WPS7514
Policy Research Working Paper 7514
Unpacking the MPI
A Decomposition Approach of Changes in Multidimensional
Poverty Headcounts
Jorge Eduardo Pérez Pérez
Carlos Rodríguez Castelán
Jose Daniel Trujillo
Daniel Valderrama
Poverty and Equity Global Practice Group
December 2015
Policy Research Working Paper 7514
Abstract
Multidimensional measures of poverty have become stan- on counterfactual simulations—to break up the changes of
dard as complementary indicators of poverty in many the multidimensional poverty headcount into the variation
countries. Multidimensional poverty calculations typically attributed to each of its dimensions. This paper examines
comprise three indices: the multidimensional headcount, the potential issues of using counterfactual simulations in
the average deprivation share among the poor, and the this framework, proposes approaches to assess these issues in
adjusted headcount ratio. While several decomposition real applications, and suggests a methodology based on rank
methodologies are available for the last index, less atten- preservation within strata, which performs positively in sim-
tion has been paid to decomposing the multidimensional ulations. The methodology is applied in the context of the
headcount, despite the attention it receives from policy recent reduction of multidimensional poverty in Colom-
makers. This paper proposes an application of existing bia, finding that the dimensions associated with education
methodologies that decompose welfare aggregates—based and health are the main drivers behind the poverty decline.
This paper is a product of the Poverty and Equity Global Practice Group. It is part of a larger effort by the World Bank to
provide open access to its research and make a contribution to development policy discussions around the world. Policy
Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at
jorge_perez@brown.edu, crodriguezc@worldbank.org, jdtrujillos@dane.gov.co, and dvalderramagonza@worldbank.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
Unpacking the MPI: A Decomposition
Approach of Changes in Multidimensional
Poverty Headcounts∗
Jorge Eduardo P´ erez†
erez P´
Carlos Rodr´ an‡
ıguez Castel´
e Daniel Trujillo§
Jos´
Daniel Valderrama¶
Keywords Multidimensional Poverty, Decomposition and Non Parametric Meth-
ods, Shapley Value, Poverty Transitions
JEL Codes I32, I38, C14
∗
´ Wendy Cunningham, James Foster, and partici-
The authors would like to thank Paola Ballon,
pants of the 2014 Summer Initiative for Research on Poverty, Inequality and Gender of the Poverty
Global Practice at the World Bank, and the 2015 LACEA meeting for their useful comments and sug-
gestions. The views, ﬁndings, and conclusions expressed in this paper are entirely those of the au-
thors and do not necessarily reﬂect those of the World Bank, its Executive Board or member country
governments. P´ erez P´erez gratefully acknowledges ﬁnancial support from Fulbright-Colciencias.The
do-ﬁle code that implement this decomposition could be provided upon request
†
Corresponding author. Department of Economics, Brown University.
Email: jorge perez@brown.edu
‡
World Bank - Poverty Global Practice. Email: crodriguezc@worldbank.org
§
DANE. Email: jdtrujillos@dane.gov.co
¶
World Bank - Poverty Global Practice. Email: dvalderramagonza@worldbank.org
1 Introduction
Calculating multidimensional measures of poverty has become commonplace in many
developing countries, as a way to complement traditional monetary indicators. The
breadth of multidimensional measures, compared to traditional approaches, is viewed
as an advantage by researchers, enabling them to aggregate a large number of welfare-
associated variables into a single measure. Multidimensional measures are also at-
tractive to governments, as they are conducive to a more precise understanding of
the determinants of individual welfare and quality of life, allowing the identiﬁcation
of those ﬁelds where public policy may have a larger impact.
Despite this popularity, multidimensional measures are not free from the criti-
cism faced by other approaches to measuring poverty. As with any indicator that
attempts to summarize a complex phenomenon into a single index, multidimen-
sional poverty measures may be hard to interpret if unaccompanied by additional
information. In this sense, rather than focusing solely on the measure that aggre-
gates across dimensions, analyses of multidimensional poverty also tend to contain
separate information about each dimension. This, however, increases the number of
indicators to be tracked, and reduces the usefulness of the summary measures.
These difﬁculties become compounded when tracking the evolution of multidi-
mensional measures across time. An exact identiﬁcation of the contribution of each
dimension to the evolution of multidimensional measures would require tracking
the transition of individuals in and out of poverty, and the dimensions by individ-
ual. This exhaustive analysis would defeat the purpose of aggregation of poverty
measures.
Decomposition approaches, whereby poverty measures are split up into the con-
tribution of broadly deﬁned determinants of interest, are a reasonable compromise
between a comprehensive analysis and a completely aggregated one. As expressed
in Ferreira and Lugo (2013), these approaches aim for a “middle ground”, which can
be informative of multidimensional poverty as well as expeditious. While these ap-
proaches are deeply rooted in the traditional monetary poverty research, they have
been less developed in the literature of multidimensional measurement. Apablaza
and Yalonetzky (2013) propose approaches to decompose multidimensional poverty
measures statically or dynamically. Yet, their focus is on two out of the three mea-
sures of multidimensional poverty: the average deprivation share among the poor
2
and the adjusted headcount ratio. This is natural since these indicators have addi-
tive separability properties that make them easy to decompose, as opposed to the
multidimensional headcount, which is nonlinear in terms of deprivation scores. De-
compositions of the multidimensional headcount ratio, in particular dynamic ones,
appear to be lacking in attention.
Overlooking the multidimensional headcount ratio is conspicuous, considering
that this indicator is particularly important on its own. From an academic perspec-
tive, focusing on the evolution of the adjusted headcount ratio as opposed to the
multidimensional headcount ratio disregards variations in poverty that come only
from changes in the number of poor and arise only in the identiﬁcation step. Using
datasets form several countries, Apablaza et al. (2010) and Apablaza and Yalonetzky
(2013) show that declines in the adjusted headcount ratio are mostly due to changes
in the multidimensional headcount and not to changes in intensity. From a policy
perspective, the multidimensional headcount is frequently used by governments as
”the rate of multidimensional poverty”—instead of the adjusted headcount ratio—
as it is easily communicable and also comparable to the monetary rate of poverty.
In this paper, we propose an approach to decompose variations in the multidi-
mensional headcount ratio into changes attributed to different categories of dimen-
sions. Our approach builds on counterfactual simulation approaches, which have
traditionally been used to decompose poverty and inequality indicators. These ap-
proaches were ﬁrst proposed by Barros et al. (2006), and then extended in a series of
papers by Azevedo et al. (2012b, 2013a,b). The approach relies, ﬁrst, on expressing
the indicator of interest as a function of the distribution of its determinants over a
ﬁnite population. Then it replaces these distributions by counterfactual ones, where
one of the determinants has been altered.
Our approach allows us to estimate the extent to which an observed change in
a dimension—or category of dimensions—can explain the observed change in the
multidimensional headcount ratio. In the presence of panel data, it allows doing
this without separately tracking the transitions in and out of poverty of each indi-
vidual. In repeated cross-section data—when such tracking is impossible—the ap-
proach constitutes a good approximation of how much each dimension contributes
to the headcount’s change. In both cases, it summarizes separate information on the
headcount by dimension, weights and incidence.
Our paper contributes to the literature on the decomposition of poverty mea-
sures, providing a guideline to split up the changes of the multidimensional poverty
3
headcount into the variation attributed to each of its dimensions. By combining our
approach to decompose changes in the multidimensional headcount, and the ap-
proach to decompose changes in the average deprivation share among the poor of
Apablaza and Yalonetzky (2013), a decomposition of changes in the adjusted head-
count ratio by dimension can be obtained.
