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

2 Canazei January 2009 Copula-based orderings of Dependence Koen Decancq  Why multidimensional approach?

3 Canazei January 2009 Copula-based orderings of Dependence Koen Decancq 1. Introduction  Individual well-being is multidimensional  What about well-being of a society? Two approaches: IncomeLifeEduc Anna Boris Catharina WBWB WAWA WCWC W soc

4 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): IncomeLifeEduc Anna Boris Catharina LifeGDPEducHDI soc

5 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): IncomeLifeEduc Anna Boris Catharina LifeGDPEducHDI soc

6 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): IncomeLifeEduc Anna Boris Catharina LifeGDPEducHDI soc

7 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

8 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)

9 Canazei January 2009 Copula-based orderings of Dependence Koen Decancq 3. Copula and Dependence (1)  x j : achievement on dim. j; X j : Random variable  F j : Marginal distribution function of good j: for all goods x j in  :  Probability integral transform: P j =F j (X j ) 1 0 x1x1 F 1 (x 1 ) income Anna5000 Boris13000 Catharina3500

10 Canazei January 2009 Copula-based orderings of Dependence Koen Decancq 3. Copula and Dependence (2)  x=(x 1,…,x m ): achievement vector; X=(X 1,…,X m ): random vector of achievements.  p=(p 1,…,p m ): position vector; P=(P 1,…,P m ): 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 :

11 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 F 1, …, F m. 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

12 Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

13 Canazei January 2009 Copula-based orderings of Dependence Koen Decancq 3. Why is the copula so useful? (2)  But less popular in welfare economics: Dardanoni and Lambert (2001): horizontal inequality Fournier (2001): correlation between incomes of spouses Bonhomme and Robin (2006): mobility Abul Naga and Geoffard (2006): multidimensional inequality measures Quinn (2007): dependence between health and income

14 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)=p 1 *…*p m : independence copula Walzer: perfect complex equal society

15 Canazei January 2009 Copula-based orderings of Dependence Koen Decancq 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 :

16 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

17 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 C F and C G and joint survival functions C F and C G, G is more dependent than F, if for all p in [0,1] m : C F (p) ≤ C G (p) and C F (p) ≤ C G (p)

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

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

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

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

22 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

23 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 (p 1,p 2 ) and (p 1,p 2 ) and subtracts probability mass ε from grade vectors (p 1,p 2 ) and (p 1,p 2 )

24 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 (p 1,p 2 ) and (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 B m with an even number of grades p j = p j, and subtracts probability mass ε from all vertices of B m with an odd number of grades p j = p j.

25 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 B m with an even number of grades p j = p j, and subtracts probability mass ε from all vertices of B m with an odd number of grades p j = p j.

26 Canazei January 2009 Copula-based orderings of Dependence Koen Decancq 5. Dependence Increasing Rearrangements (4 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 p j = p j, and subtracts probability mass ε from all vertices of Bm with an odd number of grades p j = p j.

27 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 : k = evenk = odd Positive rearr.C F (p) ≤ C G (p) Negative rearr.C F (p) ≥ C G (p) C F (p) ≤ C G (p) C F (p) ≥ C G (p) C F (p) ≤ C G (p) C F (p) ≥ C G (p)

28 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 : k = evenk = odd Positive rearr.C F (p) ≤ C G (p) Negative rearr.C F (p) ≥ C G (p) C F (p) ≤ C G (p) C F (p) ≥ C G (p) C F (p) ≤ C G (p) C F (p) ≥ C G (p)

29 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

30 Canazei January 2009 Copula-based orderings of Dependence Koen Decancq  axioms

31 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:

32 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 C X One from independent society (copula C ┴ ) Then D ┴ (C X ) is the probability of outranking between these individuals  After normalization:

33 Canazei January 2009 Copula-based orderings of Dependence Koen Decancq  Another member (kendall’s tau)

34 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

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

36 Canazei January 2009 Copula-based orderings of Dependence Koen Decancq 7. Empirical illustration: russia between  Question: What happens with the dependence between the dimensions of well-being in Russia during this period?  Household data from RLMS ( )  The same individuals (1577) are ordered according to: DimensionPrimary Ordering Var.Secondary Ordering Var. Material well- being. Equivalized incomeIndividual Income HealthObj. Health indicator EducationYears of schoolingNumber of additional courses

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

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

39 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.