Copula-Based Orderings of Dependence between Dimensions of Well-being

Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

1. Introduction • Individual well-being is multidimensional • What about well-being of a society? Two approaches: WA WB WC Wsoc Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

1. Introduction • Individual well-being is multidimensional • What about well-being of a society? Alternative approach (Human Development Index): GDP Life Educ HDIsoc Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

Outline • Introduction • Why is the measurement of Dependence relevant? • Copula and Dependence • A partial ordering of Dependence • Dependence Increasing Rearrangements • A complete ordering of Dependence • Illustration based on Russian Data • Conclusion Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

2. Why is Dependence between Dimensions of Well-being Relevant? • Dependence and Theories of Distributive Justice: The notion of Complex Inequality • Walzer (1983) • Miller and Walzer (1995) • Dependence and Sociological Literature: The notion of Status Consistency • Lenski (1954) • Dependence and Multidimensional Inequality: • Atkinson and Bourguignon (1982) • Dardanoni (1995) • Tsui (1999) Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

3. Copula and Dependence (1) • xj: achievement on dim. j; Xj: Random variable • Fj: Marginal distribution function of good j: for all goods xjin : • Probability integral transform: Pj=Fj(Xj) F1(x1) 1 0.66 0.33 0 3500 5000 13000 x1 Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

3. Copula and Dependence (2) • x=(x1,…,xm): achievement vector; X=(X1,…,Xm): random vector of achievements. • p=(p1,…,pm): position vector; P=(P1,…,Pm): random vector of positions. • Joint distribution function: for all bundles xin m: • A copula function is a joint distribution function whose support is [0,1]m and whose marginal distributions are standard uniform. For all pin [0,1]m: Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

3. Why is the copula so useful? (1) • Theorem by Sklar (1959) Let F be a joint distribution function with margins F1, …, Fm. Then there exist a copula C such that for all xin 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 Koen Decancq

3. Why is the copula so useful? (3) • Fréchet-Hoeffding bounds If C is a copula, then for all p in [0,1]m : C-(p) ≤ 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)=p1*…*pm: independence copula Walzer: perfect complex equal society Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

3. The survival copula • Joint survival function: for all bundles xin 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 pin [0,1]m : Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

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 Koen Decancq

4. Partial dependence ordering: 2 dimensions Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

1 p 1 1 4 Partial dependence ordering: 3 dimensions Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

1 u p 1 1 4 Partial dependence ordering: 3 dimensions Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

1 u u p 1 1 4 Partial dependence ordering: 3 dimensions Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

5. Dependence Increasing Rearrangements (2 dimensions) • A positive 2-rearrangement of a copula function C, adds strictly positive probability mass ε to position vectors (p1,p2) and (p1,p2) and subtracts probability mass ε from grade vectors (p1,p2) and (p1,p2) Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

5. Dependence Increasing Rearrangements (2 dimensions) • A positive 2-rearrangement of a copula function C, adds strictly positive probability mass ε to position vectors (p1,p2) and (p1,p2) and subtracts probability mass ε from grade vectors (p1,p2) and (p1,p2) • 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 Koen Decancq

5. Dependence Increasing Rearrangements (3 dimensions) • 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 Koen Decancq

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. 5. Dependence Increasing Rearrangements (4 dimensions) Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

5. Dependence Increasing Rearrangements (generalization) G has been reached from F by a finite sequence of the following k-rearrangements, iff for all p in [0,1]m: CF(p) ≥ CG(p) CF(p) ≤ CG(p) Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

6. Complete dependence ordering: measures of dependence • 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 Koen Decancq

6. Complete dependence ordering: a measure of dependence • An member of the class considered : • Interpretation: Draw randomly two individuals: • One from society with copula CX • 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 Koen Decancq

7. Empirical illustration: russia between 1995-2003 Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

7. Empirical illustration: russia between 1995-2003 • 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: Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

7. Empirical illustration: Partial dependence ordering Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

7. Empirical illustration: Complete dependence ordering Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

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 Koen Decancq

Copula-Based Orderings of Dependence between Dimensions of Well-being