Multivariate Methods

Welfare is a “many splendored thing”

• One of the indisputable aspects of empirical welfare measurement is that, whatever it is we are trying to measure be it poverty, inequality or polarization, it probably has many dimensions.

• There have been some very recent attempts to capture these features.

Multivariate Considerations

• Atkinson and Bourguignon extended SD techniques to more than one dimension (2).

• Life gets much more complicated here both theoretically (largely due to the conditions for concavity and convexity in multidimensional functions are much more complicated) and for empirical practicality reasons (curse of dimensionality).

The 3 variable concave casesufficient condition ∆F(x1,x2,x3) ≤ 0

for all x1,x2,x3

The Convex case: (a counter- cumulative version of the above)

Ibbott, Crawford and Anderson

• Generalize Anderson Univariate Method to two dimensions. Ibbott limited to 1st order dominance, Crawford considers up to 3rd order.

• Find difficulties with the curse of dimensionality problem (number of parameters increasing “exponentially” with dimensionality).

• Anderson considers K-S test in three dimensions for 1st order dominance. Does not exhibit curse of dimensionality problem but difficult to consider higher orders.

McCaig and Yatchew

• Extends the application of Hall and Yatchew integrated squared difference approach to multivariate environments.

• Promising because it does not appear to suffer from curse of dimensionality problem that bedevils the Ibbott and Crawford approach and it can address higher orders of dominance which the K-S approach used in Anderson has not yet been developed to do.

The Poverty Frontier issue

• Define a cut-off for each good• Set Intersection (agent in poverty if below the

cut-off for all goods) - the perfect substitute case• Set Union (agent in poverty if below the cut-off

for at least one good) - the perfect complements case

• Peculiar consequences when the number of goods gets large.

• Trade-offs between goods?

Duclos, Sahn and Younger

• Duclos J-Y., Sahn, D., and S.D. Younger (2006) “Robust Multidimensional Poverty Comparisons” The Economic Journal 116 943-968.

• Extends the Davidson and Duclos incomplete moments technique to multivariate environments.

• Shows the Atkinson result about poverty measurement also applies in multivariate situations which is really important because the multivariate poverty frontier is even more difficult to establish than in the univariate case.

The Distance Function Approach(aggregation function {T:leisure

time Y:income} W=T0.5Y0.5)Figure 2

(2)

(1)

(4) (3)

(6)

(5)

(2)

(1)

(4) (3)

(6)

(5)

Table 1

Household D(xi ,W)

Low ref. High ref (1) 0.70 1.13 (2) 0.62 1.00 (3) 0.70 1.14 (4) 0.91 1.47 (5) 0.79 1.28 (6) 1.00 1.61

The Poverty Boundary and the “Rawlsian Frontier”

• The “Rawlsian Frontier” T = (minW)2/Y where minW is the welfare of the poorest agent.

• The “Poverty Boundary” T = (W*)2/Y where W* is the welfare below which an agent is deemed poor.

• Different Aggregators →Different Frontiers and Boundaries. An estimation problem.

The Lower Convex Hull the “Rawlsian Frontier” and the Union

and Intersection Sets.Figure 4. The Lower Convex Hull, 25%, 50% and 75% poverty boundaries and median poverty cutoffs for two Goods which define the Union and Intersection sets.

The Welfare (Deprivation) Measure

• D(xi) min d { d : W(dxi) = W*, d > 0, for all monotone, quasi-concave W}

• Monotonicity: attributes are such that if the agent had more of any of them their well-being would not decrease.

• Quasi-concavity: as the level of some attribute rises, well-being rises at a non-increasing rate.

• The lower bound of this index is calculated by setting W* to be the lower convex hull of the data which becomes our “Empirical Rawlsian Frontier”.

• D: the factor by which the vector x has to be scaled down along a ray from the origin in order for it to be equal to a point on the Rawlsian Frontier. D measures deprivation (i.e. D=1 implies x on the frontier, D < 1 implies x is above the frontier) and –D measures welfare

Implementation

• Observe agents in two societies, pool the data and compute the Rawlsian Frontier and the distance values for each agent.

• Scale up the Rawlsian Frontier to generate a poverty frontier (e.g. choose a scaling factor such that 25% of the pooled sample is below the frontier)

• Compare the distance values for the two societies (e.g. membership of the Rawlsian Frontier, % of a particular society below the poverty frontier, average distance of agents in a particular society below the frontier etc.)

