1. ophi hdca ss11 multidimensional stochastic dominance gy

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Brief introduction to multidimensional stochastic dominance

Brief introduction to multidimensional stochasticdominance

Gaston Yalonetzky

Oxford Poverty and Human Development Initiative, University of Oxford

OPHI-HDCA Summer School, Delft, 24 August - 3 September

2011.

We are grateful to the World Bank, two anonymous donors andOPHI for financial support
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Table of contents

Introduction

Stochastic dominance conditions: for univariate to multivariatesettings

Multidimensional stochastic dominance andcomplementarity/substitutability

Relevance for poverty measurement

Tests for univariate and multivariate stochastic dominanceThe test of Barrett and Donald (2003)

Concluding remarks
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Introduction

Introduction: Multidimensional versus unidimensional

dominance

Stochastic dominance conditions provide an extreme form of robustnessfor ordinal comparisons: if they are fulfilled a comparison is robust to abroad range of parameter values and families of indices.


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Introduction


dominance


Multidimensional dominance is relevant for evaluation functions that mapfrom a multivariate space.
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Introduction


dominance


Multidimensional dominance is relevant for evaluation functions that mapfrom a multivariate space. E.g. An index of well-being that depends onseveral aspects of wellbeing.
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Introduction


dominance



In unidimensional dominance second and even third orders may beinteresting/relevant.


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Introduction


dominance



In unidimensional dominance second and even third orders may beinteresting/relevant. In multidimensional dominance second and higher

orders are not that easy to interpret.
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Introduction


dominance



In unidimensional dominance second and even third orders may beinteresting/relevant. In multidimensional dominance second and higher

orders are not that easy to interpret.

By contrast, in multidimensional dominance other things matter: thejoint distribution of the variables, as well as how they complement, orsubstitute, each other in their contributions toward the evaluationfunction (e.g. increasing wellbeing).
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Introduction

Introduction: Multidimensional dominance conditions

In this lecture we are going to review the derivation ofdominance conditions and extend it to multivariate settings.

B f d l d l h d
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Introduction



We will focus on first-order dominance and bivariatedistributions and derive the respective conditions.

B i f i d i l idi i l h i d i
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Introduction




We will discuss the logic behind some of the conditions andtheir connections to complementarity and substitutabilitybetween variables.

B ief i t d ti t ltidi e si l st h sti d i e
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Introduction





We will discuss why these conditions are also relevant forpoverty assessments.



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Introduction





We will discuss why these conditions are also relevant forpoverty assessments.

We will briefly discuss how to test these conditions.

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Stochastic dominance conditions: for univariate to multivariate settings

Traditional dominance conditions with one variable: an

example

Consider the following wellbeing function: W(x) =xmaxxmin

U(x)dF(x)

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example


U(x)dF(x)Traditional dominance conditions stem from expressing differences insocial welfare functions as sums of products of derivatives of the

individual welfare functions and functions of the cumulative (or survival)densities. Example:

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example




W =

xmaxxmin

U

x(x)F(x)dx

The condition is then: W0 U | Ux

(x) 0 F(x) 0x

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example




W =

xmaxxmin

U

x(x)F(x)dx

The condition is then: W0 U | Ux

(x) 0 F(x) 0x

The dominance condition is usually expressed in terms of distributions.

E.g. ifx :FA(x) FB(x) then we say that distribution A (first-order)

dominates B.

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First-order conditions for the bivariate case

With two variables we integrate by parts the following function:

W =

ymaxymin

xmaxxmin

U(x)f(x, y)dxdy

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W =

ymaxymin

xmaxxmin

U(x)f(x, y)dxdy

Because we have a joint density (f(x, y)) we can integrate using eithercumulative functions:

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W =

ymaxymin

xmaxxmin

U(x)f(x, y)dxdy

Because we have a joint density (f(x, y)) we can integrate using eithercumulative functions:

W =

xmax

xmin

U

x

(x, ymax)Fx(x)dx

ymaxymin

U

y(xmax, y)F

y(y)dy+

ymaxymin

xmaxxmin

2U

xy(x, y)F(x, y)dxdy



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...or survival functions:

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...or survival functions:

W = xmaxxmin

Ux

(x, ymin)Fx(x)dx

ymaxymin

U

y(xmin, y)F

y(y)dy+

ymaxymin

xmaxxmin

2U

xy(x, y)F(x, y)dxdy

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Notice the appearance of 2U

xy(x, y).


