Statistical Inference: Poverty Indices and Poverty Decompositions

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Statistical Inference Poverty Indices and
Poverty Decompositions

• Michael Lokshin
• DECRG-PO
• The World Bank

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Problem

• Poverty rate in urban areas declined by 25
  Poverty rate in rural areas declined by 25.
• Overall poverty rate in the country declined by
  more than 50.

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Solution
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Steps in poverty analysis

• H, PG, SPG
• Change in inequality
• Growth in welfare aggregate
• Regional and Urban Rural statistics
• Decompositions
• Poverty profiles
• Simulations
• Robustness check

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Decomposing changes in poverty

• Growth versus redistribution.
• What is the relative importance of growth vs.
  redistribution?
• Growth component holds relative inequalities
  (Lorenz curve) constant redistribution component
  holds mean constant
• Gains within sectors versus population shifts.
• How important are different sectors to changes in
  poverty?
• Gains within sectors, hold initial populations
  constant population shift effects, hold initial
  poverty measures constant.

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Growth and Redistribution decomposition

• Transformations

Similar decomposition could be made for other
poverty measures
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Growth and Redistribution decomposition

• Example for Brazil in 1980s.

Very little change in poverty rising
inequality Decomposition

• No change in headcount index yet two strong
  opposing effects growth (poverty reducing)
  redistribution (poverty increasing).
• Redistribution effect is dominant for PG and
  SPG.

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Sectoral decomposition of a change in poverty

• Intra-sectoral effect the contribution of
  poverty changes within sectors controlling for
  base period population shares
• Population shift effect how much of the poverty
  in the first date was reduced by the changes in
  the population shares of sectors between then and
  the second date.
• Interaction effect arises from the correlation
  between sectoral gains and population shifts the
  sign of the interaction effect tells whether
  people tented to switch to the sectors where
  poverty was falling or not.

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Sectoral decomposition Example for Indonesia
Population was moving out of the rural sector
where the poverty was falling faster negative
interaction effect.
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Poverty profiles Overview

• A decomposition of a single aggregate poverty
  number into subgroup numbers in order to
• - Begin to understand possible determinants of
  poverty
• - Help inform targeting of anti-poverty programs
  and other policies
• Additive poverty measures (e.g., FGT class) are
  useful for profiles. Additivity guarantees
  sub-group consistency
• - when poverty increases (decreases) for any
  sub-group of the population, aggregate poverty
  will also increase (decrease).

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Poverty profiles Additivity

• Suppose population is divided into m mutually
  exclusive sub-groups.
• The poverty profile is the list of poverty
  measures Pj for j1,,m.
• Aggregate poverty for additive poverty measures
• Aggregate poverty is a population weighted mean
  of the sub-group poverty measures.

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Additivity Example

• Urban population (2,2,3,4)
• Rural population (1,1,1.5,2,4)
• Zu3,Zr2,n9,nu4,nr5,
• Direct way n9 q7 Hq/n0.78

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Additive measures (Continued)

• Example of sub-group consistency
• Initial state, two equally sized groups
• Urban population Hu 0.20 Rural Hr 0.70
• Total poverty rate H 0.45
• Policy A
• Urban population Hu 0.10 Rural Hr 0.70
• Total poverty rate H 0.40
• Policy B
• Urban population Hu 0.20 Rural Hr 0.60
• Total poverty rate H 0.40
• Policy A gain goes to richer urban areas
• Policy B gain goes to poorer rural areas
• Overall poverty is unchanged but greater
  inequality between groups under Policy A

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Additive measures (Continued)

• What about this example of Policy C?
• Urban population Hu0.05 Rural Hr 0.75
• Total poverty rate H 0.40
• Policy C enhanced gain goes to richer urban
  areas, poverty in rural areas increases
• Undesirable property of additive measures
  insensitivity to the inequality between
  sub-groups in the extent of poverty

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Poverty profiles Two types

• Two main ways to present poverty profiles
• Type A Incidence of poverty for sub-groups
  defined by some characteristics (e.g., place of
  residence)
• Type B Incidence of characteristics defined by
  the poverty status.

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Poverty profiles

• Which type is more useful will depend on the
  policy question addressed.
• Geographic targeting. Select the target region
  for poverty alleviation. If one chooses South
  more money will go to poor. So Type A is
  preferable. Minimizes the poverty gap.
• Growth promotion On the other hand, if pro-poor
  growth policies can only be implemented in one
  region, the reduction in overall number of poor
  is likely to be greater if applied to the North.

