Copula-Based Orderings of Dependence between

Dimensions of Well-being

Koen DecancqDepartement of Economics - KULeuven

Canazei – January 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

21. Introduction

Individual well-being is multidimensional

What about well-being of a society?Two approaches:

Income Life EducAnna 9000 77 61Boris 1300

072 69

Catharina 3500 73 81

WB

WA

WC

Wsoc

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

31. Introduction

Individual well-being is multidimensional

What about well-being of a society?Alternative approach (Human Development Index):

Income Life EducAnna 9000 77 61Boris 1300

072 69

Catharina 3500 73 81

LifeGDP Educ HDIsoc

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

41. Introduction

Individual well-being is multidimensional

What about well-being of a society?Alternative approach (Human Development Index):

Income Life EducAnna 9000 77 61Boris 1300

072 69

Catharina 3500 73 81

LifeGDP Educ HDIsoc

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

51. Introduction

Individual well-being is multidimensional

What about well-being of a society?Alternative approach (Human Development Index):

Income Life EducAnna 1300

077 81

Boris 9000 73 69Catharina 3500 72 61

LifeGDP Educ HDIsoc

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

6Outline

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

72. 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

83. Copula and Dependence (1)

xj: achievement on dim. j; Xj: Random variable

Fj: Marginal distribution function of good j: for all goods xj in :

Probability integral transform: Pj=Fj(Xj)

1

0 x1

F1(x1

) 0.66

0.33

3500

5000

13000

incomeAnna 5000Boris 1300

0Catharina 3500

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

93. 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 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

103. 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 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 Koen Decancq

133. 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): comonotonicWalzer: Caste societiesDardanoni: after unfair rearrangement

C-(p): countermonotonicFair 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

143. 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 Koen Decancq

15Outline

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

164. 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

17

0

Position in

Dimension 1

1

1

p

Position in

Dimension 2

4. Partial dependence ordering: 2 dimensions

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

184 Partial dependence ordering: 3 dimensions

1

1

1

p

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

194 Partial dependence ordering: 3 dimensions

1

1

1

up

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

204 Partial dependence ordering: 3 dimensions

1

1

1

uu p

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

21Outline

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

225. 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)

0

Position in

Dimension 1

1

1 p1

p2

p1

p2

Position in

Dimension 2

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

235. Dependence Increasing Rearrangements (generalization)

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

245. Dependence Increasing Rearrangements (generalization)

0

Position in

Dimension 1

Position in

Dimension 2 1

1

Position in

Dimension 3

1

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

255. 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 = even k = oddPositive rearr.

CF(p) ≤ CG(p)CF(p) ≤ CG(p)

Negative rearr.

CF(p) ≥ CG(p)CF(p) ≥ CG(p)

CF(p) ≤ CG(p)CF(p) ≥ CG(p)

CF(p) ≤ CG(p)CF(p) ≥ CG(p)

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

265. 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 = even k = oddPositive rearr.

CF(p) ≤ CG(p)CF(p) ≤ CG(p)

Negative rearr.

CF(p) ≥ CG(p)CF(p) ≥ CG(p)

CF(p) ≤ CG(p)CF(p) ≥ CG(p)

CF(p) ≤ CG(p)CF(p) ≥ CG(p)

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

27Outline

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

286. 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

296. 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

30Outline

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

317. Empirical illustration: russia between 1995-2003

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

327. 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:Dimension Primary Ordering

Var.Secondary Ordering Var.

Material well-being.

Equivalized income Individual Income

Health Obj. Health indicator

Education Years of schooling Number of additional courses

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

347. Empirical illustration: Complete dependence ordering

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

358. 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.