacing |$f_i(x)$| with |$F_i^h(x)$| where |$F_i^h (x)=\int _0^{x}F_i^{(h-1)}(z)dz$| in (2.3), (2.7), or (2.8) and adjusting the normalising parameter accordingly, multilateral variation of higher order integrals of distribution functions could be contemplated reflecting the classic stochastic dominance criteria for more restrictive well-being structures (see Anderson et al., 2020). All of which are matters for future research. These indices provide a complete ordering of collections of distributions with respect to their differentness, as such they can be shown to satisfy some popular axioms in the inequality literature (Sen, 1995). When applied to the groups as subjects anonymity, scale and translation invariance, normalisation, and replication invariance axioms are all satisfied by these indices. When subdistributions are posited to be the atomistic equivalents of the subdistributions employed in Duclos et al. (2004) and subjected to the same transformations, they comply with the polarisation axioms posed therein. It should also be noted that although the Gini coefficient has problems with negative values (see Manero, 2017), the discussion was confined to distributions defined on the positive orthant. However, the measures proposed here are not subject to this difficulty and are well defined on all support types. Finally, it is of interest to understand how the DisGini coefficient is affected by the expansion of the number of groups under consideration, Appendix A demonstrates that DisGini will increase or diminish as it is exceeded by or exceeds the weighted sum of the new group’s transvariations with respect to the existing groups in the analysis. 3. ESTIMATION AND DISTRIBUTION THEORY 3.1. Estimation and standard errors of MGT Nonparametric estimation of MGT facilitates analysis of the collection of distributions over their full range revealing the extent of their similarity and differentness without reliance on the limited purview of summary statistics or visual perceptions. In the case of discrete and categorical variables, estimation of category membership probabilities and their sampling distributions is straightforward following Rao (1973). Suppose there are C categories |$\Gamma _{c}$| indexed |$c=1,\ldots ,C$| with a C vector of category membership probabilities p with typical element |$p_{c}$| and let x be a T vector of independent observations with typical element |$x_{i}$| so that |$p_{c}=\Pr \left( x_{i}\in \Gamma _{c}\right)$|⁠. Then, |$\hat{p}_{c}$|⁠, the estimate of |$p_{c}$|⁠, may be obtained by letting |$z_{i,c}=1\, \, \text{when}\, \, x_{i}\in \Gamma _{c}\, \, \text{and}\, \, 0\, \, \text{otherwise}$|⁠, so |$\widehat{p}_{c,k}=\frac{1}{T} \sum _{i=1}^{T}z_{i,c}.$| In this case the vector |$\hat{p}$| is asymptotically normal with large sample variance equal to |$1/T$| times |$V=\mathrm{diag} (p_{1},\ldots ,p_{C})-pp^{^{\intercal }}.$| We then let $$\begin{eqnarray} \widehat{\theta }_{\mathit {KT}}=\frac{1}{K}\sum _{c=1}^{C}\left( \max \left\lbrace \hat{p}_{c,1},\hat{p}_{c,2},\ldots ,\hat{p}_{c,K}\right\rbrace -\min \left\lbrace \hat{ p}_{c,1},\hat{p}_{c,2},\ldots ,\hat{p}_{c,K}\right\rbrace \right)\! . \end{eqnarray}$$ We present now the estimator for the case where |$X_{k}$| are continuously distributed with Lebesgue density |$f_{k},\, k=1,\ldots ,K,\,$| with common support |$\mathbb {R}$|⁠. Suppose that we observe independent random samples from the |$k^{th}$| population |$X_{kt},\, t=1,\ldots ,T_{k}$|⁠. We define the kernel estimates: $$\begin{eqnarray} \widehat{f_{k}}\left( x\right) =\frac{1}{T_{k}}\sum _{h=1}^{T_{k}}\mathbb {K} _{b}\left( x-X_{k,h}\right)\! ,\quad k=1,\ldots ,K\!, \end{eqnarray}$$(3.1) where |$\mathbb {K}$| is a (potentially d dimensioned multivariate) kernel with |$\mathbb {K}_{b}(.)