Parallel session IARIW 2014

1. Background
New Developments
Summary
INEQUALITY AND POVERTY IN AFRICA
Regional Updates and Estimation of a Panel of Income
Distributions
D. Chotikapanich,1 W. Griffiths,2 G. Hajargasht2 and C. Xu2
1Monash University, Australia
2University of Melbourne, Australia
33rd IARIW General Conference, Rotterdam, 2014
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

2. Background
New Developments
Summary
Outline
1 Background
Income Distribution and Inequality in Africa
Estimation of Parametric Income Distributions from Grouped
Data
2 New Developments
Interpolation and Extrapolation of Income Densities
Results
Possible Extensions
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

3. Background
New Developments
Summary
Income Distribution and Inequality in Africa
Estimation of Parametric Income Distributions from Grouped Data
Outline
1 Background
Income Distribution and Inequality in Africa
Estimation of Parametric Income Distributions from Grouped
Data
2 New Developments
Interpolation and Extrapolation of Income Densities
Results
Possible Extensions
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

4. Background
New Developments
Summary
Income Distribution and Inequality in Africa
Estimation of Parametric Income Distributions from Grouped Data
Income Distribution in Sub-Saharan Africa
Income distribution in Sub-Saharan Africa (SSA) is critical to
understanding world inequality and poverty.
High economic growth in the region - approx 5% in 2013.
Nonetheless growth is uneven, and the number of absolute
poor (below $1USD or $2USD per day) has increased.
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

5. Background
New Developments
Summary
Income Distribution and Inequality in Africa
Estimation of Parametric Income Distributions from Grouped Data
Income Distribution in Sub-Saharan Africa
Uneven growth is key concept.
Growth can vary (i) within countries, (ii) between countries,
and (iii) over time.
A panel approach is therefore desirable for modeling income
distribution.
Can assess (and decompose) inequality, poverty and pro-poor
growth.
Ultimate goal is to produce panel for all countries in all recent
years.
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

6. Background
New Developments
Summary
Income Distribution and Inequality in Africa
Estimation of Parametric Income Distributions from Grouped Data
Data Limitations
Determining regional or world distribution of income is
complicated by data limitations.
Data are in grouped (normally decile form).
Mix of income and consumption.
Many missing time periods.
Dealing with these limitations requires some methodological
innovations.
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

7. Background
New Developments
Summary
Income Distribution and Inequality in Africa
Estimation of Parametric Income Distributions from Grouped Data
Outline
1 Background
Income Distribution and Inequality in Africa
Estimation of Parametric Income Distributions from Grouped
Data
2 New Developments
Interpolation and Extrapolation of Income Densities
Results
Possible Extensions
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

8. Background
New Developments
Summary
Income Distribution and Inequality in Africa
Estimation of Parametric Income Distributions from Grouped Data
Obtaining Parametric Densities
Methodology comes from Chotikapanich et al (2007) and
Hajargasht et al (2012).
Assume y1, y2, ..., yT (unobserved) individual incomes drawn
from parametric density f (y;).
These are grouped into N income classes {z0, z1}, {z1, z2},...,
{zN−1, zN}.
Data on population shares c and class mean incomes
(y1, y2, ..., yN).
Task is to estimate and z1, ..., zN−1.
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

9. Background
New Developments
Summary
Income Distribution and Inequality in Africa
Estimation of Parametric Income Distributions from Grouped Data
GMM Estimation
Use the population shares (e.g. deciles) and class mean
incomes as moment conditions.
H(P) = 1 h (yi , ) where E [H()] = 0 and
Tt
1T
= =
z1, ..., zN−1,00.
Can write as
H() =
2
6666666664
c1 − k1 ()
...
cN−1 − kN−1 ()
~y1 − μ1 ()
...
~yN − μN ()
3
7777777775
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

10. Background
New Developments
Summary
Income Distribution and Inequality in Africa
Estimation of Parametric Income Distributions from Grouped Data
GMM Estimation
Can estimate ^
= argmin H()0WH() where W is the
!
weighting matrix.
dWhat to use for W? Obvious candidate is I, however the
latter set of moment conditions will overwhelm the former.
Optimal weighting matrix is derived (messy!) in Hajargasht et
al (2012).
Can
test validity
of
moment conditions as
TH
^
0W
^
H
^
2
N−K (essentially a test of
distributional form).
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

11. Background
New Developments
Summary
Income Distribution and Inequality in Africa
Estimation of Parametric Income Distributions from Grouped Data
Which Parametric Form?
This framework can be used to estimate any parametric
distribution.
In general the authors have focused on GB2 distributions.
This paper uses a mixture of lognormals
f (y; ) =
JX
j=1
wj
yp2j
exp
−
(ln y −

12. j )2
22
j
!
Highly flexible (8 parameters) and can have up to J modes.
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

13. Background
New Developments
Summary
Interpolation and Extrapolation of Income Densities
Results
Possible Extensions
Outline
1 Background
Income Distribution and Inequality in Africa
Estimation of Parametric Income Distributions from Grouped
Data
2 New Developments
Interpolation and Extrapolation of Income Densities
Results
Possible Extensions
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

14. Background
New Developments
Summary
Interpolation and Extrapolation of Income Densities
Results
Possible Extensions
Interpolation of Income Distributions
Frequently data for a country are missing.
E.g. If data is available for country X for years 2000 and 2004,
how do we obtain estimates for 2001, 2002, 2003 and 2004?
Decile shares can be linearly interpolated, and then the
density can be fitted to the interpolated shares.
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