To illustrate the approach, we apply the decomposition analysis in the context
of the recent decline of multidimensional poverty in Colombia, between 2008 and
2012. The results show that education and health were the largest drivers behind
the poverty decline. In this sense, the paper also contributes to the literature on
poverty in Colombia, providing insights generally absent in the analysis of standard
measures.
Section 1 summarizes the technical aspects of the multidimensional poverty mea-
surement, and sets up the analytical framework for the rest of the paper. In Section 2
we review existing approaches to decompose multidimensional poverty measures,
covering both static and dynamic analyses. The technical aspects of the counterfac-
tual simulation methodology are reviewed in section 3, in order to identify poten-
tial issues that may arise when applying this methodology to the multidimensional
headcount. These issues are further analyzed in sub-section 3.2, which provides
some practical guidelines to address them.
In section 4 the methodology is used in repeated cross-sections looking at the
recent evolution of multidimensional poverty in Colombia. Conducting different
simulations, we identify a method based on stratiﬁcation and income ranks that per-
forms well in several scenarios. Section 5 concludes.
2 Multidimensional Poverty Measures and Their
Decompositions
This section provides a review of the existing multidimensional poverty measures.
It presents a description of the methodologies used to decompose the measures by
dimensions and over time, outlining the difﬁculties associated with decomposing
changes in the adjusted headcount ratio by dimension.
4
2.1 The Multidimensional Poverty Index measures
We follow Alkire and Santos (2010) and Alkire and Foster (2011) in the presentation
of the multidimensional poverty index (MPI), with the distinction that we do not
focus on the identiﬁcation, censoring and aggregation steps. Additionally, we depart
from the deprivation matrix notation and, instead, describe the index in terms of
random variables in order to ease the transition to our discussion in the next section.
Let i = 1, 2, . . . , n index individuals. Let yi = (yi1 , yi2 , . . . , yiD ) be a vector of
achievements for individual i in dimensions d = 1, 2, . . . , D. Let cd be the deprivation
cut-off of dimension d. An individual i is said to be deprived in dimension d if yid < cd .
Now let w = (w1 , w2 , . . . , wD ) be a vector of weights given to each dimension, such
that wd ≥ 0 and D 1
d=1 wd = 1 . Individual i is said to be multidimensionally poor if
D
pi ≡ 1 wd 1 (yid < cd ) > k =1 (1)
d=1
where 1 is the indicator function; and k is called the cross-dimensional cutoff. Simply
put, an individual is multi-dimensionally poor if a weighted sum of deprivation
indicators falls below a pre-speciﬁed threshold. The amount d wd 1 (yi d < cd ) is
called the deprivation share.
From this individual measure of poverty, three population wide measures are
built. The multidimensional headcount ratio is the proportion of the population that
is multi-dimensionally poor. It measures the incidence of multidimensional poverty
over the population:
n
1
H≡ pi
n i=1
n D
1
= 1 wd 1 (yid < cd ) > k (2)
n i=1 d=1
n
Deﬁning p ≡ i=1 pi , the average deprivation share among the poor is
n D
1
A≡ pi wd 1 (yid < cd ) (3)
p i=1 d=1
1
Note that we deﬁne the weights as adding up to 1 and not to the number of dimensions.
5
which measures the intensity of poverty in the population among the multi-dimensionally
poor. Finally the adjusted headcount ratio is deﬁned as
M0 ≡ HA (4)
which adjusts the headcount by the intensity of poverty.
Alkire and Foster (2011) focus on the M0 measure, due to its desirable properties.
These include monotonicity in the number of deprived dimensions and important
decomposition properties, which we outline in the next section. The headcount ratio
H, however, tends to receive wider attention in policy circles because, as a simple
population proportion, its level is immediately comparable with traditional income-
based poverty rates.
2.2 Decomposing the measures
The three measures outlined provide a one-dimensional summary of the incidence
and intensity of poverty for the population as a whole. In order to examine the
particular determinants of poverty, these measures may be decomposed into speciﬁc
indexes designed to see which factors contribute more.
Several decomposition methodologies exist, which can be classiﬁed into two broad
categories: static and dynamic. Static methodologies decompose a single observa-
tion into contributions from cross-sectional determinants, while dynamic ones de-
compose the time-variation of the measure into the contribution of time-varying
components. While the present paper focuses in a particular type of dynamic de-
composition, static methodologies are described brieﬂy next.
Static decompositions may be of two types: group decompositions and dimen-
sional decompositions. Group decompositions are customary in the poverty mea-
surement literature and decompose poverty measures into the contributions of par-
ticular individual groups. As shown in Alkire and Foster (2011), all the measures
considered are decomposable into individual groups. For the headcount ratio, if
there are two population groups, 1 and 2, with populations n1 and n2 , the headcount
ratio is:
n1 n1
H= H1 + H2
n n
Where H1 and H2 are the headcount ratios for each group.
6
Dimensional decompositions break up measures into the contribution of each
dimension. From equation 3, it is clear that the A measure is dimensionally decom-
posable, with contributions equal to:
n
1
pi wd 1 (yid < cd ) . (5)
p i=1
Alkire and Foster (2011) also show that the M0 measure is dimensionally decom-
posable. Deﬁning the censored headcount ratio Hd as the proportion of people de-
prived in dimension d among the poor:
n
1
Hd = pi 1 (yid < cd ) (6)
n i=1
Then, M0 can be decomposed by dimensions as:
D
M0 = wd Hd (7)
d=1
Conversely, the headcount ratio H is not decomposable by dimensions, since, by
construction, H is a nonlinear function of the contribution of each dimension—as
reﬂected in equation 2.
Dynamic decompositions, on the other hand, focus on splitting up the varia-
tion of the measures over distinct periods of time into time-variation from its com-
ponents. These components may or may not be further decomposed into cross-
sectional ones.
As shown in Apablaza et al. (2010), from equation 4, a simple dynamic decom-
position of the percent variation in M0 , ∆%M0 , is:
∆%M0 = ∆%H + ∆%A + ∆%H ∆%A (8)
Static and dynamic decompositions can be combined. For instance, in the previ-
ous equation, one could decompose H and A statically in each period, and then split
up the changes in M0 into changes in the cross-sectional components previously ob-
tained. Such decompositions are available, as long as the indicator is decomposable
cross-sectionally. This approach is used by Apablaza et al. (2010), who exploit the
fact that the headcount is decomposable across groups, and that the average depri-
vation share is decomposable across dimensions, to extend this result by decompos-
7
ing the components of equation 8. They show that the percent variation in H can be
further decomposed into population changes within groups, changes in the head-
count within groups, and a multiplicative effect. Furthermore, the percent variation
in A can be decomposed into the weighted sum of percent variations of each of its
dimensional components.
However, if the indicator is not cross-sectionally decomposable, this approach
fails. This is the case when attempting to decompose the changes of the headcount
ratio H into the variation attributed to each of its dimensions. Due the nonlinearity of
H , an explicit closed-form solution for this decomposition is not feasible. However,
a counterfactual simulation methodology may be used to work around this. That is
the topic of the following section.
