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

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

• Koen Decancq
• Departement of Economics - KULeuven
• Canazei January 2009

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1. Introduction

• Individual well-being is multidimensional
• What about well-being of a society?
• Two approaches

Income Income Life Educ Educ
Anna 9000 77 61
Boris 13000 72 69
Catharina 3500 73 81
WA
WB
WC
Wsoc
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1. Introduction

• Individual well-being is multidimensional
• What about well-being of a society?
• Alternative approach (Human Development Index)

Income Income Life Educ Educ
Anna 9000 77 61
Boris 13000 72 69
Catharina 3500 73 81
Life
GDP
Educ
HDIsoc
4
1. Introduction

• Individual well-being is multidimensional
• What about well-being of a society?
• Alternative approach (Human Development Index)

Income Income Life Educ Educ
Anna 9000 77 61
Boris 13000 72 69
Catharina 3500 73 81
Life
GDP
Educ
HDIsoc
5
1. Introduction

• Individual well-being is multidimensional
• What about well-being of a society?
• Alternative approach (Human Development Index)

Income Income Life Educ Educ
Anna 13000 77 81
Boris 9000 73 69
Catharina 3500 72 61
Life
GDP
Educ
HDIsoc
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

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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)

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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 ?
• Probability integral transform PjFj(Xj)

F1(x1)
1
income income
Anna 5000
Boris 13000
Catharina 3500
0.66
0.33
0
3500
5000
13000
x1
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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 x in
  ?m
• A copula function is a joint distribution
  function whose support is 0,1m and whose
  marginal distributions are standard uniform. For
  all p in 0,1m

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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

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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

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3. Why is the copula so useful? (3)

• Fréchet-Hoeffding bounds
• If C is a copula, then for all p in 0,1m
• 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)p1pm independence copula
• Walzer perfect complex equal society

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3. The survival copula

• Joint survival function for all bundles x in ?m
• A survival copula is a joint survival function
  whose support is 0,1m and whose marginal
  distributions are standard uniform, so that for
  all p in 0,1m

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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

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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 0,1m
• CF(p) CG(p) and CF(p) CG(p)

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4. Partial dependence ordering 2 dimensions
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4 Partial dependence ordering 3 dimensions
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4 Partial dependence ordering 3 dimensions
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4 Partial dependence ordering 3 dimensions
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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

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5. Dependence Increasing Rearrangements (2
dimensions)

• A positive 2-rearrangement of a copula function
  C, adds strictly positive probability mass e to
  position vectors (p1,p2) and (p1,p2) and
  subtracts probability mass e from grade vectors
  (p1,p2) and (p1,p2)

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5. Dependence Increasing Rearrangements
(generalization)

• A positive 2-rearrangement of a copula function
  C, adds strictly positive probability mass e to
  position vectors (p1,p2) and (p1,p2) and
  subtracts probability mass e from grade vectors
  (p1,p2) and (p1,p2)
• Multidimensional generalization
• A positive k-rearrangement of a copula function
  C, adds strictly positive probability mass e to
  all vertices of hyperbox Bm with an even number
  of grades pj pj, and subtracts probability mass
  e from all vertices of Bm with an odd number of
  grades pj pj.

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5. Dependence Increasing Rearrangements
(generalization)
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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
  0,1m

k even k odd
Positive 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)
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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
  0,1m

k even k odd
Positive 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)
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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

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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

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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

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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

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7. Empirical illustration russia between
1995-2003
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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 well-being. Equivalized income Individual Income
Health Obj. Health indicator
Education Years of schooling Number of additional courses
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7. Empirical illustration Partial dependence
ordering
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7. Empirical illustration Complete dependence
ordering
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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.