Quantifying Multivariate Polarization Between Two Groups

• Esteban and Ray (1994) in a seminal paper posited a collection of axioms whose consequences should be reflected in any Polarization measure. In this and a subsequent paper (together with Duclos) they proposed indices which capture the concept in a wide range of circumstances in a univariate environment.

• The axioms are based upon a so-called Identification-Alienation nexus wherein notions of polarization are fostered jointly by an agent’s sense of increasing within-group identity or association and between-group distance or alienation (i.e. the divergence of the poor-non poor).

• Here the focus is on two distinct (but not necessarily identified) groups and the degree of polarization they experience in terms of a number of variables (Gigliano and Mosler (2008) have developed such a measure for identified groups).

• Two measures of between group polarization are considered, The Overlap Measure and the Polarization Trapezoid.

The Overlap Measure:∫∫..∫min{f(u,v,..,z)g(u,v,..,z)}dudv..dz

(Anderson, Ge and Leo (2009))

• Given the dominance condition is satisfied, one way of gauging the extent of relative poverty is to consider the overlap between the poor and non-poor distributions (The greater the overlap the less there is relative poverty).

• It is a very natural measure (always a number between 0 and 1), is readily calculated in multivariate contexts and has a well defined asymptotic sampling distribution (Anderson, Linton, Wang (2008)).

A Multivariate Poverty Index: The Area of the Polarization Trapezoid

(fpoormode+frichmode)*distance/2

Problems

• The overlap measure obviously does not work if distributions do not overlap in every dimension, or overlap in a minimal sense (the “information-less tails” problem see Davidson and Duclos (2008)) or if distributions are not separately identified.

• On the other hand it has a well defined sampling distribution (asymptotically normal, see Anderson Linton and Wang (2009)).

• The trapezoidal measure works everywhere the overlap does not but as yet we do not have its sampling distribution in the multivariate case (Update!!! We do have its distribution in the univariate case, Anderson and Linton (2009) and we’re fairly close to the multivariate result).

An Application: Poor and Non-Poor Countries and the demise of Africa

• To illustrate these ideas we consider the progress of the joint distribution of log gross national product per capita and log life expectancy for 151 countries from 1990 to 2005.

• We consider the progress of the world distribution as an entity in itself identifying the poor and non poor groups from modal points in the distribution.

• Separately identifying African nations as the poor group we compare their progress relative to the rest of the world.

Countries in The Sample.• Albania, Algeria, Angola, Argentina, Armenia, Australia, Austria, Azerbaijan,• Bahrain, Bangladesh, Belarus, Belgium, Belize, Benin, Bhutan, Bolivia, Botswana,• Brazil, Bulgaria, Burkina Faso, Burundi, Cameroon, Canada, Cape Verde, • Central African Republic, Chad, Chile, China, Colombia, Comoros, Congo, Dem. Rep.,• Congo, Rep., Costa Rica, Cote d'Ivoire, Croatia, Czech Republic, Denmark, Djibouti,• Dominican Republic, Ecuador, Egypt, El Salvador, Estonia, Ethiopia, Finland, France,• Gabon, Gambia, Georgia, Germany, Ghana, Greece, Guatemala, Guinea, Guinea-Bissau,• Guyana, Haiti, Honduras, Hong Kong (China), Hungary, Iceland, India, Indonesia, • Iran, Ireland, Israel, Italy, Jamaica, Japan, Jordan, Kazakhstan, Kenya, Korea, • Kyrgyz Republic, Lao PDR, Latvia, Lebanon, Lesotho, Liberia, Lithuania, Luxembourg,• Macao (China), Madagascar, Malawi, Malaysia, Mali, Malta, Mauritania, Mauritius, • Mexico, Micronesia (Fed. Sts.), Moldova, Mongolia, Morocco, Namibia, Nepal, • Netherlands, New Zealand, Nicaragua, Niger, Nigeria, Norway, Pakistan, Panama, • Paraguay, Peru, Philippines, Poland, Portugal, Russian Federation, Rwanda, Samoa,• Sao Tome and Principe, Saudi Arabia, Senegal, Sierra Leone, Singapore, • Slovak Republic, Slovenia, Solomon Islands, South Africa, Spain, Sri Lanka, • St. Vincent and the Grenadines, Sudan, Suriname, Swaziland, Sweden, Switzerland, • Syria, Tajikistan, Tanzania, Thailand, Togo, Tonga, Trinidad and Tobago, Tunisia, • Turkey, Uganda, Ukraine, United Arab Emirates, United Kingdom, United States, • Uruguay, Uzbekistan, Vanuatu, Venezuela, Vietnam, Yemen, Zambia, Zimbabwe.