3 Methodology
The decomposition approaches described so far are not appropriate to look into the
dimensions responsible for changes in the headcount ratio H over time. In this sec-
tion, we analyze the problem of decomposing changes in H , and describe a coun-
terfactual simulation methodology to address the issue. First, we outline the over-
all problem of decomposing changes in H . Then we summarize the counterfactual
simulation methodology based on Barros et al. (2006), Azevedo et al. (2013a) and
Azevedo et al. (2013b). The section concludes by addressing the advantages as well
as the potential caveats of applying this method to the headcount ratio H .2
Let us assume that there are two observations of the multidimensional head-
count H t , for t = 1, 2; the ﬁrst, 1, corresponds to the initial observation while 2
corresponds to the ﬁnal observation. We also observe the two associated datasets
t 1 1 1 1 1
of information on achievements yi : yi = ( yi 1 , yi2 , . . . , yiD ) , i = 1, 2, . . . n and
2 2 2 2 2
yj = yj 1 , yj 2 , . . . , yjD , j = 1, 2, . . . , n .. Additionally, we observe a set of demo-
graphic variables z1 2
i , zj . In the case of panel data, individuals can be tracked across
time, in which case variables are indexed by the same index, i, at both periods and
n1 = n2 . Otherwise, the datasets refer to repeated cross-sections. The change in the
2
The do-ﬁle code that implement this decomposition could be provided upon request
8
multidimensional headcount ratio is the difference between the two indicators 3 :
∆H = H 2 − H 1
n2 D
1
= 2 1 wd 1 yjd
2
< cd > k
n j =1 d=1
n1 D
1
− 1 wd 1 yid
1
< cd > k (9)
n1 i=1 d=1
The goal of decomposing the changes into the different dimensions is to be able
to express the change in the headcount ratio as a sum of the changes attributed to
each dimension Sd :
D
∆H = Sd (10)
d=1
Several remarks are in order. Notice, ﬁrst, that ∆H is not decomposable by di-
mensions. It is not decomposable in terms of the censored headcount ratios deﬁned
in section 2.2. Second, it is a nonlinear function of the contributions of each dimen-
sion to the average deprivation share, due to the nonlinearity of the indicator func-
tions that involve yid in equation 2: i.e., only the individuals below the dimension
speciﬁc cut-off contribute to the headcount. Third, while it is clear that the weights
play a role in the determination of the contribution of each dimension to the head-
count, they remain constant across time and are not the drivers of the changes in
H.
Note that by combining a decomposition of ∆H and the decomposition of ∆A
that arises from equation 3, a dimensional decomposition of ∆%M0 can be achieved.
As Apablaza and Yalonetzky (2013) show, ∆%A can be decomposed as:
D n D
pi
∆A = A2 − A1 = ∆ wd 1 (yid < cd ) = SA (11)
d=1 i=1
p d=1
3
In the case of panel data, this simpliﬁes to
1
n 1 D
d=1 wd 1 yid
2
< zd < k
∆H =
n i=1 −1 D
1 yid
1
d=1 wd < zd < k
9
Combining equation this with equations 8 and 10 yields:
∆%M0 = ∆%H + ∆%A + ∆%H ∆%A
D D
Sd SA
= + + ∆%H ∆%A (12)
d=1
H d=1
A
This decomposition breaks changes in the adjusted headcount ratio into changes
in the number of poor attributed to each dimension, Sd , changes in the intensity
of multidimensional poverty attributed to each dimension, SA , and an interaction
effect. The logic behind larger contributions of dimensions in explaining the changes
in intensity is very clear: if the majority of poor are no longer deprived in a particular
dimension, this dimension will contribute more to the decrease in intensity.
In the case of panel data, larger contributions of a dimension to changes in H
are also intuitive. As an illustrative example, consider a case where the change in
the headcount ratio occurs over a short period of time so that the demographic vari-
ables z remain constant. Assume that many dimensions change, but each individual
only experiences changes in one dimension. If all individuals whose deprivation
share crossed the multidimensional cut-off experienced change in the same dimen-
sion, then this dimension would be the unique contributor to the change. Indeed,
Apablaza and Yalonetzky (2013) show that the change in the headcount in the case
of panel data can be decomposed into a weighted difference in the transition proba-
bilities of moving in and out poverty. Individuals at the margin of transition may be
more prone to changes in particular dimensions, which would inﬂuence the transi-
tion probabilities and would contribute more to the variation in the headcount.
In the case of repeated cross-section data, changes in the headcount biased to-
wards a particular dimension may arise due to the inclusion of people with different
deprivation proﬁles in the sample, at one of the points of time. Thus, changes would
be biased by the inclusion of larger shares of people deprived in one dimension. In
applied work, dimensions do not change one at a time for each individual; dimen-
sions may be correlated; and cross-sectional data is the rule and not the exception,
especially in the surveys used to measure poverty in developing countries. A gen-
eral framework to decompose changes in H should thus take into account the notion
that dimensions are jointly distributed, that certain demographic proﬁles are more
likely to experience changes in particular dimensions, and that only individuals with
10
similar demographic proﬁles should be compared. This is the topic of the following
section.
3.1 The decomposition methodology
We propose the use of a counterfactual simulation methodology, ﬁrst suggested
by Barros et al. (2006), to decompose changes in H additively across dimensions.
This section describes the methodology, following closely Barros et al. (2006), and
Azevedo et al. (2012b, 2013a,b).
From equation 2, H can be written as function4 of the joint distribution of the
vector of achievements y, the weights w and the cut-offs z across the population:
H = Φ (Fy,w,z ). However, since the weights and the cut-offs do not change across
individuals, H can be considered a function only of the deprivation score by dimension,
deﬁned as:
xd = wd 1 (yid < cd ) (13)
With x = (x1 , . . . , xD ), H can be written as
H = Φ (Fx ) = Φ F(x1 ,...,xD ) (14)
Barros et al. (2006) show that, in ﬁnite populations, bivariate joint distributions
can be characterized by three functions: the two marginal distributions of each vari-
able, and a function that describes their association. If we deﬁne the ranking of in-
ı according to the random variable xd as the position of the individual in a list
dividual ˜
sorted by the value of random variable
ı })
Ryd (i) = # (i ∈ {1, . . . , n} : xid ≤ x˜ (15)
Then, according to Barros et al. (2006), for the two variables xd and xd , their joint
distribution is completely characterized by
−1
Fxd ,xd = Fxd , Fxd , Rxd Rx d
(16)
−1
Where Rxd Rx d
is the ranking according to xd of a observation with of rank
4
Notice that Φ need not be invertible.
11
Rxd according to xd . The function characterizes the rank dependence between the
two variables5 . We call this function the association between xd and xd and denote
it as C (xd , xd ). In the multivariate case, the joint distribution can be characterized
by all the marginals, along with either all the pairwise associations between the vari-
ables, or simply with the association of each variable to a reference variable r, from
which the pairwise associations can be obtained. The reference variable may either
be one of the deprivation scores by dimension or a demographic variable:
Fx = (Fx1 , Fx2 , . . . , FxD , C (x1 , r) , C (x2 , r) , . . . , C (xD , r)) (17)
With this representation in hand, Barros et al. (2006) show that decomposing
changes in a welfare aggregate into two components can be achieved by sequen-
tially changing the marginals and the association in the joint distribution. In the case
of H , the random variables considered are the deprivation scores by dimension, and
the change in equation 9 can be rewritten, using equations 14 and 17, as:
∆H = H 2 Fx2
1
, Fx2
2
, . . . , F x2
D
, C x2
1, r
2
, C x2
2, r
2
, . . . , C x2
D, r
2
− H 1 Fx1
1
, Fx1
2
, . . . , F x1
D
, C x1
1, r
1
, C x1
2, r
1
, . . . , C x2
D, r
1
(18)
A Barros et al. (2006) decomposition for ∆H in the case of two dimensions, D = 2,
would then be:
∆H = S1 + S2 + S12
S1 = H Fx2
1
, Fx2
2
, C x2 2
1 , x2 − H Fx1
1
, Fx2
2
, C x2 2
1 , x2 = H 2 − H F x1
1
, Fx2
2
, C x2 2
1 , x2
S2 = H Fx1
1
, Fx2
2
, C x2 2
1 , x2 − H F x1
1
, Fx1
2
, C x2 2
1 , x2
S12 = H Fx1
1
, Fx1
2
, C x2 2
1 , x2 − H F x1
1
, Fx1
2
, C x1 1
1 , x2 = H F x1
1
, Fx1
2
, C x2 2
1 , x2 − H1
(19)
Where H Fx1 1
, Fx2
2
, C (x2 2
1 , x2 ) is the counterfactual headcount that would be ob-
served if y1 were distributed as in period 1, but x2 and the association remained the
5
In practice, there can be ties in these rankings. This is inconsequential since, as Barros et al. (2006)
notes, this can be solved by randomizing the ranking for the tied cases.
