The document discusses two papers that examine sensitivity and robustness issues related to the Alkire-Foster multidimensional poverty index. The first paper (NRS) proposes modified versions of the index that account for inequality among the poor over dimensions and time. The second paper (TAK) analyzes disparities between monetary and multidimensional poverty measures in Vietnam and how they change over time. Both papers apply their modified indices to panel data from China and Vietnam, finding that monetary and multidimensional poverty rates differ and have different dynamics over time.

1. Duration and Multidimensionality in Poverty Measurement
Aaron Nicholas, Ranjan Ray, Kompal Sinha (NRS)
and
Disparities between monetary and multidimensional measurements of
poverty in Vietnam
Quang-Van Tran, Sabina Alkire, Stephan Klasen (TAK)
Rotterdam 2014
Gordon Anderson.

2. Both papers are concerned in some sense with sensitivity
and robustness issues surrounding versions of the Alkire –
Foster multidimensional index, but in different contexts.
The Alkire – Foster Index.
For a sample of N agents with D dimensions of potential deprivation a generalized version of the index
for the K’th level of deprivation may be written as:
N D j
 
   
w x z x
d nd d nd
 
M I c K I
     
N D z z  
   
1 1
1
1 [1]
Kj n
n d d d
Here I( ) is an indicator function (which is 1 when its argument is greater than 0 and 0 when its
argument is 0 or less), cn is a count of the number of dimensions in which the n’th agent is deprived, wd
is a weight attached to the d’th deprivation dimension, xnd is the level experienced by the n’th agent in
the d’th dimension zd is the deprivation threshold in the d’th dimension and j ≥ 0 is an FGT index
coefficient measuring various degrees of depth of poverty sensitivity.
When j >0 the index is sensitive to the depth of deprivation (NRS are interested in this aspect), when j =
0 it is essentially a “Mashup Counting Measure” in the terminology of Ravallion (2011) (TAK are
interested in this aspect).
Setting K = 0 constitutes the Union approach (deprivation in at least one dimension) to measurement,
setting it to D-1 constitutes the intersection (deprivation in all) approach. NRS set K = 0 TAK set K to a
percentage of weighted D.

3. The Coverage Problem
Diagram 1. Alkire-Foster vs U* in 2 Dimensions.
-
-
-
-
-

4. NRS “..the contribution of our proposed measure is the expansion of ways in
which to think about the depth of poverty among the poor, rather than
whom to consider poor…”
• Concerned with transfer sensitivities over K dimensions and T time periods
in depth of poverty measures (j=1)
• Think of x as indexed over N agents n, D deprivations d, and T time
periods t.
• A discussion of the sensitivity to the distribution of deprivations of
multidimensional poverty measures highlights a limitation of A-F in that a
ceteris paribus switch in achievements across poor individuals in a
dimension (or time frame) does not effect the index yet can increase
inequality amongst the poor in that dimension (or time frame).
• Prompts definition of inequality sensitivity properties (A1 Progressive
Dimensional Rearrangement and A2 Progressive Dynamic Rearrangement)
such that Mkj(x) > Mkj( ’) f ’ f m progressive
dimensional (dynamic) rearrangement.
• In turn this prompts modified versions of A-F in dimensions, time and
dimensions and time which satisfy a wide range of axioms.

5. The Proposed “Dimension” Modified AF Statistic (weights
each deprivation input according to the normalized
sum of all deprivation inputs)
 
   
       
   
N J
nj n
 
 
1 1
 

|


1



:
" "
n
n j
t
J
njt
j
n
n
d S
C
J
N
d
where S
J
note C is the poverty indicator operator
S is average deprivations

6. The Proposed “Time” Modified AF Statistic (weights
each deprivation input by the normalised sum of all
deprivation inputs over dimensions over time)
 
   
   
   
N T
 
nt n
 
1 1
 

|


1



:
" "
n
n t
j
T
nt
t
n
n
d S
C
T
N
d
where S
T
note C is the poverty indicator operator
S is average deprivations

7. No Prizes for guessing what’s coming next!
Note the importance of β for time vs dimension.
   
       
   
N T J
 
  
1 1 1
|
njt njt
 
1 1 (1 )
0 1
0 1
n
n t j
t
J T
njt njt
j t
njt
njt
d S
C
JT
N
d d
where S
J T
where
and S
 

 


 

 
 
  
 
 

8. “The Tradeoff Between Dimensional
and Durational poverty”
• To study the tradeoff Bp = Ωα,β=1/2Ωα,β=0.5 is proposed (i.e. the proportion
of overall poverty attributable to a concentration of multiple dimensions of
deprivation in particular periods).
• 퐿푝, the proportion of overall poverty attributable to a concentration of
multiple periods of deprivation in particular dimensions is equal to 1 - Bp.
• These statistics can be compared across subgroups.
• 퐵푝 can be interpreted thus: approximately 1/3 of overall poverty in the
sample can be attributed to multiple dimensions within specific periods and
the remaining 2/3can be attributed to repeated periods within specific
dimensions.

