Statistical Inference: Poverty Indices and Poverty Decompositions

Michael LokshinDECRG-PO

The World Bank

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%.

Solution:

Population Poverty rate N-poor Year 1 Urban 100 20 20 Rural 1000 40 400 Total 1100 38.18 420 Year 2 Urban 1000 15 150 Rural 100 30 30 Total 1100 16.36 180 Change Urban - 25 percent Rural - 25 percent Total - 57.14 percent

Steps in poverty analysis: H, PG, SPG Change in inequality Growth in welfare aggregate Regional and Urban Rural statistics DecompositionsDecompositions Poverty profilesPoverty profiles Simulations Robustness checkRobustness check

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.

Growth and Redistribution decomposition

Transformations:

),/(),/();2,1(),/(),/();2,1(

ReRe);2,1();2,1();2,1(

12

12

12

LmzPLmzPrDLmzPLmzPrG

componentcomponentcomponentsidualondistributiGrowth

rRrDrGPP

rr

rr

1 221 2 12 1

2 1i i i i

m mY Y Y Ym m

Similar decomposition could be made for other poverty measures

Growth and Redistribution decompositionExample for Brazil in 1980s.

1981 1988 Headcount index (H) (%) 26.5 26.5 Poverty gap index (PG) (x100) 10.1 10.7 Squared poverty gap index (SPG) (x100)

5.0 5.6

Gini index 0.58 0.62

Very little change in poverty; rising inequality

Decomposition Growth

component Redistribution

component Interaction

effect H -4.5 4.5 0.0 PG -2.3 3.2 -0.2 SPG -1.4 2.3 -0.3

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

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.

)())((

)()(

)sec()(

1212

112

11212

effectnInteractionnPP

effectshiftPopulationPnn

effecttoralIntranPPPP

iiii

iii

iii

Sectoral decomposition: Example for Indonesia

Principal Sector of Employment

Population share (1984)

Head-count index (H)

Poverty gap index (PG)

FGT P2 measure

Farming (Self-emp.) 45.0 49.8 54.6 57.4 (Laborer) 9.0 11.2 14.8 16.5 Industry (urban) 2.6 0.8 0.4 0.3 (rural) 3.3 2.8 3.1 2.7 Construction 4.1 3.2 2.6 2.2 Trade (urban) 5.4 2.2 1.6 1.4 (rural) 6.6 7.2 5.6 4.7 Transport 3.8 3.6 2.7 2.2 Services (urban) 6.5 1.0 1.0 0.9 (rural) 5.8 2.9 2.4 2.0 Total sectoral effects (incl. omitted sectors) 89.3 93.8 95.1 Contribution of population shifts 13.2 10.4 9.4 Interaction effects -2.6 -4.3 -4.5 Total 100.0 100.0 100.0 100.0 Population was moving out of the rural sector where the poverty was falling faster – negative interaction effect.

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).

Poverty profiles: Additivity

Suppose population is divided into m mutually exclusive sub-groups.

The poverty profile is the list of poverty measures Pj for j=1,…,m.

Aggregate poverty for additive poverty measures:

Aggregate poverty is a population weighted mean of the sub-group poverty measures.

jn

ijijjj

m

jjj nyzpPandnnPP

11

/),(/

Additivity: Example Urban population (2,2,3,4) Rural population (1,1,1.5,2,4) Zu=3,Zr=2,n=9,nu=4,nr=5,

Direct way: n=9; q=7; H=q/n=0.78

4

01 1

5

01

2

1

( , ) / (3, ) / 0.75

(2, ) / 0.80

/ (0.75*4 0.80*5) / 9 0.78

un

u j iu j iu ui i

r ir ri

j jj

P p z y n p y n

P p y n

P P n n

Additive measuresAdditive 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 =0.45Policy A: Urban population Hu =0.10; Rural Hr =0.70 Total poverty rate H =0.40Policy 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

Additive measuresAdditive measures (Continued)

What about this example of Policy C?: Urban populationHu=0.05; 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

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.Region Number of persons Poverty profile Poor Non-poor Type A:

% of regional population who are poor

Type B: % of total population who are poor

South 100 100 50 33 North 200 600 25 66

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.