12
same. It is important to note that counterfactuals are purely exercises to examine
changes that occur ceteris paribus, and do not intend to reﬂect equilibrium outcomes
(Azevedo et al., 2012a). To compute this counterfactual, we can calculate H from the
x1
distribution of (ˆ 2
1 , x2 ), where
−1
x
ˆ 1 = Fx 1 F x2
1
( x1 ) (20)
1
Other counterfactuals may be obtained accordingly, requiring inversion of the
distribution functions at each step. From here on, we refer to this method as the
Barros decomposition.
Azevedo et al. (2012b, 2013a,b) have proposed several improvements to the Bar-
ros decomposition to have a broader applicability. Their applications focus mainly
on the decomposition of poverty and inequality indicators, but the same improve-
ments could be applied to other measures.
For both panel data and repeated cross-section applications, these studies con-
sider the multivariate case rather than the Barros bivariate one. They propose to
keep the associations C constant and they add each variable sequentially, such that
no effect is attributed to the interactions. In terms of equation 19, this allows decom-
posing ∆H into just two components S1 and S2 , as stated in the original problem in
equation 10. Barros et al. (2006) show that the counterfactuals need to be changed in
the multivariate case in order to hold the associations constant. Thus, instead of x ˆ1
from equation 20, they build different counterfactuals x ˜d , as:
˜1
x 1
id = xi∗ d
−1
i∗ = C (ˆ
xd , xd (i)) = Rx
ˆd Rx2
d
(i) (21)
Additionally, these studies note that the Barros decomposition is path dependent:
the order in which the variables are replaced by their counterfactuals shifts the result
of the decomposition. This is addressed by using a Shapley value decomposition
approach based on Shorrocks (2013) (for details see Azevedo et al. (2012b)). Basically,
the authors compute the decomposition along each permutation of the y vector and
calculate the average of all the contributions obtained6 .
6
This has the disadvantage of making the method dependent on whether the variables are added
together. See Azevedo et al. (2012a) for details.
13
These papers also address speciﬁc problems that arise in the presence of repeated
cross-section data. The ﬁrst is the construction of the counterfactual variables x ˆd . In
the case of panel data, this is straightforward, for instance in the previous example:
˜1
x 1
i = xi .. In repeated cross-section data, however, x ˜d can only be obtained using
equations 20 and 21 if F and R are strictly monotone, but this is rarely the case.
To work around this, Azevedo et al. (2012b) assume rank preservation in income,
and then use income rank as a reference variable to track individuals across the two
periods. This amounts to replacing x ˆd with income in equation 21, and avoiding
using equation 20 completely. The counterfactual for an individual with a x2 d value
2
is an individual who has the same income rank corresponding to xd in period 1.
ˆd if rank preservation is plau-
In practice, any reference variable could substitute x
sible and its ranking function is invertible. While in some cases it may not be plau-
sible for rank preservation to hold unconditionally, it may hold conditional on the
demographic variables z. In this instance, it is necessary to stratify the data, com-
puting income ranks within strata previously deﬁned by the demographic variables
before applying equation 21.
A second problem associated with repeated cross-section data refers to unequal
sizes in the available data across time. This issue is pervasive in applied work,
since survey samples typically become larger over time. To address this, Azevedo
et al. (2012b) suggest rescaling the ranking in the larger dataset to match the smaller
dataset, which necessarily generates observations with the same ranking. Then, each
observation in the smaller dataset is matched with a randomly selected observation
with the same ranking in the rescaled larger dataset.
A third problem that may arise in this line refers to variation in the survey sam-
pling schemes across cross-section samples. This can be addressed by making the
weights compatible across samples through a reweighting procedure.
So far we have described the counterfactual simulation methodology, outlining
how it can be applied to a decomposition of the multidimensional headcount. We
have also reviewed several issues that can arise when the methodology is used in
applied work. We have yet to explore how likely it is that these issues appear when
decomposing the headcount ratio H , instead of the indicators studied in the litera-
ture so far. The next section examines these practical issues more closely.
14
3.2 Applying the decomposition methodology to H
This section discusses the practical difﬁculties of applying the methodology described—
based on counterfactual simulations—to the multidimensional headcount H . Specif-
ically, it focuses on the problems reviewed in the previous section. Our purpose is to
provide a practical guide that can be followed when applying the methodology sug-
gested in this paper. From here on, we will refer to the Barros decomposition with
the modiﬁcations by Azevedo et al. (2012b, 2013a,b) as the ASN decomposition.
The practical difﬁculties of applying the methodology can be summarized in four
issues. These are listed next, followed by a brief description of the approaches sug-
gested to address them.
1. The ASN methodology disregards interactions across dimensions. To address
this, dimensions can be grouped into categories with small interactions be-
tween them.
2. By disregarding interactions, the ASN methodology assumes that interaction
across dimensions remains constant over time. This too may be addressed if di-
mensions are grouped into categories where interaction remains constant over
time.
3. The ASN methodology is not applicable with discrete random variables as ref-
erence variables. A continuous variable, such as income, needs to be used as
reference, through the assumption of rank preservation by income. Stratiﬁca-
tion may be an option to ensure rank preservation.
4. With repeated cross-section data, rescaling datasets of unequal sizes may lead
to matching individuals across strata. To avoid this, rescaling needs to be done
within strata; while assessing the sensitivity of the methodology to the rescal-
ing process.
Each issue, and proposed approach, is presented in further detail below.
3.2.1 Disregarding interactions across dimensions
Since the multidimensional headcount H often depends on a large number of di-
mensions, the ASN decomposition is much better suited than the Barros one to de-
compose it, as it abstracts from calculating the effects of pairwise calculations, which
15
can be quite large. Moreover, as noted by Azevedo et al. (2013b), due to the path
dependence of the Barros decomposition, not all pairwise combinations would be
calculated, only those of the variables that are consecutive in the path chosen. Using
the ASN decomposition comes, however, at the cost of assuming that the interaction
across dimensions is small.
Disregarding the interaction across dimensions may not be reasonable when di-
mensions are highly related to each other. For example, two dimensions may be
based on education variables and may be very likely to vary together. It is then
necessary to group the dimensions into broad categories such that the interaction
between different categories is small.
We therefore apply the methodology by grouping dimensions into broad cate-
gories, and validating our categories such that the interaction between them is low.
