1. Statistical Inference: Poverty Indices and Poverty Decompositions Michael Lokshin DECRG-PO The World Bank

2. 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%.

3. Solution:

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

5. 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.

6. Growth and Redistribution decomposition Transformations: Similar decomposition could be made for other poverty measures

7. 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.

8. 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.

9. Sectoral decomposition: Example for Indonesia Population was moving out of the rural sector where the poverty was falling faster – negative interaction effect.

10. 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).

11. Poverty profiles: Additivity Suppose population is divided into m mutually exclusive sub-groups. The poverty profile is the list of poverty measures P j for j=1,…,m. Aggregate poverty for additive poverty measures: Aggregate poverty is a population weighted mean of the sub-group poverty measures.

12. Additivity: Example Urban population(2,2,3,4) Rural population (1,1,1.5,2,4) Z u =3,Z r =2,n=9,n u =4,n r =5, Direct way: n=9; q=7; H=q/n=0.78

13. Additive measures Additive measures (Continued) Example of sub-group consistency: Initial state, two equally sized groups: Urban population H u =0.20;Rural H r =0.70 Total poverty rate H=0.45 Policy A: Urban population H u =0.10;Rural H r =0.70 Total poverty rate H=0.40 Policy B: Urban population H u =0.20;RuralH r =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

14. Additive measures Additive measures (Continued) What about this example of Policy C?: Urban populationH u =0.05; Rural H r =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

15. 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.

16. 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.

17. Poverty profiles: Egypt regions

18. Poverty profiles: Egypt (Type A)

19. Poverty profiles: Egypt (Type B)

20. Poverty profiles by sector: Brazil, 1996 Sector of Activity f k P 0k P 1k P 2k s k Agriculture22.02 54.17 26.87 16.85 49.88 Manufacturing13.83 16.03 6.06 3.13 9.27 Construction9.64 19.49 6.70 3.36 7.86 Services31.92 10.79 3.45 1.58 14.41 Public Sector8.13 9.96 3.25 1.42 3.39 Other/Not Specified14.46 25.12 10.93 6.51 15.19 f k – Share in total population s k – Share in population of poor

21. 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.

22. 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 (P 2 ) will be affected

23. 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.

24. 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

25. 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

26. 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.)

27. 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. n A and n B Null hypothesis: H A =H B Need to calculate t-statistic:

28. Comparison of two headcount indexes (cont.) where s denotes the standard error of the sampling distribution of H A -H B and given by: if t<1.96(2.58) the difference in H cannot be considered statistically significant at the 5% (1%) level.

29. Comparison of two headcount indexes (cont.) Example: Case 1:A=(1,2,3,4); B=(1,3,4,5,6)z=3 H A =0.75; H B =0.4 Test: H A = H B Conclusion: Reject that H A =H B at 1% level.

30. Comparison of two headcount indexes (cont.) Example: Case 2:A=(1,2,3,4); B=(1,3,3,5,6)z=3 H A =0.75; H B =0.6 Test: H A = H B Conclusion: Cannot reject that H A =H B at 5% level.

31. 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

32. Alternative estimates of standard errors: Bootstrapping A computationally intensive method that generates asymptotically valid standard errors for many test-statistics. Example of H 0 for Indonesia, 1984-1999: Year198419871990199319961999 Headcount0.41510.29200.26470.20130.16250.3508 Standard error 0.00650.00530.00370.00480.00340.0046