=\mathbb {K}(./b)/b^{d},$| where b is a positive bandwidth sequence. We then estimate the unweighted multilateral transvariation index |$\theta _{K}=$|MGT defined above, $$\begin{eqnarray} \widehat{\theta }_{\mathit {KT}}= \frac{1}{K}\left( \int \max \left\lbrace \widehat{ f_{1}}\left( x\right)\! ,\widehat{f_{2}}\left( x\right)\! ,\ldots ,\widehat{f_{K} }\left( x\right) \right\rbrace dx -\ \int \min \left( \widehat{f_{1}}\left( x\right)\! ,\widehat{f_{2}}\left( x\right)\! ,\ldots ,\widehat{f_{K}}\left( x\right) \right) dx\right)\! . \end{eqnarray}$$ The integral is computed by numerical quadrature routines. The theory for |$\widehat{\theta }_{\mathit {KT}}$| follows closely the analysis in Anderson et al. (2012). We present the theory for the case where the contact sets $$\begin{eqnarray} C_{i,j}=\left\lbrace x\in \mathbb {R}^{d}:f_{i}(x)=f_{j}(x)\gt 0\right\rbrace\! , \end{eqnarray}$$ all have Lebesgue measure zero. In practice, this is perhaps the most useful case. The main case where the contact set is not of measure zero is the case where the densities are equal, which is a hypothesis of interest; however, in that case there are other tests available. For simplicity of presentation we suppose that |$T_{k}=T$| for |$k=1,\ldots ,K.$| Let |$\lambda =(\lambda _{1},\ldots ,\lambda _{d})^{\top }$| denote a vector of nonnegative integer constants. For such vector, we define |$\left|\lambda \right|=\sum _{i=1}^{d}\lambda _{i}$| and, for any function |$h(x): \, \mathbb {R}^{d}\rightarrow \mathbb {R},\, D^{\lambda }h(x)=\partial ^{|\lambda |}/(\partial x_{1}^{\lambda _{1}}\cdots \partial x_{d}^{\lambda _{d}})(h(x)),$| where |$x=(x_{1},\ldots ,x_{d})^{\top }$| and |$x^{\lambda }=\prod \limits _{j=1}^{d}x_{j}^{\lambda _{j}}.$| Assumptions (A1) K is a sth order kernel function having support in the closed ball of radius |$1/2$| centred at zero, symmetric around zero, integrates to 1, and s -times continuously differentiable on the interior of its support, where s is an integer that satisfies |$s\,\gt\, d.$| (A2) The densities |$f_{k},\, k=1,\ldots ,K$| are strictly positive, bounded and absolutely continuous with respect to Lebesgue measure and s-times continuously differentiable with uniformly bounded derivatives. For all |$\lambda \in \mathbb {N}^{d}$| with |$0\le |\lambda |\le s,$||$\int |D^{\lambda }f_{k}(x)|dx\lt \infty ,\, k=1,\ldots ,K.$| (A3) The bandwidth satisfies: (i) |$nb^{2s}\rightarrow 0,$|(ii) |$nb^{2d}\rightarrow \infty$|⁠, and (iii) |$nb^{d}/\left( \log n\right) \rightarrow \infty .$| (A4) |$\lbrace X_{ki}\!:i\ge 1,\, k=1,\ldots ,K\rbrace$| are i.i.d. with support |$\mathbb {R}^{d}\times \cdots \times \mathbb {R}^{d}.$| Define the sets |$CK_{i,\ast }$| and |$CK^{i,\ast }$|⁠: $$\begin{eqnarray} CK_{i,\ast } &=&\left\lbrace x\!:f_{i}\left( x\right) \lt f_{j}\left( x\right)\! ,\text{ for all }j=1,...,K,j\ne i\right\rbrace \\ CK^{i,\ast } &=&\left\lbrace x\!:f_{i}\left( x\right) \gt f_{j}\left( x\right)\! ,\text{ for all }j=1,...,K,j\ne i\right\rbrace\! . \end{eqnarray}$$ Let |$p_{kU}=\Pr \left( X_{k}\ {\in }\ \mathit {CK}^{i,\ast }\right)$| and |$p_{kL}=\Pr \left( X_{k}\ {\in }\ \mathit {CK}_{i,\ast }\right)\! ,$| and note that |$CK_{i,\ast }{\cap }CK^{i,\ast }={\emptyset }$| so that |$p_{kUL}=\Pr \left( X_{k}\ {\in }\ \mathit {CK}^{i,\ast }{\cap }CK_{i,\ast }\right) =0,$| and define the positive scalar $$\begin{eqnarray} v_{KT}=\frac{1}{K^{2}}\sum _{k=1}^{K}\lbrace p_{kU}\left( 1-p_{kU}\right) +p_{kL}\left( 1-p_{kL}\right) +2p_{kU}p_{kL}\rbrace . \end{eqnarray}$$ Theorem 3.1 Suppose that Assumptions A1–A4 hold. Then, we have: $$\begin{eqnarray} \sqrt{T}\left( \widehat{\theta }_{KT}-\theta _{KT}\right) \Longrightarrow N(0,v_{KT}). \end{eqnarray}$$ The limiting variance can be consistently estimated by $$\begin{eqnarray} \widehat{v}_{KT}=\frac{1}{K^{2}}\frac{1}{T}\sum _{k=1}^{K}\lbrace \widehat{p_{kU}} \left( 1-\widehat{p_{kU}}\right) +\widehat{p_{kL}}\left( 1-\widehat{p_{kL}} \right) +2\widehat{p_{kU}}\widehat{p_{kL}}\rbrace . \end{eqnarray}$$ We may construct a |$1-\alpha$| asymptotic coverage confidence interval for |$\theta _{KT}$| as |$\widehat{\theta }_{KT}\pm z_{\alpha /2}\sqrt{\widehat{v} _{KT}/T}.