This is done without loss of generality, by simply partitioning the dimensions into
disjoint categories. Formally, let us assume that we partition the dimensions into
two categories: {1, 2, . . . , d1 } and {d1 , d1 + 1. . . . , D}. We can then redeﬁne the depri-
vation scores by the dimension of equation 13 as deprivation scores by category:
dB
¯1 ≡
x xd
d=1
D
¯2 ≡
x xd (22)
d=dA+1
And carry out the decomposition over x ¯2 .
¯1 and x
The fact that the ASN methodology may be generalized to categories of dimen-
sions, should by no means be interpreted as stating that building categories is cost-
less. As Azevedo et al. (2012b) note, the ASN methodology is sensitive to aggregat-
ing the dimensions into categories, and results may vary depending on the aggrega-
tion.
In our empirical application of section 4 below, we propose the use of descriptive
statistics in order to assess whether interactions are indeed small across categories.
16
3.2.2 Interactions remaining constant over time
The ASN methodology also assumes that the interactions among variables remain
constant over time, as these are not changed when building the counterfactuals.
Therefore, in addition to being small, our empirical application requires that the
interaction between categories is constant over time. This can be achieved, as before,
by a proper deﬁnition of categories. This requirement is therefore assessed in our
empirical application, as will be shown below.
3.2.3 Income as reference variable in repeated cross-section data
Another difﬁculty is associated with the inability to use discrete random variables as
reference variables in repeated cross-section data. Given that the achievement vari-
ables yd are indicator variables, the deprivation scores by dimension xd have discrete
distributions. This implies that their distribution functions are not invertible, so that
the Barros decomposition may not be calculated. The issue does not come up when
using panel data as, in such case, the counterfactuals are simply built by tracking the
same individuals. The ranking functions Rx are also stepwise, non-invertible func-
tions, so that the ASN methodology cannot be applied, in principle, if the reference
variables are chosen from the deprivation scores by dimension.
However, as discussed in the previous section, the reference variable could be
any continuous variable. Following Azevedo et al. (2012b), the ASN methodology
can be applied, assuming rank preservation on income and thus using income as a
reference variable. As noted, if it is assumed that rank preservation holds only after
stratiﬁcation, then the rank function of income needs to be invertible within strata.
Narrowly deﬁned strata may not satisfy this assumption, so we deﬁne strata broadly
enough to have enough income variation within strata. Some strata may be empty
in one of the datasets. By deﬁnition, these strata do not satisfy rank preservation, so
they should be excluded from all calculations.
3.2.4 Rescaling of datasets
When faced with unequal sizes in the different cross-sectional data available across
time, the ASN methodology suggests rescaling the ranking of the larger dataset to
that of the smaller dataset. Similarly ranked observations are then matched to build
the counterfactuals, but if the rankings are rescaled across strata, individuals from
17
different strata may be matched. To avoid this, rankings need to be rescaled within
strata. If possible, the sensitivity of the method to the rescaling should be assessed,
for example, by recalculating the decomposition with subsamples of the original
data.
This section has outlined the practical issues of applying the ASN methodology
to the multidimensional headcount H. We now turn to two different empirical ap-
plications to illustrate the performance of the methodology, which highlight the im-
portance of the issues described.
4 Application to repeated cross section data: The case
of Colombia
We apply the proposed methodology in the context of the decline in multidimen-
sional poverty observed in Colombia between 2008 and 2012. The Colombian case
is interesting for a couple of reasons. Being a middle-income developing country,
a large share of Colombians is still deprived in the dimensions considered for cross
country calculations of MPI measures. On the other hand, the pace of decline has
varied over the years. Monetary and multidimensional poverty declined sharply
from 2003 to 2008; while over the last ﬁve years, the decline has been less sizable
though steady.
4.1 Description of the data, the MPI and trends
We use data from the Colombian Quality of Life Survey for the years 2008, 2010
and 2013. This is the survey that the National Administrative Department of Statis-
tics (Departamento Administrativo Nacional de Estad´ıstica, DANE) uses for multidimen-
sional poverty calculations. The survey included around 50,000 households in 2008;
while the sample size increased by about 5 percent and 38 percent in the next two
rounds, respectively. We were able to replicate the ofﬁcial poverty measures pub-
lished by DANE closely. For a detailed summary, see Angulo (2010).
The multidimensional poverty index for Colombia includes 15 dimensions grouped
into ﬁve broad categories: education, childhood and youth, labor, health and stan-
dard of living7 . Each of the categories has a weight of 0.2, which is distributed evenly
7
This index is broader that the calculations of Alkire and Santos (2010) for Colombia. Their selec-
18
across the dimensions within each category. Table 1 shows all the dimensions of
the index. Many dimensions are household-based: if the household is deprived in
any of the dimensions, all household members are considered deprived. The cross-
dimensional cut-off k is 1/3; that is households are considered multi-dimensionally
poor if the weighted sum of deprivation scores is larger than 1/3. For example, a
household deprived in all the dimensions within two categories receives a score of
0.4, and is considered poor. We use the categories to group the 15 dimensions, as
described in section 3.2.1.
Figure 1, panel A, shows the evolution of monetary and multidimensional poverty
measures, along with the measures described in section 2.1, for the three years con-
sidered. Monetary and multidimensional poverty headcounts have been declining
over time, almost at the same pace. By 2012, around 27 percent of the Colombian
population was estimated to be multi-dimensionally poor, while 32.7 percent is con-
sidered poor by the monetary measure. Although the headcount ratio has fallen,
the average deprivation share among the poor has remained almost constant. This
implies that the adjusted headcount ratio, which adjusts for the intensity of poverty,
has not declined as quickly as the headcount.
Panel B decomposes the decrease in the adjusted headcount ratio into the contri-
butions of changes in the multidimensional headcount –due to changes in the num-
ber of poor– and changes in the average deprivation share among the poor –due to
changes in the intensity of poverty–. For both years, changes in intensity account for
less than a quarter of the change in the adjusted headcount ratio, while most of the
decrease is attributed to changes in the headcount. This is consistent with Apablaza
et al. (2010) and Apablaza and Yalonetzky (2013), who ﬁnd this pattern for different
countries and datasets. The large contribution of the change headcount to the over-
all change in the adjusted headcount ratio, highlights the importance of analyzing
its determinants separately.
Table 2 shows the evolution of censored headcounts by dimension Hd , that is, the
headcounts of those deprived in at least one dimension within each category, and the
average number of deprivations in each category. This table shows the percentage of
the population deprived in an speciﬁc dimension within those that are multidimen-
sional poor, as opposed to uncensored headcounts, which are shown in table A.1.
Since deprivations are calculated at the household level, the numbers in table 2 are
tion criteria are detailed in Angulo (2010).
19
higher that national aggregates at the individual level.
Almost all poor individuals have low educational attainment and are not em-
ployed in the formal sector. The average poor individual is deprived in both ed-
ucation dimensions, but usually only in one dimension of standard of living. The
average number of deprivations within each category declines slightly over time for
the poor in most categories. The sharpest decline in the percentage of people with
at least one deprivation occurs in the health category, where this headcount falls by
about 10 percentage points.
In Table 3, we examine the percentage evolution of the censored headcounts
by dimension over time. The sharpest declines occur in deprivation in access to
childcare services and on the number of households living in homes built with low-
quality materials. Although most headcounts declined between 2008 and 2012, many
of them underwent a sharp decline from 2008 to 2010, followed by a rebound in the
next two year; for example, lack of access to health services loses more than half
of its initial decrease. Furthermore, long-term unemployment headcounts do not
decrease but rather experience a steep increase over the period. The trends in the
uncensored headcounts, shown in table A.1, are similar to the censored ones, but the
U-shaped patterns in some dimensions are not as stark since the number of multi-
dimensionally poor declines over time. Together, tables 2, 3 and A.1 show that the
largest movers are in the categories of health, standard of living, and childhood and
youth, although we cannot conclude that these are the biggest contributors to the
change in the multidimensional headcount.