S h d d f l
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xy(x, y). This cross-partial derivative determines

whether x and y areALEP complementsor ALEP substitutesin theircontributions to U.


S h i d i di i f i i l i i i
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whether x and y areALEP complementsor ALEP substitutesin theircontributions to U. Four conditions stem from the two previous equations:


St h sti d i e diti s f i i te t lti i te setti s
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1. A condition for monotonically increasing functions with ALEP substitute

arguments (e.g. 2U

xy(x, y) 0):

x, y : W0U | U

i (x, y) 0

2U

xy(x, y) 0

Fx(x), Fy(y), F(x, y) 0


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1. A condition for monotonically increasing functions with ALEP substitute

arguments (e.g. 2U

xy(x, y) 0):

x, y : W0U | U

i (x, y) 0

2U

xy(x, y) 0

Fx(x), Fy(y), F(x, y) 0

Notice that x, y : F(x, y) 0 suffices to ascertain W0.


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2. A condition for monotonically increasing functions with ALEPcomplement arguments (e.g.

2U

xy

(x, y)0):

x, y : W0U | U

i (x, y)0

2U

xy(x, y)0

Fx(x), Fy(y), F(x, y) 0


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2. A condition for monotonically increasing functions with ALEPcomplement arguments (e.g.

2U

xy

(x, y)0):

x, y : W0U | U

i (x, y)0

2U

xy(x, y)0

Fx(x), Fy(y), F(x, y) 0

Notice that x, y : F(x, y)0 suffices to ascertain W0.


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3. A condition for monotonically increasing functions with ALEPneutral arguments (e.g.

2Uxy(x, y) = 0):

x, y : W0U | U

i (x, y)0

2U

xy(x, y) = 0

Fx(x), Fy(y) 0




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g


3. A condition for monotonically increasing functions with ALEPneutral arguments (e.g.

2Uxy(x, y) = 0):

x, y : W0U | U

i (x, y)0

2U

xy(x, y) = 0

Fx(x), Fy(y) 0

Notice that, in this case, it is necessary to test Fi(i) 0 forevery variable (unlike in the previous cases).


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4. A condition for ALL individual welfare functions that aremonotonically increasing!

x, y : W0U | U

i (x, y)0

Fx(x), Fy(y), F(x, y)0Fx(x), Fy(y), F(x, y) 0


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4. A condition for ALL individual welfare functions that aremonotonically increasing!

x, y : W0U | U

i (x, y)0

Fx(x), Fy(y), F(x, y)0Fx(x), Fy(y), F(x, y) 0

Notice that, in this case, it is necessary and sufficient to testF(x, y)0 and F(x, y) 0.


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First-order conditions for the general multivariatecase

In the general multivariate case, first-order conditions are based on allcumulative and/or survival functions combining all variables.


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In the general multivariate case, first-order conditions are based on allcumulative and/or survival functions combining all variables. They include allmarginal distributions, Fi(i) and all cumulative and survival joints up toF(x, y, ..., w, z) and F(x, y, ...,w, z).


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In the case of cumulative distributions, the signs of the cross partial derivativesof the functions for which the conditions hold change the following way:Ui>0, Uij0 and so forth.


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In the case of survival functions, all the signs of the cross partial derivatives ofthe functions for which the conditions hold are positive (e.g. Ui>0,Uij>

0,Uijk>

0).


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In the case of survival functions, all the signs of the cross partial derivatives ofthe functions for which the conditions hold are positive (e.g. Ui>0,Uij>

0,Uijk>

0).

Hence when conditions on both the cumulative and the survival functions arefulfilled:

x, y, ..., z : W0U | U

i , 3U

ijk, ... 0


Multidimensional stochastic dominance and complementarity/substitutability
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ALEP relationships and multidimensional dominance

conditions

Why the conditions on cumulative distributions are associated withALEP substitutability and the conditions on survival functions areassociated with ALEP complementarity?


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ALEP substitution and F(x,y)0




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ALEP complementarity and F(x,y)0


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Applying dominance conditions to poverty measurement

Social poverty functions are characterized by:


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Being additive and symmetric across individuals:

P(X; z) = 1NNn=1Pn(x; z).


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P(X; z) = 1NNn=1Pn(x; z). Higher poverty is worse.


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Higher poverty is worse.

