Comparing small area estimation methods for poverty indicators in - - PowerPoint PPT Presentation

Comparing small area estimation methods for poverty indicators in the municipalities of Minas Gerais State

Solange Correa, Debora Souza, Nicia Brendolin, Viviane Quintaes and Djalma Pessoa

New Techniques and Technologies for Statistics 2015 Brussels, Belgium, 10-12 March 2015

2

Outline

• Introduction
• Small Area Estimation Methods
• Simulation Study
• Results
• Conclusions

3

Introduction

• Demand for poverty measures at muncipality level in Brazil
• In 2008, Poverty Map published jointly by the Brazilian NSI (IBGE)

and World Bank

• Approach: Elbers, Lanjouw and Lanjouw (2003) using 2002-2003

Brazilian Consumer Expenditure Survey (CES) and Census 2000

• Objective: assessment of small area models to estimate poverty

measures: – Fay and Herriot (1979 ) - FH – Elbers, Lanjouw and Lanjouw (2003) - ELL – Molina and Rao (2010) – MR

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Small Area Estimation Methods

• Fay and Herriot (1979) (FH)

𝜄 𝑒 = 𝑕 𝑍 𝑒 = 𝜄𝑒 + 𝑓𝑒 ; 𝐹(𝑓𝑒 𝜄𝑒 = 0 ; 𝑊(𝑓𝑒|𝜄𝑒) = 𝜏𝑓,𝑒

2 (known)

𝜄𝑒 = 𝒚𝑒

′ 𝜸 + 𝑤𝑒 ; 𝑒=1,...,D; 𝐹(𝑤𝑒 𝜄𝑒 = 0 ; 𝑊(𝑤𝑒|𝜄𝑒) = 𝜏𝑤 2

• EBLUP:

𝜄𝑒

𝐹𝐶𝑀𝑉𝑄 = 𝛿

𝑒𝜄 𝑒 + (1 − 𝛿 𝑒)𝜄 𝑒; 𝑒=1,...,D 𝛿 𝑒 = 𝜏 𝑤

2

𝜏 𝑓,𝑒

2 + 𝜏

𝑤

2 ; 𝜄

𝑒 = 𝒚𝑒

′ 𝜸

• Closed expression for the MSE estimator

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Small Area Estimation Methods

• Both ELL and MR methods:

− Consider nested error linear regression model 𝑧𝑒𝑘 = 𝒚𝑒𝑘

′ 𝜸 + 𝑤𝑒 + 𝑓𝑒𝑘 ; 𝑤𝑒~ 𝑗𝑗𝑒 𝑂 0, 𝜏𝑤 2 ; 𝑓𝑒𝑘 ~ 𝑗𝑗𝑒 𝑂 0, 𝜏𝑓 2 (1)

𝑘=1,..., 𝑂𝐸; 𝑒 = 1, … , 𝐸; 𝑤𝑒 and 𝑓𝑒𝑘 indep. – Combine both Census and survey data at unitlevel to predict the welfare variable 𝑧𝑒𝑘 in the Census data – ELL uses the model fitted to the sample data to predict 𝑧𝑒𝑘 in the Census data, and then aggregate (repeated L times) – MR uses the sampled data to predict the 𝑧𝑒𝑘 for nonsampled units in each area, and then aggregate (repeated L times)

6

Small Area Estimation Methods

• Both ELL and MR methods:

– Estimate MSE of poverty measures using bootstrap/Monte Carlo – ELL: understimation of MSE if model is not correct – MR: MSE estimation is computing intensive – ELL requires availability of unit level Census data close to the time period of the survey – MR requires identification in the Census data of the area sampled units

7

Simulation Study

• Areas: municipalities of Minas Gerais (MG) State
• 2008-2009 CES MG sample: 5,028 HH; 195 municipalities (out of 853);
• 400 samples selected from the 2010 MG Census (739,762 HH)

following same CES sampling design (PSU: enumeration area; SSU: household)

• Quantities to be estimated: poverty incidence and poverty gap for

each area d

• 𝑧𝑒𝑘: per capita HH income for HH j in municiplaity d
• Poverty line z=R$255 (£90); L=1,000 replications of 𝑧 vector in the

Census data (ELL and MR)