Table 4 examines the contribution of each category to the intensity of poverty.
It decomposes the average deprivation share as in equation 5, grouping dimensions
over categories. The largest contributors to the intensity of poverty are the education
and labor categories. The contributions of each category are stable over time, with
the exception of health, for which the contribution is declining.
4.2 Assessing the plausibility and performance of the
methodology
Before applying the ASN methodology to the Colombian case, it is necessary to ad-
dress the issues described in section 3.2, concerning repeated cross-section data. To
address the ﬁrst two issues—limited and stable interaction between categories—we
need to conﬁrm that in the categorization of dimensions deﬁned so far, dimensions
20
are not strongly associated across categories, and that associations remain constant
over time. To do this, we calculate the Kendall rank correlation coefﬁcients of the
deprivation scores across categories for the three years, as illustrated in table 5. We
choose Kendall correlation coefﬁcients since we are concerned about changes in the
ordinal association between the deprivation scores and the reference variable, which
turn into changes in the association between deprivation score across categories. The
results are encouraging: the correlation coefﬁcients across categories are small and
remain stable over time.
The third issue—the inability to use discrete random variables as reference vari-
ables with repeated cross-sections—requires signiﬁcant attention. We need to ad-
dress whether the continuous variable, income, is a proper reference variable for the
methodology; if rank preservation is reasonable at the sample level; and, if not, we
need to stratify the sample in order to make rank preservation more plausible. The
observational data is uninformative in these matters, as it is not longitudinal data. A
simulation exercise is carried out to address this issue, instead.
We build a panel dataset on the basis of the 2010 observations, adding an addi-
tional simulated period of data. For this added period, we simulate changes in the
deprivations of individuals, in order to replicate the actual changes that took place
in the uncensored headcount ratios by dimension between 2010 and 2012. We also
change household income to reﬂect the income growth over the period. To do this,
we calculate the variation in mean income within income deciles between the two
years from the observational data, and add a normal random variable centered in
this mean change to each individual within the income decile. The variance of this
increase is chosen to match the change in within-decile variances that occurred be-
tween both years. We change deprivations in all the categories, initially one category
at a time, and then all at once. We then assess different choices in the ASN method-
ology. We address the performance of these choices by comparing the results of
the decomposition to those that would have been obtained using the panel dataset.
Some of our choices are made with the purpose of highlighting the potential pitfalls
of applying the method without examining the issues. All calculations are made
using the software in Azevedo et al. (2012a).
Table 6 presents the results of the decompositions when deprivations in all cat-
egories are changed: this is the scenario closer to the observational data. For each
method, the value on the left corresponds to the share that each category contributes
to the change in the multidimensional headcount. The value on the right is the ab-
21
solute difference between each column and the panel method. We describe each of
the columns below.
• The “Panel” method corresponds to the decomposition using the panel struc-
ture of the simulated data. The third issue discussed does not arise here, so this
column is provided as reference.
• The “No Panel” method applies the decomposition ignoring the panel struc-
ture, and using the deprivation share as the ranking variable. We know that,
despite the fact that there are many dimensions in the index, the distribution
of this score is discrete, and so its ranking function is not invertible. Because of
this, it is not surprising that this method performs poorly.
• The “Strata” method is equivalent to the “No Panel” method, but it stratiﬁes
the sample into groups deﬁned by categorical demographic variables. The cat-
egories are deﬁned in Table A.2. This method shows that the ranking functions
of the deprivation score are also non invertible within strata, so the method
performs at least as poorly as the “No-Panel” one.
• The “Income Within” method is our preferred one. It uses income as a refer-
ence variable within strata, by stratifying the sample, sorting it within strata
by income, and building the ranking functions based on income-within-strata.
A potential pitfall of this is that income may not have enough variation within
strata, and the ranking function may become non invertible. This may happen
if strata are too narrow. However, this does not appear to be the case here,
since the method performs about as well as an artiﬁcial method that builds the
ranking variable by strata (across) and by incomes.
Overall, we ﬁnd that the last method outperforms all remaining choices in pretty
much all cases, as expected from the discussion in section 3.2. We also compute
simulations changing categories one at a time. The full results are reported in Table
A.3, and summarized in table 7. We calculate the difference of each method with
the panel one, as in Table 6. Then, we average these discrepancies across categories.
The result is then scaled by the total change in the headcount to arrive at “average
discrepancies as a share of the total change in the headcount”. The results conﬁrm
what we learned from the previous, more realistic simulation that changed all cat-
egories. The “Income Within” method outperforms the other ones, except for the
22
“Strata” method, which performs rather well. This is, however, unrealistic: since the
changes in the deprivation score in this case are small, the ranks of the deprivation
score are similar across years, which results in individuals being matched almost as
if the dataset had a panel structure.
So far, our simulations have not addressed the fourth issue—rescaling datasets
of unequal sizes within strata. We do so by repeating the exercises of Tables 6 and
7, reducing the sample size in period 1, by taking a (stratiﬁed on demographics)
random sample of this data and repeating the exercises. The results of this exercise
are presented in Table 8. The “Panel” method is no longer applicable here, so we
compute the discrepancy against the panel method with equal sample sizes of the
previous exercises. As in the previous case, the “No Panel” method performs poorly.
The “Strata” method performs well when the changes are isolated into one category,
but does poorly if changes occur in all categories.Our preferred method, the “Income
Within” one, outperforms the others with sample reduction, although, as it is only
natural, performs worse when the sample in the ﬁrst period is smaller compared to
the second period.
Having examined the potential issues with the methodology, we have found the
method that best addresses them, outperforming the other ones. In the next section,
we apply our preferred “Income Within” method to the case of the multidimensional
poverty decline in Colombia.
4.3 Results of the decomposition for the Colombian case
We present the results of applying our preferred “Income Within” method to the
case of the multidimensional poverty decline in Colombia in Table 9. These results
present several highlights that would be absent in a more standard analysis focusing
solely on the evolution of censored or uncensored headcounts.
The largest contributors to the decrease in the Colombian multidimensional head-
count ratio are the ‘education’ and ‘health’ categories. Together, they account for ap-
proximately ﬁve percentage points out of a 7.5 percent reduction between 2008 and
2012; that is, more than 60 percent of the decline. Their contribution is similar be-
tween the 2008-2010 period and the 2010-2012 one. The next contributor, ‘childhood
and youth’, is responsible for about one percentage point of the decline. The ‘labor’
category does not contribute much: this result could be expected from the analysis
of the censored and uncensored headcounts, which do not present large reductions
23
in Tables 3 and A.1. It is also intuitive that labor does not contribute much, given
the sample period analyzed, as Colombia experienced an economic slowdown due
to the global ﬁnancial crisis over these years. Nevertheless, from those same tables,
it would not have been intuitive to conclude that education was a large driver, in-
stead, more weight would have been attributed (erroneously) to standard of living.
The childhood and youth category is the only one responsible for a larger reduction
of poverty in 2010-2012 than over the previous two years.
5 Conclusions
This paper analyzes the problem of decomposing changes in the multidimensional
headcount ratio into the contributions from dimensions, or categories of dimensions.