The “World” Comparisons: the unidentified case.

Table 1. Low (Poor) Mode High (Non-Poor) Mode Year

Height Location (Income, Life expectancy)

Height Location (Income, Life expectancy)

Relative Poverty Index

1990 0.3598 5.9089 4.1202 0.0966 10.0507 4.3362 0.2531 1995 0.3421 6.2625 4.1607 0.0942 10.1059 4.3478 0.2226 2000 0.3253 6.5184 4.1795 0.0923 10.2002 4.3594 0.2022 2005 0.2960 6.8493 4.2014 0.0902 10.2647 4.3720 0.1429 Note. 1) All of the densities and overlap measures in this paper were computed using population weighted versions of the multivariate kernel density estimator K(x)=(4(1-x'x)^3)/(π)) (see Silverman (1986) equation 4.6). 2) The locations of the modes respect the dominance criterion in that poor income and life expectancy are respectively less than the non-poor income and life expectancy in every case. 3) The Poverty Polarization index indicates considerable world income convergence or de-polarization which reflects considerable progress made by China and India (who are also heavily weighted in the data).

Africa and “The Rest”• Separating out Africa and considering what I refer to as

the identified version of the comparator provides a very different story.

• When the separate distributions are identified we can consider the degree to which they overlap as a convenient measure of depolarization (Anderson Linton Wang (2008)) in multivariate environments.

• Unfortunately the overlap measure will not work if the distributions do not intersect in a dimension (whereas the poverty polarization measure will always work).

• The modal point in the Rest of the World distribution is clearly the poor group of our earlier analysis absent the African nations.

Results for the Africa-Rest Comparisons. The Dominance

Relation

For the “poverty” interpretation of the polarization index it is necessary for the African Outcomes to be dominated by the rest of the world outcomes. The following is a Multivariate Dominance test of that proposition which has to be established by showing that Africa does not dominate the rest but the rest does dominate Africa (see Anderson (2007) for details). Table 2. Kolmgorov Smirnov Tests for Dominance Comparisons

Africa Dominates Rest of World

Rest of World Dominates Africa

Year

Max -D (P(√n*D < λ)

Max D (P(√n*D < λ)

2005 2000 1995 1990

0.000628 0.001723 7.52e-005 7.68e-005

7.89e-007 5.94e-006 1.13e-008 1.18e-008

0.526846 0.475490 0.415322 0.315112

0.426004 0.363762 0.291769 0.180115

Africa’s Overlap with the Rest of the World

Table 3. Overlap Tests for De-Polarization. 2005 Overlap Standard Error

1990 Overlap Standard Error

Standard Normal Test

0.4311 0.0403 0.6443 0.0390 -3.7268 One attraction of the overlap test is that it has a known distribution (Anderson, Linton and Wang (2008)) which here is used to test and reject the hypothesis of convergence or depolarization of Africa with the rest of the world.

Africa’s Poverty Polarization.

Table 3. The Polarization Index. Year Euclidean

Distance (fp(xmp)+fr(xmp))/2 BIPOL Population

share of Africa

SPPI (%)

2005 0.2941 0.3460 0.1018 12.71% 1.2939 2000 0.2489 0.3566 0.0888 13.75% 1.2211 1995 0.2327 0.3617 0.0841 12.94% 1.0270 1990 0.1413 0.3769 0.0533 10.88% 0.5799 Note: 1) the Alienation component of the index has increased substantially over the period whereas the Within Group association component has diminished somewhat over the period. 2) However their net effect on BIPOL has seen it increase over the period 3) This suggests that the representative agent has experienced slightly diminished within group identity together with increased between group alienation combining for an increased sense of poverty polarization. 4) Variations in Africa’s share of the population in the sample have failed to mask the overall upward trend in Africa’s contribution to world poverty.

Conclusions

• Simple to calculate “Poverty Polarization” measures have been proposed which circumvent the need to define poverty frontiers and which are amenable to calculation in multivariate environments.

• The advantage of the Polarization Trapezoid over overlap measures is that a) it does not require intersection of the comparator distributions in all dimensions and b) it can be used in mixture distributions when the poor and non poor are not identified.

• Application to world income and life expectancy distributions revealed an overall improvement in the lot of the poor but when applied specifically to Africa it revealed that Africa’s relative position is deteriorating.