9. Application
• The proposed dynamic measure of multidimensional poverty is
applied to a panel data set from China from 1993-2009.
• The data came from the China Health and Nutrition Survey (CHNS).
• Formed 2 samples (both a balanced panel) the former with
primarily qualitative data (13 dimensions , α irrelevant), the latter
with quantitative data (3 dimensions, α=1).
• The sensitivity to choices of β is considered across a collection of
subgroups (Male-Female, Provinces, Urban Rural), and across
dynamic and dimensional specifications over the two samples.
• Two measures are considered one Ω , which allows the ranking of
subgroups according to the highest average poverty score per
person while the other Ω(Ns), gives the percentage contribution of
each particular subgroup to the overall poverty score. Results are
presented in 4 tables.

10. Specific Comments on NRS
• Would have like to have seen something on distinguishing
between sustained deprivation over time vs in and out of
deprivation, a notion of time preference (two ways to do
it).
• Possible to modify AF with cross product terms to allow for
complimentary and substitutable deprivations (Fleurbaye).
• Allow cutoffs Fj to have a time dimension.
• Ultimately we do not get a great deal of mileage out of
varying β (the dimension vs time mix parameter).
• Would have been interesting to see how the “coverage”
changed wrt changing β. What were the characteristics of
the poor when duration mattered as opposed to when it
did not? But then:-

12. Spearmans Rank Correlation of results for provincial
rankings (All other coeffs were = 1).
• Spearman rank coeffs for tables 2 and 3, for various models A (no
transfer sensitivity) B time and dimension transfer sensitivity (β =
0.5) C no depth component
• Sample 1 Sample 2
• Ω(Ns) A v B 0.89285714 0.91836735
• A v C 0.89285714 0.91836735
• B v C 1.0000000 1.0000000
• Ω A v B 1.0000000 1.0000000
• A v C 1.0000000 1.0000000
• B v C 1.0000000 1.0000000
• Standard errors 0.25821704 0.25821704
• final table rank coeff, (std error) 0.98901099 (0.18258702)
• Conclusion varying B does not have any effect on the ranks.

13. TAK is concerned with A-F coverage versus a simple monetary
measure of poverty in the context of basic A-F count measures (j=0)
the coverage is also examined over time.
• Use panel data from over 2000 Vietnamese households collected in 2007,
2008 and 2010 to identify which sub-groups of the population are
monetary poor and/or multidimensionally poor they analyze the dynamics
of those two measures of poverty over time.
• Probit models and transition matrices are the primary source of analysis.
• The results show that there is much disparity between the monetary and
multidimensional measures of poverty. Also, the disparity varies across
sub-groups of the population depending on households' characteristics
and their access to markets.
• Both measures improve over time but monetary poverty is more time
variant. Household and head charectersitics determine the dynamics of
monetary and also MD poverty and Health is a key driver (among the
dimensions) of MD poverty transitions.
• The authors conclude that Economic growth seems to provide relief in the
monetary dimension but not so much in the multidimensional realm
during the early growth years.

15. Some Details
• This study defines a person as being multi-dimensionally
poor if he or she is deprived in at
least 30 percent of the dimensions. The poverty
rate at this cutoff is approximately equal to the
poverty rate measured by consumption at $2.00
in 2007 ($1.67 consumption cutoff and 38% of
dimensions are also used for the same reason).
• A slight modification to [1] with the average
number of deprivations among the poor (an
intensity of poverty measure) replacing the depth
of poverty component ((z-x)/z)j.

16. 16
Deprivation and the multidimensionally poor, percent
Indicators Raw headcount
(Population deprived )
Weight Censored headcount
(Population poor and
deprived )
Contribution
to MPI
2007 2008 2010 2008 2008
Nutrition 30.6 28.2 29.7 16.7% 21.3 26.9
Functioning 30.3 21.7 26.0 16.7% 16.6 24.0
Schooling 11.1 10.2 8.8 16.7% 8.3 9.5
Attendance 5.1 4.9 5.1 16.7% 3.7 5.6
Cooking fuel 82.8 80.0 68.3 5.6% 34.4 10.7
Sanitation 79.2 76.8 66.3 5.6% 33.9 9.3
Drink water 81.1 75.8 69.7 5.6% 32.1 9.4
Electricity 2.2 1.1 1.1 5.6% 0.4 0.2
Housing 7.2 6.0 5.7 5.6% 2.9 1.7
Assets 12.4 9.3 6.6 5.6% 5.3 2.7
Total / Average 100% 16.1 100

