This note extends the Duclos Esteban Ray Polarization Measure to situations in which agents are characterized in many dimensions where the characteristics are both discrete and continuous. It provides a formula for its asymptotic variance and illustrates its use in an application to a sample of Chinese Urban Households where household income, living space and educational status of the household head are the agent characteristics.Polarization Multidimensional estimation Inference

Gordon Anderson

English

Research Papers in Economics

University of Toronto 
Department of Economics 
 
November 05, 2010
By Gordon Anderson
Polarization Measurement and Inference in Many Dimensions:
A Note
Working Paper 414 
 
 
 
 
Polarization Measurement and Inference in Many Dimensions: 
                                            A Note 
 
 
 
 
                                   Gordon Anderson 
 
 
 
                            Department of Economics. 
                               University of Toronto 
 
 
 
                                       October 2010 
 
 
 
                                    Very preliminary 
 
 
  
Many thanks Oliver Linton for help with multivariate Kernel Estimation and to Jean-
Yves Duclos and Joan Esteban for helpful comments on an earlier draft of this note.Introduction. 
 
The functionings and capabilities approach to wellbeing measurement (Sen (1992)) has 
given considerable impetus to multidimensional analyses (Grusky and Kanbur (2006)). 
Much work has been done on extending one dimensional wellbeing measures to many 
dimensions in the context of poverty (Duclos, Sahn and Younger (2006)) and inequality 
measurement (Maassoumi (1987), (1999) Koshevoy and .Mosler (1997), Tsui (1995) 
Anderson (2008)). While Anderson (2008) and Gigliarano and Mosler (2008) provide 
multivariate bi-polarization measures for two identified groups, multivariate polarization 
measures have not been developed for the more general non-identified many group case. 
Here the most popular general polarization index (Esteban and Ray (1994), Duclos, 
Esteban and Ray (2004)) is extended to many discrete and continuous dimensions. Its 
implementation is illustrated with an application to Chinese urban household data drawn 
from six provinces in the years 1987 and 2001 (years spanning the growth and 
urbanization period subsequent to the economic reforms). 
 
The Extension. 
 
The multivariate generalization of the Duclos Esteban and Ray (2004) (DER) 
Polarization index is, like DER, based upon the sample equivalents of the population 
concepts. For scalar continuous x with distribution function F(x) the DER index
1 is given 
by: 
 
                                        () | | () () [ 1 ] Pf x y x d F y d F x

  
 
Which is asymptotically normally distributed with an asymptotic variance V given by: 
 
()
00
v a r ( 1 )() | | | | () () () 2| | | | () () [ 2 ] fy
y
V fy y xd Fx y fx d Fx y x fx d Fx
 
 
  
      
  
Development of the polarization index was founded on a set of axioms that such an index 
should obey, the axioms
2 concern changes (squeezes and slides) in the uni-modal sub 
distributions in the mixture distribution that is f(x). The resultant index reflects the two 
primary factors that underlay polarization, the alienation or distance between groups 
(given by |y-x|) and the association within a group (given by f(x)
α+1). Here α is a 
polarization sensitivity parameter
3 chosen by the investigator such that 0.25  ≤ α ≤ 1 with 
higher values of α corresponding to increased sensitivity. The same axioms can be 
applied when x is a vector. 
                                                 
1 A similar discrete variable index is provided in Esteban and Ray (1994). 
2 Essentially concentrations (squeezes) of the sub distributions and increases in the distance between them 
should not reduce the polarization measure. 
3 Note when α = 0 the index is in essence twice the Gini coefficient thus a similar value in the following 
would provide a multivariate version of a Gini like coefficient and its variance. Let wi and zi be jointly distributed vectors describing the status of the i’th agent with wi 
having dimension k x 1 and zi having dimension h x 1 with i =1,..,n being the elements of 
the sample, where w are continuous variables and z are discrete variables. The continuous 
variables all reflect wellbeing positively and for convenience are defined on R
+ and the 
discrete variables are ordered integers reflecting positive wellbeing in the same fashion
4. 
The joint density of the w’s for a given configuration of z’s is fz(w|z) and the joint 
probability of the z’s is p(z) so that the joint density of the w’s and z’s for the i’th agent 
with discrete variables zi is given by f(wi,zi) = fi(wi |zi )p(zi). Let xi be the stacked 
vector wi | zi then the dimension normalized Euclidean distance between agents i and j 
||xi-xj|| is well defined and may be written as: 
 
                                            
2
1
()
|| ||
Q
iq jq
q
ij
xx
xx
Q

 
   
 
where xiq is the q’th element of the vector xi where Q = k+h. For notational convenience 
denote the first k continuous components of the vector x as x{c}. 
 
