CopulaBased Orderings of Dependence between Dimensions of Wellbeing
Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Canazei – January 2009
2 1. Introduction § Individual well-being is multidimensional § What about well-being of a society? Two approaches: Income Life Educ Anna 9000 77 61 Boris 13000 72 69 3500 73 81 Catharina WA WB WC Wsoc Canazei January 2009 Copula-based orderings of Dependence Koen Decancq 2
3 1. Introduction § Individual well-being is multidimensional § What about well-being of a society? Alternative approach (Human Development Index): Income Life Educ Anna 9000 77 61 Boris 13000 72 69 3500 73 81 Catharina GDP Life Educ HDIsoc Canazei January 2009 Copula-based orderings of Dependence Koen Decancq 3
4 1. Introduction § Individual well-being is multidimensional § What about well-being of a society? Alternative approach (Human Development Index): Income Life Educ Anna 9000 77 61 Boris 13000 72 69 3500 73 81 Catharina GDP Life Educ HDIsoc Canazei January 2009 Copula-based orderings of Dependence Koen Decancq 4
5 1. Introduction § Individual well-being is multidimensional § What about well-being of a society? Alternative approach (Human Development Index): Income Life Educ Anna 13000 77 81 Boris 9000 73 69 Catharina 3500 72 61 GDP Life Educ HDIsoc Canazei January 2009 Copula-based orderings of Dependence Koen Decancq 5
6 Outline § § § § Introduction Why is the measurement of Dependence relevant? Copula and Dependence A partial ordering of Dependence Increasing Rearrangements A complete ordering of Dependence Illustration based on Russian Data Conclusion Canazei January 2009 Copula-based orderings of Dependence Koen Decancq 6
2. Why is Dependence between Dimensions of Well-being Relevant? 7 § Dependence and Theories of Distributive Justice: The notion of Complex Inequality l l Walzer (1983) Miller and Walzer (1995) § Dependence and Sociological Literature: The notion of Status Consistency l Lenski (1954) § Dependence and Multidimensional Inequality: l l l Atkinson and Bourguignon (1982) Dardanoni (1995) Tsui (1999) Canazei January 2009 Copula-based orderings of Dependence Koen Decancq 7
8 3. Copula and Dependence (1) § xj: achievement on dim. j; Xj: Random variable § Fj: Marginal distribution function of good j: for all goods xj in : F 1(x 1) 1 income 0. 66 Anna 5000 0. 33 Boris 13000 0 Catharina 3500 5000 3500 13000 x 1 § Probability integral transform: Pj=Fj(Xj) Canazei January 2009 Copula-based orderings of Dependence Koen Decancq 8
9 3. Copula and Dependence (2) § x=(x 1, …, xm): achievement vector; X=(X 1, …, Xm): random vector of achievements. § p=(p 1, …, pm): position vector; P=(P 1, …, Pm): random vector of positions. § Joint distribution function: for all bundles x in m: § A copula function is a joint distribution function whose support is [0, 1]m and whose marginal distributions are standard uniform. For all p in [0, 1]m: Canazei January 2009 Copula-based orderings of Dependence Koen Decancq 9
10 3. Why is the copula so useful? (1) § Theorem by Sklar (1959) Let F be a joint distribution function with margins F 1, …, Fm. Then there exist a copula C such that for all x in m: § The copula joins the marginal distributions to the joint distribution § In other words: it allows to focus on the dependence alone § Many applications in multidimensional risk and financial modeling Canazei January 2009 Copula-based orderings of Dependence 10 Koen Decancq
3. Why is the copula so useful? (3) 13 § Fréchet-Hoeffding bounds If C is a copula, then for all p in [0, 1]m : C-(p) ≤ C+(p). § C+(p): comonotonic Walzer: Caste societies Dardanoni: after unfair rearrangement § C-(p): countermonotonic Fair allocation literature: satisfies ‘No dominance’ equity criterion § C ┴(p)=p 1*…*pm: independence copula Walzer: perfect complex equal society Canazei January 2009 Copula-based orderings of Dependence 13 Koen Decancq
14 3. The survival copula § Joint survival function: for all bundles x in m § A survival copula is a joint survival function whose support is [0, 1]m and whose marginal distributions are standard uniform, so that for all p in [0, 1]m : Canazei January 2009 Copula-based orderings of Dependence 14 Koen Decancq
15 Outline § § § § Introduction Why is the measurement of Dependence relevant? Copula and Dependence A partial ordering of Dependence Increasing Rearrangements A complete ordering of Dependence Illustration based on Russian Data Conclusion Canazei January 2009 Copula-based orderings of Dependence 15 Koen Decancq
16 4. A Partial dependence ordering § Recall: dependence captures the alignment between the positions of the individuals § Formal definition (Joe, 1990): For all distribution functions F and G, with copulas CF and CG and joint survival functions CF and CG, G is more dependent than F, if for all p in [0, 1]m: CF(p) ≤ CG(p) and CF(p) ≤ CG(p) Canazei January 2009 Copula-based orderings of Dependence 16 Koen Decancq
4. Partial dependence ordering: 2 dimensions Canazei January 2009 Copula-based orderings of Dependence 17 17 Koen Decancq