$| The distributional properties of MGT-W can be derived as above by working with |$w_{k}\widehat{f_{k}}\left( x\right)$| in place of |$\widehat{f_{k}} \left( x\right)$| and modifying |$g_{KT}\left( K\right)$| accordingly as in (2.4). 3.2. Estimation and standard error of DCV As the weighted average of K subgroup distribution–overall distribution transvariations, the DCV can be estimated as: $$\begin{eqnarray} \widehat{\theta }_{DCV}=\frac{1}{(1-\sum _{k=1}^K w_k^2)}\sum _{k=1}^{K}w_{k} \left\lbrace 1-\int _{a}^{b}\min \left( \widehat{f_{k}}\left( x\right) ,\widehat{f}\left( x\right) \right) \mathit {dx}\right\rbrace , \end{eqnarray}$$(3.2) where |$\widehat{f_{k}}\left( x\right)$| are kernel estimates of |$f_{k}(x),\, \, \, k=1,\ldots ,K$|⁠, |$w_{k}$|’s are known weights, and |$\widehat{f}\left( x\right)$| is the kernel estimate of the overall distribution |$f\left(x\right)=\sum _{k=1}^Kw_kf_k(x), \sum _{k=1}^K w_k=1 \ \text{and} \ w_k{\ge }0 \ \text{for all} \ k$| (or it could be some prespecified overall target distribution). Maintaining the assumptions of the previous MGT analysis, in this case define the sets |$CK_{k,O }$| and |$CK^{k,O}$|⁠: $$\begin{eqnarray} CK_{k,O } &=&\left\lbrace x: f_{k}\left( x\right) \lt f\left( x\right) \right\rbrace\! , \, k=1,\ldots ,K \\ CK^{k,O } &=&\left\lbrace x: f_{k}\left( x\right) \gt f\left( x\right) \right\rbrace\! , \, k=1,\ldots ,K\! . \end{eqnarray}$$ Let |$p_{kU}=\Pr \left( X_{k}\ {\in }\ \mathit {CK}^{k,O }\right)$| and |$p_{kL}=\Pr \left( X_{k}\ {\in }\ \mathit {CK}_{k,O }\right)$|⁠, and define the positive scalar $$\begin{eqnarray} v_{DCV}=\frac{1}{(1-\sum _{k=1}^K w_k^2)^2} \sum _{k=1}^{K} w_k^2 \left(p_{kU}\left( 1-p_{kU}\right) + p_{kL}\left( 1-p_{kL}\right) +2p_{kU}p_{kL}\right)\!. \end{eqnarray}$$ Then in a similar fashion to Theorem 3.1 above, it may be shown that: $$\begin{eqnarray} \sqrt{T}\left( \widehat{\theta }_{DCV}-\theta _{DCV}\right) \Longrightarrow N(0,v_{DCV}). \end{eqnarray}$$ The limiting variance can be estimated and the asymptotic coverage confidence interval can be computed as above. The representative agent (equally weighted) case can be considered by setting |$w_k=1/K \ \text{for all} \ k$|⁠. 3.3. Estimation and standard errors of DisGini We estimate the Distributional Gini Index (DisGini or DG) over K distributions by: $$\begin{eqnarray} \widehat{\theta }_{DG}=\frac{1}{(1-\sum _{k=1}^K w_k^2)}\sum _{i=1}^{K}\sum _{j=1}^{K}w_{i}w_{j} \left\lbrace 1-\int _{a}^{b}\min \left( \widehat{f_{i}}\left( x\right)\! ,\widehat{ f_{j}}\left( x\right) \right) \mathit {dx}\right\rbrace\! , \end{eqnarray}$$(3.3) where |$\widehat{f_{k}}\left( x\right)$| are kernel estimates of |$f_{k}(x),\, \, \, k=1,\ldots ,K$|⁠, and the |$w_{k}$|’s are known weights. Define the pairwise sets |$C_{i,j}\, \, i,j=1,..,K\, \, i{\ne }j$| as: $$\begin{eqnarray} C_{i,j}=\lbrace x:f_{i}\left( x\right) \lt f_{j}\left( x\right) \rbrace , \end{eqnarray}$$ and let $$\begin{eqnarray} v_{\mathit {DG}} &=& \sum _{i=1}^{K}\sum _{j\gt i}w_{i}^{2}w_{j}^{2}\left( \Pr \left( X_{i}{\in }C_{\mathit {ij}}\right) -\Pr \left( X_{i}{\in }C_{\mathit {ij }}\right) ^{2}\right) \\ && + \ 2\sum _{i=1}^{K}\sum _{j\gt i}\sum _{k\gt j\gt i}w_{i}^{2}w_{j}w_{k}\left( \Pr \left( X_{i}{\in }C_{\mathit {ij}}{\cap }C_{\mathit {ik}}\right) -\Pr \left( X_{i}{ \in }C_{\mathit {ij}}\right) \Pr \left( X_{i}{\in }C_{\mathit {ik}}\right) \right)\! . \end{eqnarray}$$ Theorem 3.2 Suppose that Assumptions A1–A4 hold. Then, we have: $$\begin{eqnarray} \sqrt{T}\left( \widehat{\theta }_{DG}-\theta _{DG}\right) \Longrightarrow N(0,v_{DG}). \end{eqnarray}$$ The limiting variance may be consistently estimated by replacing the population quantities by their sample analogues. 