15. Background
New Developments
Summary
Interpolation and Extrapolation of Income Densities
Results
Possible Extensions
Extrapolation of Income Distributions
Suppose we have data for a country for 2000 and 2004, but
we wish to model the distribution for 2004 onwards.
Look to similar countries that have similar distributions.
Measure similarity with symmetric KL divergence
D (f (yA/μA) k f (yB/μB)) =
Z 1
0
(f (yA/μA) − f (yB/μB)) ln f (yA/μA)
f (yB/μB)
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

16. Background
New Developments
Summary
Interpolation and Extrapolation of Income Densities
Results
Possible Extensions
Extrapolation of Income Distributions
Use the SKL divergence to determine which countries are
similar (e.g. in 2004)
Can then measure the change in density for similar countries
from one period to the next (e.g. 2004 to 2005).
Use the SKL divergence as weights.
Change from 2004 to 2005 is then the weighted average of
those in similar countries.
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

17. Background
New Developments
Summary
Interpolation and Extrapolation of Income Densities
Results
Possible Extensions
Outline
1 Background
Income Distribution and Inequality in Africa
Estimation of Parametric Income Distributions from Grouped
Data
2 New Developments
Interpolation and Extrapolation of Income Densities
Results
Possible Extensions
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

18. Background
New Developments
Summary
Interpolation and Extrapolation of Income Densities
Results
Possible Extensions
Results
Focus on only 10 countries (40 studied).
These are Central African Republic; Ethiopia; Ghana; Kenya;
Mozambique; Nigeria; Senegal; Sierra Leone; South Africa
and Tanzania.
Estimates are generally very stable over time. This is a good
sign.
Results from Ethiopia bounce around a bit, but this is a
feature of the data (not model driven).
Reductions in poverty in most cases, no real trend in
inequality.
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

19. Background
New Developments
Summary
Interpolation and Extrapolation of Income Densities
Results
Possible Extensions
Outline
1 Background
Income Distribution and Inequality in Africa
Estimation of Parametric Income Distributions from Grouped
Data
2 New Developments
Interpolation and Extrapolation of Income Densities
Results
Possible Extensions
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

20. Background
New Developments
Summary
Interpolation and Extrapolation of Income Densities
Results
Possible Extensions
Distance Measures
Symmetric Kullback Leibler Divergence is logical choice of
distance measure.
Can be very tail sensitive.
Similar looking distributions can have large divergences due to
differing behavior at either tail.
Bhattacharyya distance might be less sensitive.
B (f (yA/μA) k f (yB/μB)) =
Z 1
0
q
f (yA/μA) f (yB/μB)
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

21. Background
New Developments
Summary
Interpolation and Extrapolation of Income Densities
Results
Possible Extensions
Latent Classes
The success of mixtures of two or three lognormals in
modeling income in SSA might be due to the presence of
latent classes withing the income distribution.
E.g. perhaps one lognormal is picking up subsistence
agriculture, while another models everything else.
Difficult to know if this is the case but interesting to
speculate, especially if distributional change is occurring in
one mixing density but not the other(s).
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

22. Background
New Developments
Summary
Interpolation and Extrapolation of Income Densities
Results
Possible Extensions
Different Parametric Forms
There has been great effort expended in using highly flexible
specifications.
One option not tried is a mixture of continuous and discrete.
Income microdata frequently contains zeros, which are not
permitted.
Consider
Y
(
0 with probability p 2 [0, 1]
f (y; ) with probability 1 − p
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

23. Background
New Developments
Summary
Interpolation and Extrapolation of Income Densities
Results
Possible Extensions
Different Parametric Forms
Figure : Lognormal with point mass at zero
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

24. Background
New Developments
Summary
Interpolation and Extrapolation of Income Densities
Results
Possible Extensions
Consumption and Income
One difficulty with this type of study is that it combines
income and consumption data.
Consumption is typically more equal than income, so the
variable matters.
The distance function approach could be used to construct
“shadow” income and consumption distributions for each
country.
Then determine the world distribution of both these variables
separately.
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

25. Background
New Developments
Summary
Interpolation and Extrapolation of Income Densities
Results
Possible Extensions
Consumption and Income
Draw data from micro-data sets that contain both income and
consumption data.
Obtain these sets for a range of different countries.
Determine the distributional change between income and
consumption for sets of countries (e.g. South East Asia).
Use distance measures to determine weights.
Estimate the correction to be applied to convert a
consumption density into an income density and vice versa,
based on a distance-weighted average.
Do this for all countries.
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

26. Background
New Developments
Summary
Interpolation and Extrapolation of Income Densities
Results
Possible Extensions
Consumption and Income
Let f (x) denote a normalized income distribution and f (c)
be normalized consumption.
Wish to estimate f (c1 ) based on
I =
n
f (x 1 ) , f (x 2 ) , ..., f
x q
o
and
C =
n
f (c2 ) , f (c3 ) , ..., f
cq
o
.
Determine gp (x; c) = f (x p ) − f
cp
for countries p = 2...q
noting that R1
0 g (x; c) = 0 in each case.
Determine pairwise entropies
D (f (x1 )
k f (x2 )),
D (f (x1 ) k f (x3 )),..., D
f (x1 ) k f (xq )
.
Set wp / D
f (x 1 ) k f (x p )
where Pwp = 1.
Estimate ~f (c1 ) = f (x 1 ) +
Pwpgp (x; c)
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA

27. Background
New Developments
Summary
Summary
Highly sophisticated estimation of income densities.
Interpolation and extrapolation methods allow for missing
values to be imputed.
These techniques allow for the world inequality, poverty and
pro-poor growth rates to be determined at much higher levels
of detail.
D. Chotikapanich, W. Griffiths, G. Hajargasht and C. Xu INEQUALITY AND POVERTY IN AFRICA