copula-based orderings of dependence between dimensions of well-being koen decancq departement of...

34
Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Post on 21-Dec-2015

216 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Copula-Based Orderings of

Dependence between

Dimensions of Well-being

Koen DecancqDepartement of Economics - KULeuven

Oxford – June 2009

Page 2: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

31. 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

Catharina 3500 73 81

WB

WA

WC

Wsoc

Page 3: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

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 Educ

Anna 9000 77 61

Boris 13000

72 69

Catharina 3500 73 81

LifeGDP Educ HDIsoc

Page 4: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

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 Educ

Anna 9000 77 61

Boris 13000

72 69

Catharina 3500 73 81

LifeGDP Educ HDIsoc

Page 5: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

61. 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

LifeGDP Educ HDIsoc

Page 6: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

7Outline

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

Page 7: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

82. 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)

Page 8: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

93. 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

income

Anna 5000

Boris 13000

Catharina 3500

Page 9: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

103. 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:

Page 10: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

113. 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

Page 11: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

143. 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

Page 12: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

153. 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 :

Page 13: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

16Outline

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

Page 14: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

174. 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)

Page 15: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

18

0

Position in

Dimension 1

1

1

p

Position in

Dimension 2

4. Partial dependence ordering: 2 dimensions

Page 16: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

194 Partial dependence ordering: 3 dimensions

1

1

1

p

Page 17: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

204 Partial dependence ordering: 3 dimensions

1

1

1

up

Page 18: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

214 Partial dependence ordering: 3 dimensions

1

1

1

uu p

Page 19: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

22Outline

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

Page 20: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

235. 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

Page 21: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

245. 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)

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.

Page 22: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

255. 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 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.

0

Position in

Dimension 1

Position in

Dimension 2 1

1

Position in

Dimension 3

1

Page 23: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

265. 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 pj = pj, and subtracts probability mass ε from all vertices of Bm with an odd number of grades pj = pj.

Page 24: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

275. 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 = 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)

Page 25: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

285. 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 = 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)

Page 26: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

29Outline

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

Page 27: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

316. 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:

Page 28: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

326. 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:

Page 29: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

34Outline

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

Page 30: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

357. Empirical illustration: russia between 1995-2003

Page 31: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

367. 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

Page 32: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

377. Empirical illustration: Partial dependence ordering

Page 33: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

387. Empirical illustration: Complete dependence ordering

Page 34: Copula-Based Orderings of Dependence between Dimensions of Well-being Koen Decancq Departement of Economics - KULeuven Oxford – June 2009

Canazei January 2009 Copula-based orderings of Dependence Koen Decancq

398. 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.