4. AN EMPIRICAL EXAMPLE 4.1. Household income distributions in the Eurozone The efficacy of the new techniques is first illustrated in a study of the twenty-first-century evolution of household income inequality in the Eurozone. Milanovic (2011) noted that growing divergence between constituencies within a federation can be a catalyst for the deterioration of its cohesion and the recent rise of economic nationalism in Europe has given cause for concern regarding the European Unions coherence (Krastev, 2012; Webber, 2018; Lindberg, 2019). Formed to promote commonality of well-being among its constituents, there is interest in seeing whether the European nations household income distributions are converging. The growth and convergence literature suggests that variation of average incomes across constituencies is of interest since it speaks directly to the question of whether the distribution of income across economies is becoming more or less equitable (Quah, 1993). However, deterioration of cohesiveness has much to do with the extent to which economic well-being differs across constituencies, the sense in which such differences are perceived by agents within those constituencies and the relative importance of those constituencies. In this context, cohesiveness is more than just a matter of whether or not constituencies have similar average incomes, it is more a matter of whether or not they have common income distributions. When member nations are also unequal with relatively similar income levels and distributions, there is a commonality of the situation among member constituents which promotes cohesion, whereas a more divisive and alienated situation arises when such inequalities and income levels are not so equally shared in a more segmented society. The cohesiveness of a union of economies is therefore related to the extent to which its respective nation income distributions are segmenting or converging. The new measures are employed to address these distinctions within nations in the Eurozone. Viewed as an entity, the overall Eurozone household income distribution |$f\left(x\right)$| is a mixture of the household income distributions |$f_k\left(x\right)k=1,...,K$|of its K constituent nations where the weights |$w_k$| correspond to relative population sizes (see equation 2.5). Stochastic processes are frequently used to rationalise distributional structures and Gibrat’s Law of Proportional Effects and some of its modifications (Gabaix, 1999; Reed, 2001) have been foundational in providing a theoretical rationale for expecting increasing income inequality. The law posits that household incomes in subgroup k follow a stochastic process which, in its simplest form in period t, has the form: $$\begin{eqnarray} x_{k,t}=\left(1+\delta _{k,t}\right)x_{k,t-1} , \end{eqnarray}$$ where |$\delta _{k,t}$| is a random variable with mean |$\delta _{k}$| (which is small relative to one in absolute value) and variance |$\sigma _{k}^{2}$|⁠. The law predicts that, given a starting value |$x_{0}$| and letting |$X=\text{ ln}(x)$|⁠, after T periods |$X_{kT}$| will have a mean equal to |$X_{0}+T\left( \delta _{k}+0.5\sigma _{k}^{2}\right)$| and variance equal to |$T\sigma _{k}^{2}$|⁠, respectively, i.e., log income variation that grows through time. Following Modigliani and Brumberg (1954), classical economic models