We examine the potential use of decompositions based on counterfactual simulations
to break up changes in the multidimensional headcount; outlining potential issues
with the methodology.
We propose and examine different options to address the caveats of the method-
ology, identifying a method to address these issues that performs well in simulations.
The paper presents the application of this method to decompose the recent decline of
poverty in Colombia, ﬁnding that health and education are the largest contributors
to the decline.
Our proposed decomposition provides a useful way to estimate the extent in
which each category contributes to the change in the headcount in the absence of
panel data, without tracking which individuals cross the multidimensional poverty
cut-off and which dimensions changed for each of those individuals. This method-
ology can be a useful complement to the analysis of multidimensional poverty that
focuses on a wide range of indicators, such as those suggested by Ferreira and Lugo
(2013). The exploration of further tools to decompose multidimensional poverty
measurements based on non-scalar indexes, such as multidimensional distributions,
appears as a fruitful avenue for future research.
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26
Figures and tables
Figure 1: Trends in monetary and multidimensional poverty measures
Panel A Panel B
Poverty Indicators Decomposition of Change in M0
Adjusted Headcount Ratio
.5
1.5
100
.448
.431 10.8
.42 .425
.4
21.4
80
.372
.345
.327
60
Percentage
.302
.3
.27
40
.2
.154 20
.13
.115
77.1 90.9
.1
0
-1.7
2008 2010 2012
Year 2008-2010 2010-2012
Years
H - Multidimensional Pov. Headcount
A - Average Deprivation Share (poor) Contribution of H
M0 - Adjusted Headcount Ratio Contribution of A
Monetary Pov. headcount Interaction
Source: DANE. Author’s calculations
27
Table 1: Categories and dimensions of the Colombian multidimensional poverty
index.
Category Dimension Deprived if
Education Educational achievement Any person older than 15 years
has less than 9 years of schooling.
Literacy Any person older than 15 years
and illiterate.
Childhood and youth School attendance Any child 6 to 16 years old does
not attend school.
Children behind grade Any child 7 to 17 years old is
behind the normal grade for his
age.
Access to child care services Any child 0 to 5 years old doesn’t
have access to health, nutrition or
education.
Child labour Any child 12 to 17 years old
works.
Employment Long term unemployment Any economically active member
has been unemployed for 12
months or more.
Formal employment Any employed household
members is not afﬁliated to a
pension fund.
Health Health insurance Any person older than 5 years
does not have health insurance.
Health services Any person who fell sick or ill in
the last 30 days did not look for
specialized services.
Standard of living Water system Urban: Household not connected
to public water system. Rural:
Household obtains water used for
cooking from wells, rainwater,
spring source, water tanks, water
carriers or other sources.
Sewage Urban: Household not connected
to public sewer system. Rural:
Household uses a toilet without a
sewer connection, a latrine or
simply does not have a sewage
system.
Floors Households has dirt ﬂoors.
Walls Rural: The household’s exterior
walls are made of vegetable, zinc,
cloth, cardboard or waste
materials or if no exterior walls
exist. Urban: Walls made of rural
materials or untreated wood,
boards or planks.
Overcrowding Urban: There are 3 people or more
per room. Rural: More than 3
people per room.
Source: Angulo (2010). Deprivations are measured at the household level: all members of the
household are considered deprived if one of the members is deprived in a dimension.
28
Table 2: Multidimensional poverty measures, censored headcount ratios by
dimension and average deprivation shares by category
Indicator Weights 2008 2010 2012
H : Multidimensional headcount ratio (%) 34.47 30.38 26.97
A : Average deprivation share among the poor 0.448 0.432 0.425
M0 : Adjusted headcount ratio 0.154 0.131 0.115
Education .2
Educational Achievement (%) 0.1 96.30 94.26 94.24
Literacy (%) 0.1 43.37 44.76 44.56
At least 1 component (%) 96.36 94.78 94.96
Average deprivation share in category 0.698 0.695 0.694
Childhood and youth .2
School Attendance (%) 0.05 20.06 20.23 18.48
Children behind grade (%) 0.05 71.27 72.51 71.69
Access to child care services (%) 0.05 29.38 27.91 21.38
Child labour (%) 0.05 17.67 16.42 14.58
At least 1 component (%) 81.90 82.12 79.00
Average deprivation share in category 0.346 0.343 0.315
Labour .2
Long term unemployment (%) 0.1 10.12 10.61 11.14
Formal employment (%) 0.1 99.03 99.25 99.18
At least 1 component (%) 99.12 99.31 99.19
Average deprivation share in category 0.546 0.549 0.552
Health .2
Health insurance (%) 0.1 53.25 47.78 44.96
Health services (%) 0.1 22.99 16.97 19.81
At least 1 component (%) 63.51 57.39 55.93
Average deprivation share in category 0.381 0.324 0.324
Standard of living .2
Water system (%) 0.04 30.27 27.56 30.26
Sewage (%) 0.04 32.01 29.21 29.64
Floors (%) 0.04 23.44 20.75 19.46
Walls (%) 0.04 7.75 7.62 5.76
Overcrowding (%) 0.04 40.04 38.38 35.29
At least 1 component (%) 68.65 66.00 66.70
Average deprivation share in category 0.267 0.247 0.241
Source: Author’s calculations. Headcounts are censored, i.e. calculated over the poor. See table A.1 for the uncensored headcount levels,
calculated over the whole sample. “At least 1 component” denotes censored headcount ratios of individuals deprived in at least one
dimension within the category. “Average deprivation share in category” is the average over poor individuals of the number of deprivations
divided by the number of dimensions in the category. See table 3 for the changes of censored headcounts over time.
29
Table 3: Changes of censored headcount ratios over time
Indicator 2008-2010 2010-2012 2008-2012
Education
Educational Achievement (% Change) -2.11 -0.02 -2.13
Literacy (% Change) 3.20 -0.46 2.73
Childhood and youth
School Attendance (% Change) 0.87 -8.67 -7.88
Children behind grade (% Change) 1.74 -1.12 0.60
Access to child care services (% Change) -4.99 -23.40 -27.23
Child labour (% Change) -7.07 -11.19 -17.47
Labour
Long term unemployment (% Change) 4.79 5.03 10.07
Formal employment (% Change) 0.22 -0.07 0.15
Health
Health insurance (% Change) -10.27 -5.91 -15.57
Health services (% Change) -26.20 16.75 -13.83
Standard of living
Water system (% Change) -8.96 9.81 -0.03
Sewage (% Change) -8.75 1.48 -7.40
Floors (% Change) -11.49 -6.22 -16.99
Walls (% Change) -1.69 -24.46 -25.74
Overcrowding (% Change) -4.15 -8.06 -11.88
Source: Author’s calculations. See table 2 for the censored headcount levels.
30
Table 4: Multidimensional measures and average deprivation shares by category
Indicator 2008 2010 2012
H : Multidimensional headcount ratio (%) 34.47 30.38 26.97
A: Average deprivation share among the poor 0.448 0.432 0.425
M0 :Adjusted headcount ratio 0.154 0.131 0.115
Education 0.140 0.139 0.139
Childhood and youth 0.069 0.069 0.063
Labour 0.109 0.110 0.110
Health 0.076 0.065 0.065
Standard of living 0.053 0.049 0.048
Source: Author’s calculations. Average deprivation shares by category are calculated by calculating
the average deprivation shares by dimension as in equation 5, then adding over categories.