Poverty profiles: Egypt regions

5

3010

20

8

10

21

1554

20

2 5

% of poor % of population

Border

Upper EgyptRural

Upper EgyptUrban

Lower EgyptRural

Lower EgyptUrban

Metropolitan

Poverty profiles: Egypt (Type A)

Poverty measurements by gender of individual

29.1054 7.5857 2.838363411 63411 63411

27.1285 7.0133 2.625061876 61876 61876

28.1290 7.3030 2.7330125287 125287 125287

52.4891 13.2851 4.656051304 51304 51304

49.9152 12.4107 4.288349526 49526 49526

51.2248 12.8556 4.4754100830 100830 100830

39.5633 10.1346 3.6512114715 114715 114715

37.2588 9.4128 3.3645111402 111402 111402

38.4279 9.7790 3.5100226117 226117 226117

MeanNMeanNMeanNMeanNMeanNMeanNMeanNMeanNMeanN

Sex of PersonMale

Female

Total

Male

Female

Total

Male

Female

Total

AREAURBAN

RURAL

Total

PO P1 P2

Poverty profiles: Egypt (Type B)

Egypt 2000 (Type B)

Distribution of poor and non poor by gender and area

18456 16786 35242 52.4% 47.6% 100.0% 44955 45090 90045 49.9% 50.1% 100.0% 63411 61876 125287 50.6% 49.4% 100.0% 26929 24721 51650 52.1% 47.9% 100.0% 24375 24805 49180 49.6% 50.4% 100.0% 51304 49526 100830 50.9% 49.1% 100.0%

Count % within Poor Count % within Non-Poor Count % within Total Count % within Poor Count % within Non-Poor Count % within Total

poor

non- poor

Total

poor

non- poor

Total

AREA URBAN

RURAL

Male Female Sex of Person

Total

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

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:

1. Sampling error – selected sample is not representative of underlying population or sample size very small in reference to total population.

2. Refusal bias – certain sub-groups are more likely to refuse survey interview than other groups.

3. Instrument mis-design – survey instrument misses relevant dimension of welfare.

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-sensitive measures (P2) will be affected

Measurement Errors (cont.) Case 2: Errors in the mean of the distribution. It’s

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 => 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 -> higher PG, SPG.

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 => measures of poverty will be unaffected.

Head count index is usually less sensitive to some common forms of the measurement errors

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:

Hypothesis testing

( (1 ) / )S H H n

nHHHHnHHH /)1(96.1/)1(96.1

Example: (1…1,2…2,3…3,4…4), z=3, H=0.75, n=4000

Calculate 95% confidence interval:

On very small samples, the approximation might not be the best.

Hypothesis testing (cont.)

764.0736.0014.075.0014.075.0

4000/)75.01(75.096.175.04000/)75.01(75.096.175.0

HHH

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: HA=HB Need to calculate t-statistic:

sHHt BA )(

Comparison of two headcount indexes (cont.)

sHHt BA )(

where s denotes the standard error of the sampling distribution of HA-HB and given by:

)()()11)(1(

BA

bBAA

BA nnHnHnHand

nnHHs

if t<1.96(2.58) the difference in H cannot be considered statistically significant at the 5% (1%) level.

Comparison of two headcount indexes (cont.)

Example:Case 1: A=(1,2,3,4); B=(1,3,4,5,6) z=3

HA=0.75; HB=0.4

Test: HA = HB

Conclusion: Reject that HA=HB at 1% level.

95

)54()4.0*575.0*4(

)()(

BA

bBAA

nnHnHnH

11.0)51

41)(

951(

95)11)(1(

BA nnHHs

58.218.311.0

40.075.0)(

sHHt BA

Comparison of two headcount indexes (cont.)

Example:Case 2: A=(1,2,3,4); B=(1,3,3,5,6) z=3

HA=0.75; HB=0.6

Test: HA = HB

Conclusion: Cannot reject that HA=HB at 5% level.

32

)54()6.0*575.0*4(

)()(

BA

bBAA

nnHnHnH

96.15.11.0

60.075.0)(

sHHt BA

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

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 0.4151 0.2920 0.2647 0.2013 0.1625 0.3508

Standard error 0.0065 0.0053 0.0037 0.0048 0.0034 0.0046