of income (Hall, 1978) use this idea to predict increasingly unequal income distributions (Battistin et al., 2009; Blundell and Preston, 1998; Browning and Lusardi, 1996). When applied to the |$k=1,\ldots ,K$| constituent societies in the Eurozone, clearly different configurations of pairs (⁠|$\delta _{k},\sigma _{k}^{2})\, \, \, \text{for}\, \, k=1,\ldots ,K$| will yield collections of distributions that could be converging or diverging, segmenting or increasingly overlapping, becoming more or less equal in distribution. Multilateral comparisons, as presented in equations (2.4), (2.7), and (2.8), can be estimated using |$w_k=1/K$| or |$w_k$| proportional to the population size of nation k. The first case resembles the unweighted inequality between nations in which each country is taken as the unit of observation, disregarding its size. This unweighted version of the measures can be construed as a representative agent model, recording the juxtaposition of nation income distributions directly without respect to their relative importance or impact in the overall income distribution. In the second case each country is weighted by its population. The unit of observation is then a person instead of a country. This weighted version gives insight into distributional differences of the Eurozone as an entity, with small populations given low weight and large populations high weight. The data source used is the European Union Survey on Income and Living Conditions (EU-SILC).5 To analyse the evolution of the Euro area income distribution over time, four temporally equi-spaced waves, 2006, 2009, 2012, and 2015 were chosen. Since data for Malta are only available from the 2008 wave, this country is excluded from analysis leaving 18 Eurozone countries. Income is the total household net disposable annual income (in thousands Euro) obtained by aggregation of all income sources from all household members net of direct taxes and social contributions.6 Assuming consumption economies of scale in cohabitation and incomes are age and size adjusted using the modified OECD equivalence scale. Given significant disparities in the cost of living between countries, the PPP (purchasing power parity) index for the household final consumption expenditure is used to adjust household incomes. As an entity, the Eurozone had overall household income Gini coefficients of 0.305, 0.313, 0.317, and 0.335 for the years 2006, 2009, 2012, and 2015 respectively, suggesting ever-increasing household income disparities in the area over the period. In light of concerns regarding European disintegration, questions arise as to the extent to which such inequalities are equally shared across its various nations, which prompts an investigation into the juxtaposition of the income distributions of the Eurozone’s constituent nations.7 Table 1 reports the unweighted MGT, the DCV, and the DisGini. The income densities |$f_k(x)$| are kernel estimated using the Sheather and Jones (1991) bandwidth as smoothing parameter and take into account the weighting scheme of the EU-SILC survey. The measures can yield insights into the progress of distributional inequalities over the era, tending towards 0 as distributions converge and tending towards 1 as they segment or diverge. All the unweighted indices record a decline over the whole period with respect to 2006. Thus, in a representative agent view of the world, similar to that pursued in the sigma convergence literature wherein nations are equally weighted (Quah, 1993), the multilateral results present significant evidence of nation income distribution convergence. Table 1. Unweighted MGT, DCV, and DisGini coefficients: nation group analysis. Year . MGT . DCV . DisGini . 