Table 5: Rank correlations between deprivation scores across categories
Education Childhood and youth Labour Health Standard of living
2008
Education 1.000 0.209 0.201 0.121 0.288
Childhood and youth 0.209 1.000 0.063 0.150 0.240
Labour 0.201 0.063 1.000 0.112 0.100
Health 0.121 0.150 0.112 1.000 0.081
Standard of living 0.288 0.240 0.100 0.081 1.000
2010
Education 1.000 0.177 0.217 0.061 0.241
Childhood and youth 0.177 1.000 0.024 0.112 0.189
Labour 0.217 0.024 1.000 0.104 0.074
Health 0.061 0.112 0.104 1.000 0.055
Standard of living 0.241 0.189 0.074 0.055 1.000
2012
Education 1.000 0.170 0.205 0.066 0.248
Childhood and youth 0.170 1.000 0.021 0.108 0.190
Labour 0.205 0.021 1.000 0.095 0.082
Health 0.066 0.108 0.095 1.000 0.040
Standard of living 0.248 0.190 0.082 0.040 1.000
Source: Author’s calculations.
31
Table 6: Results of the decomposition in a simulated panel with changes in all of the
categories
Panel No Panel Strata Income within
Value Value |∆| vs Value |∆| vs Value |∆| vs
Panel Panel Panel
Education -0.618 -0.571 0.046 -0.513 0.104 -0.597 0.020
Chilhood and Youth -0.142 -0.485 0.343 -0.450 0.308 -0.082 0.060
Labour -0.034 0.023 0.057 -0.085 0.050 -0.066 0.032
Health -0.812 -0.738 0.074 -0.977 0.165 -0.943 0.131
Standard of living -0.117 0.049 0.166 0.302 0.419 -0.035 0.082
Source: Author’s calculations. See the text for a description of the table.
Table 7: Performance of simulations when changing categories one by one: average
discrepancy as share of change in headcount
Panel No Panel Strata Income
within
Education 0.000 0.749 0.886 0.346
Chilhood and Youth 0.000 3.070 1.833 1.507
Labour 0.000 11.083 0.499 4.245
Health 0.000 0.816 0.707 0.408
Standard of living 0.000 3.089 1.430 1.965
All 0.000 0.398 0.607 0.188
Source: Author’s calculations. See the text for a description of the table.
32
Table 8: Performance of simulations with samples of unequal sizes
No Panel Strata Income
within
85 % subsample in ﬁrst period
Education 1.035 0.715 0.526
Chilhood and Youth 3.778 1.167 3.646
Labour 8.179 3.440 8.511
Health 0.566 0.656 0.588
Standard of living 4.528 1.623 3.983
All 0.475 0.588 0.244
70 % subsample in ﬁrst period
Education 1.102 1.031 0.796
Chilhood and Youth 4.641 3.899 4.602
Labour 12.3 10.52 11.75
Health 1.068 0.823 0.773
Standard of living 5.648 4.613 5.318
All 0.544 0.704 0.379
Source: Author’s calculations. See the text for a description of the table.
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Table 9: Decomposition of the multidimensional headcount ratio in Colombia
2008-2012
Indicator 2008- 2010- 2008-
2010 2012 2012
Education
Change due to category (Percentage points) -1.124 -1.034 -2.465
Percentage contribution of category (%) 27.51 30.24 32.85
Childhood and youth
Change due to category (Percentage points) -0.484 -0.663 -1.152
Percentage contribution of category (%) 11.85 19.40 15.35
Labour
Change due to category (Percentage points) -0.334 -0.100 -0.405
Percentage contribution of category (%) 8.18 2.93 5.39
Health
Change due to category (Percentage points) -1.418 -1.007 -2.372
Percentage contribution of category (%) 34.72 29.46 31.61
Standard of living
Change due to category (Percentage points) -0.725 -0.614 -1.111
Percentage contribution of category (%) 17.74 17.97 14.80
Total -4.09 -3.42 -7.50
34
Appendix
Table A.1: Evolution of uncensored headcount ratios by dimension
Indicator 2008 2010 2012 2008- 2010- 2008-
2010 2012 2012
(%) (%) (%) (∆ %) (∆ %) (∆ %)
Education
Educational Achievement 62.70 58.67 56.24 -6.43 -4.13 -10.30
Literacy 17.42 16.04 14.19 -7.92 -11.52 -18.53
Childhood and youth
School Attendance 8.04 7.09 6.23 -11.80 -12.08 -22.45
Children behind grade 45.10 47.00 44.64 4.21 -5.03 -1.03
Access to child care services 17.06 16.28 12.92 -4.57 -20.67 -24.29
Child labour 7.78 6.60 5.39 -15.13 -18.35 -30.71
Labour
Long term unemployment 6.73 6.77 6.63 0.53 -2.07 -1.55
Formal employment 83.72 83.55 82.90 -0.20 -0.79 -0.99
Health
Health insurance 27.72 24.34 21.13 -12.18 -13.20 -23.77
Health services 10.79 7.81 7.89 -27.61 0.98 -26.90
Standard of living
Water system 14.56 12.84 13.22 -11.86 3.00 -9.22
Sewage 15.67 13.33 13.51 -14.93 1.31 -13.81
Floors 9.22 7.68 6.78 -16.76 -11.72 -26.52
Walls 3.43 3.22 2.37 -6.11 -26.50 -30.99
Overcrowding 22.17 20.76 18.36 -6.38 -11.52 -17.16
Source: Author’s calculations. Headcounts are uncensored, i.e. calculated over the whole sample.
See table 2 for the censored headcount levels, calculated among the poor.
35
Table A.2: Demographic variables used for stratiﬁcation
Variable Description
Decile Income Decile
Household head gender 1 if the household head is male, 0 otherwise
Education 1 if the household head has no education, 2 if he has
primary, 3 secondary and 4 tertiary.
Household size 1 if household has more than 4 people, 0 otherwise.
Kids 1 if household has kids, 0 otherwise.
Urban 1 if household is located in a urban area, 0 otherwise
36
Table A.3: Results of the decomposition in a simulated panel with changes in each
one of the categories
Panel No Panel Strata Income
within
Simulated change in education
Education -0.678 -0.551 -0.574 -0.722
Chilhood and Youth 0.000 -0.240 -0.123 0.050
Labour 0.000 -0.013 -0.006 0.091
Health 0.000 0.079 -0.096 -0.098
Standard of living 0.000 0.047 0.120 0.001
Simulated change in
chilhood and youth
Education 0.000 0.005 0.016 -0.009
Chilhood and Youth -0.115 -0.291 -0.155 -0.108
Labour 0.000 0.067 0.011 0.098
Health 0.000 0.095 -0.027 -0.099
Standard of living 0.000 0.009 0.039 0.003
Simulated change in labour
Education 0.000 0.149 0.005 -0.019
Chilhood and Youth 0.000 -0.144 -0.001 0.037
Labour -0.047 0.035 -0.053 0.052
Health 0.000 -0.114 -0.005 -0.110
Standard of living 0.000 0.027 0.007 -0.007
Simulated change in health
Education 0.000 0.206 0.094 0.016
Chilhood and Youth 0.000 -0.315 -0.137 0.045
Labour 0.000 -0.019 0.019 0.102
Health -0.818 -0.724 -0.905 -0.981
Standard of living 0.000 0.035 0.112 0.000
Simulated change in
standard of living
Education 0.000 0.111 0.005 -0.012
Chilhood and Youth 0.000 -0.135 -0.025 0.031
Labour 0.000 0.029 -0.005 0.095
Health 0.000 -0.047 -0.020 -0.110
Standard of living -0.087 -0.046 -0.043 -0.092
37