Monotonicity: Higher xshould not increase poverty:Pn(x;z)

x 0


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Higher poverty is worse.

Monotonicity: Higher xshould not increase poverty:Pn(x;z)

x 0

So if we define W(X; z) =P(X;Z) then we can apply the abovementioned conditions to poverty comparisons as well!

Brief introduction to multidimensional stochastic dominanceTests for univariate and multivariate stochastic dominance

The test of Barrett and Donald (2003)
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This test is a type of Kolmogorov-Smirnov test of

homogeneity of distributions.


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This test is a type of Kolmogorov-Smirnov test of

homogeneity of distributions. The logic is very intuitive: dominance conditions are based on

comparing integrals ofFA(z) with FB(z) for a range ofz,depending on the order of dominance (e.g. for first order, justcompare cumulative or survival distributions).


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There are four possible outcomes (illustrated with first-order):

1. A dominates B:FA

(z

)FB

(z

)z

z|FA


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1. A dominates B: FA(z) FB(z)z z|FA


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The generic hypothesis subject to test is the following (illustratedwith first-order):

H0 :FA(z) FB(z)z[zmin, zmax]


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H1 :z[z,min, zmax]|FA(z)>FB(z)


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H1 :z[z,min, zmax]|FA(z)>FB(z)

Results of these tests require interpretation in order to ascertainany of the four possibilities.


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1. Test:

H0 (FA FB) vs

H1 (FA> FB)

A B or

equal

2. Test:

H0(FB FA) vs

H1(FB

> FA

)A = B

Notreject

A B

Rejec

t

Switch

Not

reject

B A or

crossing

2. Test:

H0(FB FA) vs

H1(FB> FA)

A and B cross

Reject

B A

Not r

eject

Switch

Rejec

t


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The test proceeds as follows (illustrated with first-order):


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For a range of z, estimate: FA(z) FB(z)


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Estimate the supremum and multiply by a function of the

sample sizes:S= ( NMN+M)1/2supz[FA(z) FB(z)]


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sample sizes:S= ( NMN+M)1/2supz[FA(z) FB(z)]S is the statistic we need. We now need to know how likely it

is for this value to appear under the null hypothesis.


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sample sizes:S= ( NMN+M)1/2supz[FA(z) FB(z)]S is the statistic we need. We now need to know how likely it

is for this value to appear under the null hypothesis.

There are different procedures to derive the distribution under

the null hypothesis. Barrett and Donald (2003) develop twotypes. We are going to show one of them: a bootstrapmethod.


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The bootstrap method in the test of Barrett and Donald

(2003)

1. We pool the two samples of A and B (a total of N+Mobservations) and draw several subsamples with replacement

(R):


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(2003)


(R):2. For each subsample (say 1000) we estimate:

Sr = ( NMN+M)

1/2supz[FA(z; r) FB(z; r)]


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(2003)



Sr = ( NMN+M)


3. The p-value is then approximated by:

pA,B

1RRr=11(Sr>S)


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(2003)



Sr = ( NMN+M)


3. The p-value is then approximated by:

pA,B

1RRr=11(Sr>S)

4. If, say, pA,B


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Concluding remarks

Stochastic dominance conditions are useful to ascertain whether anordinalcomparison is robust to changes in the parameters or members ofa family of evaluation functions.

Brief introduction to multidimensional stochastic dominanceConcluding remarks
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Concluding remarks


When dominance conditions are not fulfilled then the comparison dependson the choice of parameters (e.g. poverty lines, risk/inequality aversion,

etc.).

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Concluding remarks


When dominance conditions are not fulfilled then the comparison dependson the choice of parameters (e.g. poverty lines, risk/inequality aversion,etc.).

One could restrict the dominance analysis to smaller sets of parameters(or members of families of indices), but this needs to be done carefully,lest significant parts of the domain of interest may be left out.


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Concluding remarks




When one is concerned for cardinal comparisons, e.g. between countriesor across time, then dominance conditions are not that useful. Sensitivityanalysis is required.


C l di k
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Concluding remarks




When one is concerned for cardinal comparisons, e.g. between countriesor across time, then dominance conditions are not that useful. Sensitivityanalysis is required.

We have assumed the variables are continuous. But these results can beeasily extended to ordinal variables (e.g. see Yalonetzky, 2011). Furtherextensions to combinations of continuous and ordinal variables arepossible.
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1. ophi hdca ss11 multidimensional stochastic dominance gy

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