17. 17
Marginal effects of probit models at 38% and $1.67, 2008
2007 2008 2010
MN poor MD poor MN poor MD poor MN poor MD poor
Household size 0.0513*** -0.00155 0.0310*** -0.0117** 0.0326*** -0.00590
(0.00634) (0.00566) (0.00453) (0.00499) (0.00481) (0.00499)
Minority groups 0.431*** 0.113*** 0.372*** 0.0940*** 0.413*** 0.0807***
(0.0352) (0.0327) (0.0388) (0.0299) (0.0389) (0.0293)
Primary school -0.0667** -0.0929*** -0.0498** -0.0793*** -0.0727*** -0.0607***
(0.0310) (0.0250) (0.0202) (0.0208) (0.0208) (0.0225)
Middle school -0.199*** -0.303*** -0.135*** -0.254*** -0.227*** -0.244***
(0.0331) (0.0301) (0.0259) (0.0275) (0.0288) (0.0289)
Secondary+ -0.252*** -0.223*** (omitted) -0.166*** -0.149*** -0.148***
(0.0159) (0.0125) (0.0113) (0.0111) (0.0140)
Coastal -0.0477 0.0405 0.0141 0.0283 -0.0623*** -0.0105
(0.0296) (0.0297) (0.0239) (0.0262) (0.0208) (0.0242)
Plain 0.0335 0.0133 -0.00368 0.00105 -0.0443** -0.0373*
(0.0278) (0.0252) (0.0210) (0.0221) (0.0202) (0.0206)
Provincial FE YES YES YES YES YES YES
Observations 1,865 1,865 1,761 1,866 1,866 1,866
Pr (Yit =1) = it + βitXit + it

18. Monetary and multidimensional poverty transition matrices
MN poor Monetary poor 2010 Multidimensionally poor 2010 MD poor
2007 Ext. Mod. N on. Total Ext. Mod. N on. Total 2007
Ext. 8.1 9.8 3.9 21.8 9.0 8.7 4.4 22.0 Ext.
Mod. 3.3 13.7 17.9 34.9 6.0 13.8 14.9 34.6 Mod.
Non. 1.1 6.1 36.1 43.3 2.1 12.1 29.2 43.4 Non.
Total 12.5 29.6 57.9 100.0 17.1 34.5 48.4 100.0 Total
Ext.: extremely poor, at $1.48 a day in monetary dimension and 31 percent in
multidimensional measure.
Mod.: moderately poor, at $1.48-$2.46 in monetary measure and 19-31 percent in
multidimensional measure.
Non. refers to non-poor, which refers to $2.46 in monetary measure and 19 percent
in multidimensional measure.

19. Specific Comments on TAK
• ? Income index, is household income adjusted for household size. ?
• Transition matrices in Table 1.7 (call them T) are not transition matrices,
they are joint probability matrices (PM): T=PM*P-1 where P is a diagonal
matrix with period t-1 category probabilities on the diagonal
(sumsum(PM)=1, sumsum(T)=K).
• ? Potential endogeneity problem with education variable in the probit
regressions (education determines dimensionally poor).
• Quite straight forward to test for common parameters in the probit
equations .
y X 0
1 1
y e
y X X
2 2 2


 
    
  V

20. Think about the characteristics of
transitions
• For Time Transitions
• I(p7 ∩ p10) = xβ1 + e1
• I(np7 ∩ p10) = xβ2 + e2
• I(p7 ∩ np10) = xβ3 + e3
• I(np7 ∩ np10) = xβ4 + e4
• (note one of these equations is redundant, coefficients add up)
• For Money v Dimension Poor
• I(pM ∩ pD) = xβ1 + e1
• I(npM ∩ pD) = xβ2 + e2
• I(pM ∩ npD) = xβ3 + e3
• I(npM ∩ npD) = xβ4 + e4
• (note one of these equations is redundant, coefficients add up)

21. Final Thoughts on Multidimensional methods.
• T “ f D m ” is a problem familiar to non-parametric
econometricians, increasing the demands placed on data.
• Problems arise in part because density surfaces become flatter and in part
because notionally similar points in K dimensional space become further
apart as K increases.
• e.g. for the joint density of K i.i.d. standard normal variables, where 0 is the
K x 1 null vector, and the peak f(0)=1/(2π)K/2 which goes to 0 as K
increases and the Euclidean distance between 0 and 1 (the unit vector) is
√K g K
• Mass at the center of the distribution empties out as dimensions increase
(K w N N w g “Em S P m”)
• T “f g” f m m m ff
distinguish between them i.e. changing circumstances in many dimensions
will be a lot less apparent than in a few dimensions, we need to be more
circumspect in how we approach multidimensionality. There is a cost to
“ g m ”