2006 0.135 0.270 0.386 (0.004) (0.001) (0.003) 2009 0.114 0.226 0.321 (0.004) (0.001) (0.003) 2012 0.106 0.234 0.326 (0.004) (0.001) (0.003) 2015 0.107 0.254 0.341 (0.004) (0.001) (0.003) Year . MGT . DCV . DisGini . 2006 0.135 0.270 0.386 (0.004) (0.001) (0.003) 2009 0.114 0.226 0.321 (0.004) (0.001) (0.003) 2012 0.106 0.234 0.326 (0.004) (0.001) (0.003) 2015 0.107 0.254 0.341 (0.004) (0.001) (0.003) Note: Asymptotic standard errors are in brackets. Open in new tab Table 1. Unweighted MGT, DCV, and DisGini coefficients: nation group analysis. Year . MGT . DCV . DisGini . 2006 0.135 0.270 0.386 (0.004) (0.001) (0.003) 2009 0.114 0.226 0.321 (0.004) (0.001) (0.003) 2012 0.106 0.234 0.326 (0.004) (0.001) (0.003) 2015 0.107 0.254 0.341 (0.004) (0.001) (0.003) Year . MGT . DCV . DisGini . 2006 0.135 0.270 0.386 (0.004) (0.001) (0.003) 2009 0.114 0.226 0.321 (0.004) (0.001) (0.003) 2012 0.106 0.234 0.326 (0.004) (0.001) (0.003) 2015 0.107 0.254 0.341 (0.004) (0.001) (0.003) Note: Asymptotic standard errors are in brackets. Open in new tab Table 2 reports the weighted multilateral Gini transvariation (MGT-W), the weighted distributional coefficient of variation (DCV-W), and the weighted distributional Gini coefficient (DisGini-W). Looking at the patterns under the population-weighted version of the statistics, quite different stories emerge. The population-weighted indices, after a slight dip in 2009, show a significant increase, indicating increasing distributional divergence in terms of increasingly segmented nations. Table 2. Population-weighted MGT, DCV, and DisGini coefficients: nation group analysis. Year . MGT-W . DCV-W . DisGini-W . 2006 0.291 0.107 0.237 (0.005) (0.002) (0.003) 2009 0.288 0.103 0.223 (0.005) (0.002) (0.003) 2012 0.323 0.135 0.282 (0.005) (0.002) (0.003) 2015 0.349 0.173 0.361 (0.005) (0.002) (0.003) Year . MGT-W . DCV-W . DisGini-W . 2006 0.291 0.107 0.237 (0.005) (0.002) (0.003) 2009 0.288 0.103 0.223 (0.005) (0.002) (0.003) 2012 0.323 0.135 0.282 (0.005) (0.002) (0.003) 2015 0.349 0.173 0.361 (0.005) (0.002) (0.003) Note: Asymptotic standard errors are in brackets. Open in new tab Table 2. Population-weighted MGT, DCV, and DisGini coefficients: nation group analysis. Year . MGT-W . DCV-W . DisGini-W . 2006 0.291 0.107 0.237 (0.005) (0.002) (0.003) 2009 0.288 0.103 0.223 (0.005) (0.002) (0.003) 2012 0.323 0.135 0.282 (0.005) (0.002) (0.003) 2015 0.349 0.173 0.361 (0.005) (0.002) (0.003) Year . MGT-W . DCV-W . DisGini-W . 2006 0.291 0.107 0.237 (0.005) (0.002) (0.003) 2009 0.288 0.103 0.223 (0.005) (0.002) (0.003) 2012 0.323 0.135 0.282 (0.005) (0.002) (0.003) 2015 0.349 0.173 0.361 (0.005) (0.002) (0.003) Note: Asymptotic standard errors are in brackets. Open in new tab Taken together, the weighted and unweighted versions of the statistics reveal that lesser populated nations of the Eurozone are exhibiting a convergence pattern whereas nations with larger populations appear to be segmenting. A further insight on the extent to which each country is converging to, or diverging from, the Eurozone norm is given by |$\text{GT}_{ko}$|⁠, |$k=1,...\, K\!,$|the bilateral transvariations between each country k and the overall Eurozone distribution. These magnitudes can be visualised in a radar chart whose spokes are the respective country/overall distribution transvariations. Figure 1 reports the corresponding radar chart, a decomposed distributional coefficient of variation as it were. Figure 1. Open in new tabDownload slide Radar chart of bilateral transvariation of each country with respect to Eurozone. The centre of the wheel corresponds complete overlapping with the Eurozone distribution. Moving to the periphery reflects less commonality with the Eurozone. Countries are clockwise ordered starting with the largest positive