From the above, a multivariate version of [1] is given by: 
                      {} {} ((| ) ( ) ) | | | | ( ) ( )[ 1 ] zc
zx zy
Pf w z p z y x d F y d F x



   c a
Here summation is over the domain of each element of the z vector and integration is 
over the domain of each element of the w vector. As in the univariate case the alienation 
or distance between groups is given by ||y-x|| and the association within a group given by 
f(w,z)
α  in exactly the same fashion. By employing kernel estimates of the conditional 
multivariate distributions and sample estimates of the population proportions p(z) the 
sample equivalents, given n observations on Q variables in an n x Q matrix X with 
typical element xiq i = 1,.., n, q = 1,..,Q and typical row xi the index can be seen to be
5: 
                                 
 2
11 1
2
(( | )() ) ( )
Q nn
ii i i q j q
ij q
fw zpz x x
P
nQ


 


 
   
The multivariate version of [2] is given by: 
 
0
()
0
(1 )( ( | ) ( )) || || ( | ) ( )
var [2 ]
|| || ( ( | ) ( )) ( | ) ( ) 2 || || ( ( | ) ( )) ( | ) ( )
z
fy
zz y
fwzPz y xd FwzPz
V a
y fwz P z d Fwz P z y x fwz P z d Fwz P z






 
   
    
    
 

 
 
                                                 
4 For example the continuously measured variables may represent levels of consumption, leisure and 
housing stock whereas the discretely measured variables may reflect educational, health or freedom status. 
5 In DER the columns of X are mean standardized and assumed to reside in the positive orthant.  Where after ordering the vectors xi on ||xi|| as xi
o, the first, second and third terms of the 
i’th element of the variance vector may be respectively estimated in an obvious fashion 
as: 
                                             



2
11
2
1 1
2
1
(|) ( ) ( )
(|) ( )
(|) ( ) ( )
2
Q n
ii i i k j k
jk
n Q
ii i ik
j k
Q n
oo
ii i i k j k
ji k
fw zPz x x
nQ
fw zPz x
Qn
fw zPz x x
nQ




 




 

 
 
Essentially the generalization simply involves employing the dimension normalized 
Euclidean norm for |y-x| and |y| when they are Q dimensioned vectors together with 
multivariate kernel estimates of f(w|z)p(z) for f(x) and f(y). 
 
An Application. 
 
There is a suspicion that the economic reforms (including the one child policy) in China 
together with the massive urbanization over the period changed fundamentally the nature 
of urban households. Data on two independent surveys of urban households from three 
coastal and three interior provinces
6 in China for the years 1987 (for which there were 
3651 observations) and 2001 (for which there were 4297 observations) a period over 
which the reforms took effect. The data were used to generate observations on log adult 
equivalent household income (at constant prices), adult equivalent
7 living space (in 
square meters) and an integer index of the education level of the head of household. 
Table 1 presents the summary statistics. Considerable increases in both equivalent 
incomes and living space (due in part to growth and in part to reductions in family size) 
and educational attainment are evident. To calculate the polarization statistic the 
continuous multivariate mean standardized pdf’s were estimated using a multivariate 
standard normal kernel with a window width h = 1.06*σ(x).*n
-(1/(4+k))  (Silverman (1986)). 
The seven outcome educational scale was condensed to a three outcome scale, 1, 2 and 3 
corresponding to high, medium and low educational attainments. Table 2 reports the 
polarization indices and standard errors for the continuous univariate measures as per 
DER and Table 3 reports the multivariate measures.  
 
                                                 
6 The coastal provinces were Jilin, Shandong and Guangdong the interior, Sichuan, Shaanxi and Hubei . 
7 Equivalization was effected using the square root rule (Brady and Barber (1948)). Table 1. Sumamary Statistics (1987 n=3651, 2001 n = 4297) 
  1987 2001 1987 2001 1987  2001 
 equivalized 
log income  
equivalized 
log income 
at 1987 
prices. 
equivalized 
living 
space (sq 
meters)  
equivalized 
house 
space(sq 
meters)  
Education 
of 
household 
head 
Education 
of 
household 
head 
Mean       
Median    
Std Dev 
4.8227503    
4.8578656    
0.4194296 
8.8212479    
8.9074170    
0.8489056 
17.117842    
15.205262    
9.4884068 
24.720038    
22.516660    
12.621758 
3.1035333  
4.0000000  
1.5753986 
3.4263440  
4.0000000  
1.6137025
 
