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 September2011.

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

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

Introduction

Introduction: Multidimensional versus unidimensionaldominance

I 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.

I 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.

I In unidimensional dominance second and even third orders may beinteresting/relevant. In multidimensional dominance second and higherorders are not that easy to interpret.

I 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”).

Introduction

I 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.

Introduction

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

In multidimensional dominance second and higherorders are not that easy to interpret.

Introduction

Introduction: Multidimensional dominance conditions

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

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

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

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

I We will briefly discuss how to test these conditions.

Introduction

Stochastic dominance conditions: for univariate to multivariate settings

Traditional dominance conditions with one variable: anexample

Consider the following wellbeing function: W (x) =∫ xmax

xminU(x)dF (x)

Traditional dominance conditions stem from expressing differences insocial welfare functions as sums of products of derivatives of theindividual welfare functions and functions of the cumulative (or survival)densities. Example:

∆W = −∫ xmax

δx(x)∆F (x)dx

The condition is then: ∆W ≥ 0 ∀U | δUδx (x) ≥ 0↔ ∆F (x) ≤ 0∀x

The dominance condition is usually expressed in terms of distributions.

E.g. if ∀x : FA(x) ≤ FB(x) then we say that ”distribution A (first-order)

dominates B”.

xminU(x)dF (x)

∆W = −∫ xmax

δx(x)∆F (x)dx

dominates B”.

xminU(x)dF (x)

∆W = −∫ xmax

δx(x)∆F (x)dx

dominates B”.

xminU(x)dF (x)

∆W = −∫ xmax

δx(x)∆F (x)dx

dominates B”.

First-order conditions for the bivariate case

With two variables we integrate by parts the following function:

∫ ymax

∫ xmax

U(x)f (x , y)dxdy

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

∆W = −∫ xmax

δx(x , ymax)∆F x(x)dx

−∫ ymax

δy(xmax , y)∆F y (y)dy +

∫ ymax

∫ xmax

δxδy(x , y)∆F (x , y)dxdy

∫ ymax

∫ xmax

U(x)f (x , y)dxdy

∆W = −∫ xmax

−∫ ymax

∫ ymax

∫ xmax

∫ ymax

∫ xmax

U(x)f (x , y)dxdy

∆W = −∫ xmax

−∫ ymax

∫ ymax

∫ xmax

...or survival functions:

∆W =

∫ xmax

δx(x , ymin)∆F x(x)dx

∫ ymax

δy(xmin, y)∆F y (y)dy+

∫ ymax

∫ xmax

...or survival functions:

∆W =

∫ xmax

δx(x , ymin)∆F x(x)dx

∫ ymax

δy(xmin, y)∆F y (y)dy+

∫ ymax

∫ xmax

Notice the appearance of δ2Uδxδy

(x , y).

This cross-partial derivative determineswhether x and y are ALEP complements or ALEP substitutes in theircontributions to U. Four conditions stem from the two previous equations:

1. A condition for monotonically increasing functions with ALEP substitute

arguments (e.g. δ2Uδxδy

(x , y) ≤ 0):

∀x , y : ∆W ≥ 0∀U | δUδi

(x , y) ≥ 0 ∧ δ2U

δxδy(x , y) ≤ 0↔

∆F x(x),∆F y (y),∆F (x , y) ≤ 0

Notice that ∀x , y : ∆F (x , y) ≤ 0 suffices to ascertain ∆W ≥ 0.

(x , y). This cross-partial derivative determineswhether x and y are ALEP complements or ALEP substitutes in theircontributions to U.

Four conditions stem from the two previous equations:

(x , y) ≤ 0):

∀x , y : ∆W ≥ 0∀U | δUδi

(x , y) ≥ 0 ∧ δ2U

δxδy(x , y) ≤ 0↔

∆F x(x),∆F y (y),∆F (x , y) ≤ 0

(x , y). This cross-partial derivative determineswhether x and y are ALEP complements or ALEP substitutes in theircontributions to U. Four conditions stem from the two previous equations:

(x , y) ≤ 0):

∀x , y : ∆W ≥ 0∀U | δUδi

(x , y) ≥ 0 ∧ δ2U

δxδy(x , y) ≤ 0↔

∆F x(x),∆F y (y),∆F (x , y) ≤ 0

(x , y) ≤ 0):

∀x , y : ∆W ≥ 0∀U | δUδi

(x , y) ≥ 0 ∧ δ2U

δxδy(x , y) ≤ 0↔

∆F x(x),∆F y (y),∆F (x , y) ≤ 0

(x , y) ≤ 0):

∀x , y : ∆W ≥ 0∀U | δUδi

(x , y) ≥ 0 ∧ δ2U

δxδy(x , y) ≤ 0↔

∆F x(x),∆F y (y),∆F (x , y) ≤ 0

2. A condition for monotonically increasing functions with ALEPcomplement arguments (e.g. δ2U

δxδy (x , y) ≥ 0):

∀x , y : ∆W ≥ 0∀U | δUδi

(x , y) ≥ 0 ∧ δ2U

δxδy(x , y) ≥ 0↔

∆F x(x),∆F y (y),∆F (x , y) ≥ 0

Notice that ∀x , y : ∆F (x , y) ≥ 0 suffices to ascertain ∆W ≥ 0.