difference between 2006 and 2015 (indicating convergence) and ending with the largest negative difference (indicating divergence). Figure 1. Open in new tabDownload slide Radar chart of bilateral transvariation of each country with respect to Eurozone. The centre of the wheel corresponds complete overlapping with the Eurozone distribution. Moving to the periphery reflects less commonality with the Eurozone. Countries are clockwise ordered starting with the largest positive difference between 2006 and 2015 (indicating convergence) and ending with the largest negative difference (indicating divergence). The centre of the chart corresponds to zero transvariation where all subgroups have identical distributions. The closer a point is on a nation’s spoke to the periphery, the higher is the transvariation of that nation’s income distribution with respect to the whole Eurozone distribution (a value equal to 1 means complete segmentation, i.e., the two distribution are far apart). The points have been colour-coded by year, so that intuitively nations with green dots (year 2015) nearer the centre than black dots (year 2006) are converging to the Eurozone distribution over the observation period, whereas nations with green dots outside of the black dots are diverging from the Eurozone distribution. The bilateral nation–overall transvariations range from 0.03 for Italy to 0.68 for Slovakia in the year 2006. The pattern of this bilateral index shows a process of convergence towards the Euro area distribution for eastern European countries (notably low population countries) and significant divergence from the Eurozone distribution for Spain, Finland, France, and Greece. Figures in the appendices show the evolution of the income distributions of constituent nations and their overlapping with respect to the Eurozone distribution in 2006 and in 2015. Summing up, what emerges is a collection of distributions that result in a Eurozone with an increasingly unequal overall income distribution comprised of an increasingly similar (i.e., convergent) collection of unweighted distributions that, when population weighted, become divergent as a collection. 5. CONCLUSIONS When comparing collections of groups, simple first and second order moment multilateral comparisons can overlook substantive differences between groups that a more comprehensive multilateral distributional comparison can reveal. Here, some new tools for the multilateral comparison of many distributions in univariate or multivariate, discrete and continuous, weighted and unweighted environments have been introduced. Based on extensions of Gini’s transvariation measure, new multilateral transvariation measures and more comprehensive Gini-like distributional difference measures, together with their asymptotic distributions, have been developed, namely the multilateral Gini transvariation (MGT), the distributional coefficient of variation (DCV), and the distributional Gini coefficient (DisGini). The DCV is a scaled weighted sum of subgroup versus overall distribution transvariations, the magnitude of which can be represented as a polygon within a radar chart which in turn has prompted definition of the notion of comprehensive inequality reduction (increase), the consequence of all subgroups converging to (diverging from) the overall distribution. Assessing distributional differences in categorical—noncardinal environments is particularly challenging and these techniques have been shown to overcome these challenges in these situations. The measures have been exemplified in applications which study national household income distributions in the Eurozone in the twenty-first century and income and health inequalities and the ageing process in China. Table D1. Distributional Gini coefficients. Age group analysis when Dibao recipients are separately identified and when they are not. Age groups . DisGini . DisGini . . (Dibao recipients separately identified) . (Dibao recipients not identified) . 