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
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
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
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
6
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?
7
 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
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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 :
F1(x1) 1
income
0.66
Anna
5000
0.33
Boris
13000
Catharina
0
3500 5000
3500
13000 x1
 Probability integral transform: Pj=Fj(Xj)
Canazei January 2009
Copula-based orderings of Dependence
Koen Decancq
3. Copula and Dependence (2)
9
 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
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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 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
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).
 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
14
 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
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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
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
Koen Decancq
4. Partial dependence ordering: 2 dimensions
17
Position in
Dimension 2
1
p
Position in
Dimension 1
0
Canazei January 2009
1
Copula-based orderings of Dependence
Koen Decancq
4 Partial dependence ordering: 3 dimensions
18
1
p
1
1
Canazei January 2009
Copula-based orderings of Dependence
Koen Decancq
4 Partial dependence ordering: 3 dimensions
19
1
up
1
1
Canazei January 2009
Copula-based orderings of Dependence
Koen Decancq
4 Partial dependence ordering: 3 dimensions
20
1
up
1
1
Canazei January 2009
Copula-based orderings of Dependence
Koen Decancq
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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
22
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)
Position in
Dimension 2
1
p2
p2
Position in
Dimension 1
0
Canazei January 2009
p1
p1
1
Copula-based orderings of Dependence
Koen Decancq
23
5. 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
5. Dependence Increasing Rearrangements (generalization)
24
Position in
1
Dimension 2
Position in
0
Dimension 1
1
1
Position in
Dimension 3
Canazei January 2009
Copula-based orderings of Dependence
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)
CF(p) ≥ CG(p)
CF(p) ≥ CG(p)
CF(p) ≤ CG(p)
CF(p) ≤ CG(p)
CF(p) ≥ CG(p)
Copula-based orderings of Dependence
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)
CF(p) ≥ CG(p)
CF(p) ≥ CG(p)
CF(p) ≤ CG(p)
CF(p) ≤ CG(p)
CF(p) ≥ CG(p)
Copula-based orderings of Dependence
Koen Decancq
27
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
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
Koen Decancq
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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
30
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
7. Empirical illustration: russia between 1995-2003
Canazei January 2009
Copula-based orderings of Dependence
31
Koen Decancq
32
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:
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
Copula-based orderings of Dependence
Koen Decancq
7. Empirical illustration: Complete dependence ordering
Canazei January 2009
Copula-based orderings of Dependence
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Koen Decancq
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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