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.