45–50 0.3645 0.1085 (0.0015) (0.0014) 50–60 0.2625 0.0945 (0.0014) (0.0013) 60–70 0.3145 0.1420 (0.0014) (0.0015) |$\gt $|70 0.3943 0.1552 (0.0014) (0.0015) Age groups . DisGini . DisGini . . (Dibao recipients separately identified) . (Dibao recipients not identified) . 45–50 0.3645 0.1085 (0.0015) (0.0014) 50–60 0.2625 0.0945 (0.0014) (0.0013) 60–70 0.3145 0.1420 (0.0014) (0.0015) |$\gt $|70 0.3943 0.1552 (0.0014) (0.0015) Note: Asymptotic standard errors in brackets. Open in new tab Table D1. Distributional Gini coefficients. Age group analysis when Dibao recipients are separately identified and when they are not. Age groups . DisGini . DisGini . . (Dibao recipients separately identified) . (Dibao recipients not identified) . 45–50 0.3645 0.1085 (0.0015) (0.0014) 50–60 0.2625 0.0945 (0.0014) (0.0013) 60–70 0.3145 0.1420 (0.0014) (0.0015) |$\gt $|70 0.3943 0.1552 (0.0014) (0.0015) Age groups . DisGini . DisGini . . (Dibao recipients separately identified) . (Dibao recipients not identified) . 45–50 0.3645 0.1085 (0.0015) (0.0014) 50–60 0.2625 0.0945 (0.0014) (0.0013) 60–70 0.3145 0.1420 (0.0014) (0.0015) |$\gt $|70 0.3943 0.1552 (0.0014) (0.0015) Note: Asymptotic standard errors in brackets. Open in new tab ACKNOWLEDGEMENTS This work was supported by the 2019 Research Grant of the Center for Distributive Justice at the Institute of Economic Research, Seoul National University. We would like to thank participants of workshops at the Universities of Waterloo, Toronto, Cambridge, Guelph, the Canadian Economics Association Meetings, the International Association for Research in Income and Wealth, and ECINEQ. Footnotes 1 Outcomes of two or more distinct groups are compared and contrasted in equality of opportunity, mobility, and well-being literature (e.g., Blackorby and Donaldson, 1978; Arrow et al., 2000; Herrnstein and Murray, 1994; Peragine et al., 2014; Roemer, 1998; Weymark, 2003). The financial returns of a collection of portfolios are compared on a combined mean–variance basis (Markowitz, 1952; Bali et al., 2013; Banz, 1981; Basu, 1983; Jegadeesh, 1990). Within and between firm and industry, wage inequalities have been explored in the industrial organisation and labour literatures (Abowd et al., 2018; Card et al., 2018; Song et al., 2019). In treatment effect, event and matching study and policy evaluation literatures (Angrist and Krueger, 2001) assessment is based on comparisons of conditional means across outcome states. Recent developments in the measurement and analysis of subjective well-being (Kahneman and Krueger, 2006) involve multilateral comparisons of groups in the context of ordinal categorical data. 2 See Pittau and Zelli (2017) for an overview of Gini’s original concepts of transvariation. 3 Since translation to discrete and categorical paradigms is straightforward, discussion is confined to the continuous paradigm for brevity purposes. 4 The Gini bilateral transvariation can be regarded as the total variation distance between two probability measures (it is the normalised |$\ell _{1}$| distance between them). 5 Version estatCROS 2019ki9, released in May 2019. 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TI - On unit free assessment of the extent of multilateral distributional variation
JO - The Econometrics Journal
DO - 10.1093/ectj/utab003
DA - 2021-09-10
UR - https://www.deepdyve.com/lp/oxford-university-press/on-unit-free-assessment-of-the-extent-of-multilateral-distributional-Uq8YPCmEWA
SP - 502
EP - 518
VL - 24
IS - 3
DP - DeepDyve
ER -