Table 2. Univariate Measures (Standard Errors in brackets) 
Sensitivity 
Parameter 
(α) 
Income 1987  Income 2001  Housing 1987  Housing 2001 
0.25  0.11731620      
(0.00031190748) 
0.13209502      
(0.00029271751)
0.41820690      
(0.00072261910)
0.43142905      
(0.00066703302)
0.5  0.15241276      
(0.00031033800) 
0.16609740      
(0.00027844521) 
0.36340574      
(0.00039098378)
0.37452838      
(0.00036208622)
0.75  0.20407695      
(0.00035301313) 
0.21356717      
(0.00029870682)
0.32713093       
(0.00026857448)
0.33492727       
(0.00024186942)
1.0  0.27893908       
(0.00043703932) 
0.27933600       
(0.00034867352)
0.30070666       
(0.00020994170)  
0.30512450       
(0.00018199596) 
 
For all values of the polarization sensitivity parameter the index shows an increase for 
both income and house space variables however, based upon the samples in the two years 
being independent of one another, it is seldom significant at usual levels of significance.  
 
Table 3. Multivariate Measures (Standard Errors in brackets) 
Sensitivity 
Parameter α 
Income and 
Housing 1987 
Income and 
housing 2001 
Income,Housing 
and Edu  1987 
Income,Housing 
and Edu  2001 
0.25  0.41364256       
(0.001067345) 
0.41695109       
(0.00070989495)
0.39928822      
(0.00096876272)
  0.39936785      
(0.00085576967) 
0.5  0.48998936       
(0.001102657) 
0.47347036       
(0.00074058380) 
  0.39315119       
(0.00097944440)
0.36022864       
(0.00086639805)
0.75  0.61216969       
(0.0011584829) 
0.55999669       
(0.00078619134) 
0.40427540       
(0.00099784998)
0.33319929       
(0.00088564705)
1.0  0.79130432       
(0.0012435013) 
0.68103002       
(0.00085357946)
0.42945411       
(0.0010262811) 
0.31408093       
(0.00091355524)
 
The joint continuous distributions exhibit different effects to the univariate cases with 
depolarization being the norm in almost all cases and significantly so at higher orders of 
polarization sensitivity. This reflects the homogenization of urban households over the 
period of the reforms.  References. 
 
Anderson, G.J. (2008) “The Empirical Assessment of Multidimensional Welfare, 
Inequality and Poverty: Sample Weighted Generalizations of the Kolmorogorov-Smirnov 
Two Sample Test for Stochastic Dominance.” Journal of Economic Inequality 6 73-87. 
 
Anderson G.J. (2010) “Polarization of the Poor: Multivariate Relative Poverty 
Measurement Sans Frontiers” Forthcoming Review of Income and Wealth  
 
Brady D.S. and H.A. Barber (1948) “The Pattern of Food Expenditures” Review of 
Economics and Statistics 30 198-206. 
 
Duclos J-Y, J.Esteban and D. Ray (2004) “Polarization: Concepts, Measurement, 
Estimation” Econometrica 72 pp. 1737-1772. 
 
Duclos J-Y., Sahn, D., and S.D. Younger (2006) “Robust Multidimensional Poverty 
Comparisons” The Economic Journal 116 943-968. 
 
Esteban, J.-M. and Ray, D. (1994). “On the measurement of polarization.” Econometrica  
62, 819–51. 
Gigliarano C. and K. Mosler  “Constructing Indices of Multivariate Polarization.” 
Forthcoming Journal of Economic Inequality 2008. 
 
Grusky, D., and R. Kanbur (eds.) (2006): Poverty and Inequality. Stanford Univer- 
sity Press: Stanford, California. 
 
Koshevoy G.A. and K.Mosler (1997) “Multivariate Gini Indices” Journal of Multivariate 
Analysis 60 252-276. 
 
Maasoumi E. (1986) “The Measurement and Decomposition of Multidimensional 
Inequality” Econometrica 54 771-779 
 
Maasoumi E. (1999) “Multidimensioned Approaches to Welfare Analysis” chapter 15 
Handbook of Income Inequality Measurement J Silber editor Kluwer Academic 
Publishers Boston. 
 
Sen, A. K. (1992): Inequality Reexamined. Harvard University Press: Cambridge, Mas- 
sachusetts. 
 
Silverman B.W. (1986) Density estimation for Statistics and Data Analysis. Chapman 
Hall. 
 
Tsui K-Y (1995) “Multidimensional Generalizations of the Relative and Absolute 
Inequality Indices: The Atkinson-Kolm-Sen Approach. Journal of Economic Theory 67 
pp 251-265. 

Polarization Measurement and Inference in Many Dimensions: A Note

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