4 Partial dependence ordering: 3 dimensions 18 1 p 1 1 Canazei January 2009 Copula-based orderings of Dependence 18 Koen Decancq
4 Partial dependence ordering: 3 dimensions 19 1 up 1 1 Canazei January 2009 Copula-based orderings of Dependence 19 Koen Decancq
4 Partial dependence ordering: 3 dimensions 20 1 up 1 1 Canazei January 2009 Copula-based orderings of Dependence 20 Koen Decancq
21 Outline § § § § Introduction Why is the measurement of Dependence relevant? Copula and Dependence A partial ordering of Dependence Increasing Rearrangements A complete ordering of Dependence Illustration based on Russian Data Conclusion Canazei January 2009 Copula-based orderings of Dependence 21 Koen Decancq
5. Dependence Increasing Rearrangements (2 dimensions) 22 § A positive 2 -rearrangement of a copula function C, adds strictly positive probability mass ε to position vectors (p 1, p 2) and subtracts probability mass ε from grade vectors (p 1, p 2) and (p 1, p 2) Canazei January 2009 Copula-based orderings of Dependence 22 Koen Decancq
5. Dependence Increasing Rearrangements (generalization) 23 § A positive 2 -rearrangement of a copula function C, adds strictly positive probability mass ε to position vectors (p 1, p 2) and subtracts probability mass ε from grade vectors (p 1, p 2) and (p 1, p 2) § Multidimensional generalization: § A positive k-rearrangement of a copula function C, adds strictly positive probability mass ε to all vertices of hyperbox Bm with an even number of grades pj = pj, and subtracts probability mass ε from all vertices of Bm with an odd number of grades pj = pj. Canazei January 2009 Copula-based orderings of Dependence 23 Koen Decancq
5. Dependence Increasing Rearrangements (generalization) Canazei January 2009 Copula-based orderings of Dependence 24 24 Koen Decancq
5. Dependence Increasing Rearrangements (generalization) 25 G has been reached from F by a finite sequence of the following k -rearrangements, iff for all p in [0, 1]m : Positive rearr. Negative rearr. Canazei January 2009 k = even k = odd CF(p) ≤ CG(p) CF(p) ≥ CG(p) CF(p) ≤ CG(p) Copula-based orderings of Dependence CF(p) ≤ CG(p) CF(p) ≥ CG(p) 25 Koen Decancq
5. Dependence Increasing Rearrangements (generalization) 26 G has been reached from F by a finite sequence of the following k -rearrangements, iff for all p in [0, 1]m : Positive rearr. Negative rearr. Canazei January 2009 k = even k = odd CF(p) ≤ CG(p) CF(p) ≥ CG(p) CF(p) ≤ CG(p) Copula-based orderings of Dependence CF(p) ≤ CG(p) CF(p) ≥ CG(p) 26 Koen Decancq
27 Outline § § § § Introduction Why is the measurement of Dependence relevant? Copula and Dependence A partial ordering of Dependence Increasing Rearrangements A complete ordering of Dependence Illustration based on Russian Data Conclusion Canazei January 2009 Copula-based orderings of Dependence 27 Koen Decancq
6. Complete dependence ordering: measures of dependence 28 § We look for a measure of dependence D(. ) that is increasing in the partial dependence ordering § Consider the following class: with for all even k ≤ m: Canazei January 2009 Copula-based orderings of Dependence 28 Koen Decancq
6. Complete dependence ordering: a measure of dependence 29 § An member of the class considered : § Interpretation: Draw randomly two individuals: l One from society with copula C X l One from independent society (copula C ┴) Then D┴(CX) is the probability of outranking between these individuals § After normalization: Canazei January 2009 Copula-based orderings of Dependence 29 Koen Decancq
30 Outline § § § § Introduction Why is the measurement of Dependence relevant? Copula and Dependence A partial ordering of Dependence Increasing Rearrangements A complete ordering of Dependence Illustration based on Russian Data Conclusion Canazei January 2009 Copula-based orderings of Dependence 30 Koen Decancq
7. Empirical illustration: russia between 1995 -2003 Canazei January 2009 Copula-based orderings of Dependence 31 31 Koen Decancq
7. Empirical illustration: russia between 1995 -2003 32 § Question: What happens with the dependence between the dimensions of well-being in Russia during this period? § Household data from RLMS (1995 -2003) § The same individuals (1577) are ordered according to: Dimension Primary Ordering Var. Secondary Ordering Var. Material wellbeing. Equivalized income Individual Income Health Obj. Health indicator Education Years of schooling Canazei January 2009 Number of additional courses Copula-based orderings of Dependence 32 Koen Decancq
7. Empirical illustration: Complete dependence ordering Canazei January 2009 Copula-based orderings of Dependence 34 34 Koen Decancq
35 8. Conclusion § The copula is a useful tool to describe and measure dependence between the dimensions. § The obtained copula-based measures are applicable. § Russian dependence is not stable during transition. Hence we should be careful in interpreting the HDI as well-being measure. Canazei January 2009 Copula-based orderings of Dependence 35 Koen Decancq
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