2. A condition for monotonically increasing functions with ALEPcomplement arguments (e.g. δ2U

δxδy (x , y) ≥ 0):

∀x , y : ∆W ≥ 0∀U | δUδi

(x , y) ≥ 0 ∧ δ2U

δxδy(x , y) ≥ 0↔

∆F x(x),∆F y (y),∆F (x , y) ≥ 0

Notice that ∀x , y : ∆F (x , y) ≥ 0 suffices to ascertain ∆W ≥ 0.

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

2Uδxδy (x , y) = 0):

∀x , y : ∆W ≥ 0∀U | δUδi

(x , y) ≥ 0 ∧ δ2U

δxδy(x , y) = 0↔

∆F x(x),∆F y (y) ≤ 0

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

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

2Uδxδy (x , y) = 0):

∀x , y : ∆W ≥ 0∀U | δUδi

(x , y) ≥ 0 ∧ δ2U

δxδy(x , y) = 0↔

∆F x(x),∆F y (y) ≤ 0

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

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

∀x , y : ∆W ≥ 0∀U | δUδi

(x , y) ≥ 0↔

∆F x(x),∆F y (y),∆F (x , y) ≤ 0∧∆F x(x),∆F y (y),∆F (x , y) ≥ 0

Notice that, in this case, it is necessary and sufficient to test∆F (x , y) ≤ 0 and ∆F (x , y) ≥ 0.

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

∀x , y : ∆W ≥ 0∀U | δUδi

(x , y) ≥ 0↔

∆F x(x),∆F y (y),∆F (x , y) ≤ 0∧∆F x(x),∆F y (y),∆F (x , y) ≥ 0

Notice that, in this case, it is necessary and sufficient to test∆F (x , y) ≤ 0 and ∆F (x , y) ≥ 0.

First-order conditions for the general multivariate case

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

They include allmarginal distributions, F i (i) and all cumulative and survival joints up toF (x , y , ...,w , z) and F (x , y , ...,w , z).

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, Uij < 0 (ALEP substitution), Uijk > 0 and so forth.

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 : ∆W ≥ 0∀U | δUδi,δ3U

δiδjδk, ... ≥ 0

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

∀x , y , ..., z : ∆W ≥ 0∀U | δUδi,δ3U

Multidimensional stochastic dominance and complementarity/substitutability

ALEP relationships and multidimensional dominanceconditions

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

ALEP substitution and ∆F (x , y) ≤ 0

��

� ��

��

ALEP complementarity and ∆F (x , y) ≥ 0

��

Applying dominance conditions to poverty measurement

Social poverty functions are characterized by:

I Being additive and symmetric across individuals:P(X ; z) = 1

∑Nn=1 Pn(x ; z).

I Higher poverty is worse.

I Monotonicity: Higher x should not increase poverty:δPn(x ;z)

δx ≤ 0

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

∑Nn=1 Pn(x ; z).

δx ≤ 0

∑Nn=1 Pn(x ; z).

δx ≤ 0

∑Nn=1 Pn(x ; z).

δx ≤ 0

∑Nn=1 Pn(x ; z).

δx ≤ 0

Tests for univariate and multivariate stochastic dominance

The test of Barrett and Donald (2003)

I This test is a type of Kolmogorov-Smirnov test ofhomogeneity of distributions.

I The logic is very intuitive: dominance conditions are based oncomparing integrals of FA(z) with FB(z) for a range of z ,depending on the order of dominance (e.g. for first order, justcompare cumulative or survival distributions).

I This test is a type of Kolmogorov-Smirnov test ofhomogeneity of distributions.

I The logic is very intuitive: dominance conditions are based oncomparing integrals of FA(z) with FB(z) for a range of z ,depending on the order of dominance (e.g. for first order, justcompare cumulative or survival distributions).

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

1. A dominates B: FA(z) ≤ FB(z)∀z ∧ ∃z |FA < FB(z)

2. B dominates A: FB(z) ≤ FA(z)∀z ∧ ∃z |FB < FA(z)

3. No dominance because A=B: FA(z) = FB(z)∀z4. No dominance because A and B cross:∃z1|FA(z1) < FB(z1) ∧ ∃z2|FB(z2) < FA(z2)

3. No dominance because A=B: FA(z) = FB(z)∀z

4. No dominance because A and B cross:∃z1|FA(z1) < FB(z1) ∧ ∃z2|FB(z2) < FA(z2)

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

H0 : FA(z) ≤ FB(z)∀z ∈ [zmin, zmax ]

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

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

1. Test:

H0(FA ≤ FB ) vs

H1(FA > FB )

A < B or

2. Test:

H0(FB ≤ FA) vs

H1(FB > FA)

Not reject

Switch

reject

B < A or

crossing

2. Test:

H0(FB ≤ FA) vs

H1(FB > FA)

A and B cross

Reject

Not rej

Switch

Reject

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

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

I Estimate the supremum and multiply by a function of thesample sizes: S = ( NM

N+M )1/2supz [FA(z)− FB(z)]

I S is the statistic we need. We now need to know how likely itis for this value to appear under the null hypothesis.

I There are different procedures to derive the distribution underthe null hypothesis. Barrett and Donald (2003) develop twotypes. We are going to show one of them: a bootstrapmethod.

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):

2. For each subsample (say 1000) we estimate:Sr = ( NM

N+M )1/2supz [FA(z ; r)− FB(z ; r)]

3. The p-value is then approximated by:pA,B ' 1

∑Rr=1 1(Sr > S)

4. If, say, pA,B < 0.01 we reject the null hypothesis: under thehypothesis a value like S is very unlikely.

∑Rr=1 1(Sr > S)

Concluding remarks

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

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

I 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.

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

I 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.

Concluding remarks

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