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Poverty profiles Egypt regions
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Poverty profiles Egypt (Type A)
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Poverty profiles Egypt (Type B)
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Poverty profiles by sector Brazil, 1996

• Sector of Activity fk P0k P1k P2k sk
• Agriculture 22.02 54.17 26.87 16.85 49.88
• Manufacturing 13.83 16.03 6.06 3.13 9.27
• Construction 9.64 19.49 6.70 3.36 7.86
• Services 31.92 10.79 3.45 1.58 14.41
• Public Sector 8.13 9.96 3.25 1.42 3.39
• Other/Not Specified 14.46 25.12 10.93 6.51
  15.19
• fk Share in total population
• sk Share in population of poor

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Precision of poverty estimates

• Poverty profiles imply a comparison across
  poverty measures of sub-groups.
• How do we know if observed differences in survey
  measures reflect true differences in population?
  Some potential sources of errors in surveys
  include
• Sampling error selected sample is not
  representative of underlying population or sample
  size very small in reference to total population.
• Refusal bias certain sub-groups are more likely
  to refuse survey interview than other groups.
• Instrument mis-design survey instrument misses
  relevant dimension of welfare.

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Measurement Errors

• Poverty measures could be sensitive to certain
  sorts of measurement errors in underlying
  parameters and quite robust to others.
• Case 1 If welfare indicator contains an additive
  random error with zero mean then the expected
  value of headcount index will be unbiased. One
  will predict the same H with the noisy data as
  with a precise data. However, this will not be
  true for other indicators. Any distribution-sensit
  ive measures (P2) will be affected

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Measurement Errors (cont.)

• Case 2 Errors in the mean of the distribution.
  Its being estimated that often the elasticity of
  H with respect to the mean is around 2.
  (Indonesia for Urban elasticity of H is 2.1, of
  PG is 2.9 and for SPG is 3.4). Thus 5
  underestimation of the mean of consumption
  translates into 10 overestimation of the H and
  gt 10 more poor.
• Case 3 Change in the distribution. Surveys might
  overestimate consumption of the poor and
  underestimate consumption of the rich. Hard to
  say about H. For PG and SPG, under-estimation of
  consumption of the poor -gt higher PG, SPG.

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Measurement Errors (cont.)

• Case 4. Comparison over time. Errors in rate of
  inflation. This may affect consumption of
  everyone in the same way (No change in the
  distribution). Affects both the mean and the
  poverty line gt measures of poverty will be
  unaffected.
• Head count index is usually less sensitive to
  some common forms of the measurement errors

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Hypothesis testing

• Straight-forward for additive poverty measures
  and simple random samples.
• Standard error of the sample distribution of the
  head-count index (given by binomial normal
  distribution standard for population
  proportions) is
• So, there is a 95 chance that the true value of
  H lies in the interval

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Hypothesis testing (cont.)

• Example (11,22,33,44), z3, H0.75, n4000
• Calculate 95 confidence interval
• On very small samples, the approximation might
  not be the best.

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Comparison of two headcount indexes

• Suppose you measure poverty in these two samples.
  How to test whether poverty in the first sample
  is different from the poverty in the second
  sample.
• Two distributions A and B. nA and nB
• Null hypothesis HAHB
• Need to calculate t-statistic

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Comparison of two headcount indexes (cont.)
where s denotes the standard error of the
sampling distribution of HA-HB and given by
if tlt1.96(2.58) the difference in H cannot be
considered statistically significant at the 5
(1) level.
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Comparison of two headcount indexes (cont.)

• Example
• Case 1 A(1,2,3,4) B(1,3,4,5,6) z3
• HA0.75 HB0.4
• Test HA HB
• Conclusion Reject that HAHB at 1 level.

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Comparison of two headcount indexes (cont.)

• Example
• Case 2 A(1,2,3,4) B(1,3,3,5,6) z3
• HA0.75 HB0.6
• Test HA HB
• Conclusion Cannot reject that HAHB at 5 level.

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Precision of poverty estimates

• Recommendation Quantitative poverty comparison
  which fails the above test must be considered
  ambiguous.
• These methods could be extended to other additive
  poverty measures.
• Kakwani (1990) has derived formulae for the
  standard errors for other additive measures
  including FGT. Limitations
• - One might prefer to treat the poverty line as a
  random variable
• - These formulae ignore the imprecision that
  arises when used on grouped data
• There are no general results to handle these
  problems

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Alternative estimates of standard errors
Bootstrapping

• A computationally intensive method that generates
  asymptotically valid standard errors for many
  test-statistics.
• Example of H0 for Indonesia, 1984-1999

Year 1984 1987 1990 1993 1996 1999 Headcount 0.41
51 0.2920 0.2647 0.2013 0.1625 0.3508 Standard
error 0.0065 0.0053 0.0037 0.0048 0.0034 0.0046