𝐺𝐻𝑈

𝑑𝑒 = 1

𝑂𝑒 𝐺𝐻𝑈

𝑑𝑒𝑘 = 1

𝑂𝑒 𝑨 − 𝐹𝑒𝑘 𝑨

𝑑

𝐽(

𝑂𝑒 𝑘=1 𝑂𝑒 𝑘=1

𝐹𝑒𝑘 < 𝑨), 𝑑 = 0,1,2, 𝑒 = 1, … , 𝐸

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Simulation Study

• For each method and area, the following measures were computed:

– Mean of poverty measures: – Relative bias: – Relative mean square error: – Spearman's rank correlation coefficient (area ranking)

• ri(FH;POP), ri(ELL;POP), ri(MR;POP)

– Comparison of MSE estimation for all 3 methods is in progress (MR requires bootstrap)   



400 1

400 / 1

i m icd m cd

f f  

 

pop cd i pop cd m icd m cd

F F f RB / 400 / 1

400 1

 

 

 

 

 

 

 

400 1 2.

/ 400 / 1

i pop cd pop cd m icd m cd

F F f RMSE

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Results

• Poverty Incidence - Relative Bias and Relative MSE

Estimate (%) Relative Bias (%) Relative MSE (%) Population FH ELL MR FH ELL MR FH ELL MR Minimum 7.32 5.33 11.29 9.81 -43.61 -24.02 -21.34 1.20 0.04 0.05 1º Quartile 18.02 24.37 22.66 25.68 13.91 3.01 8.14 6.30 0.46 1.05 Median 26.99 33.20 29.94 32.18 24.71 12.50 19.79 12.12 1.85 4.27 Mean 29.07 35.97 32.16 34.30 28.11 17.42 27.49 21.13 7.71 15.40 3º Quartile 39.15 46.90 40.51 41.92 38.01 27.60 40.36 23.21 8.12 16.99 Maximum 65.25 76.79 65.45 66.36 145.20 134.10 177.00 326.40 186.70 320.30

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Results

• Poverty Gap - Relative Bias and Relative MSE

Estimate (%) Relative Bias (%) Relative MSE (%) Population ELL MR ELL MR ELL MR Minimum 1.93 3.46 3.22 -45.15

• 36.11

0.12 0.07 1º Quartile 6.21 7.39 9.19

• 7.99

3.82 0.68 1.12 Median 10.35 10.54 12.38 4.22 19.73 2.03 4.62 Mean 12.41 12.32 14.05 9.52 30.00 7.88 23.94 3º Quartile 17.25 16.34 17.92 21.11 47.77 5.83 23.89 Maximum 40.26 35.78 36.81 182.83 261.60 350.13 713.31

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Results

• Poverty Incidence - Distributions of the Spearman's rank correlation

coefficient for municipalities

• Similar results for the Poverty Gap

Rank Correlation Min. 1º Quartile Median Mean 3º Quartile Max. ri(FH,POP) 0.77 0.85 0.87 0.86 0.88 0.92 r1(ELL,POP) 0.95 0.96 0.97 0.96 0.97 0.97 ri(MR,POP) 0.95 0.96 0.96 0.96 0.96 0.97 ri(ELL,MR) 0.97 0.99 0.99 0.99 0.99 1.00 ri(FH,ELL) 0.78 0.85 0.87 0.86 0.88 0.92 ri(FH,MR) 0.78 0.85 0.87 0.86 0.88 0.92

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Conclusions

• All methods overestimate the poverty incidence
• ELL shows better performance
• FH shows the higher relative bias and relative MSE
• In general, relative bias and relative MSE higher for municipalities

with low poverty incidence

• ELL method shows better agreement with the poverty

incidence/poverty gap population ranking

• Same results found for groups of municipalities (microregions)

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Conclusions

• Difficulties to apply FH to real data:

– Explanatory variables at area level required (from Census or administrative registers) – MSE estimation only for sampled areas

• Difficulties to apply ELL and MR methods to real data:

– Explanatory variables at unit level available at both Census and survey data – MR: not possible to identify in the Census data the HH selected in the sample – Large number of areas with no sample

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References

• IBGE (2008) Mapa de pobreza e desigualdade: municípios brasileiros 2003,

Rio de Janeiro: IBGE.

• C. Elbers, J. Lanjouw and P. Lanjouw (2003) Micro-level estimation of

poverty and inequality, Econometrica 71, 355-364.

• I. Molina and J. Rao (2010) Small area estimation of poverty indicators,

Canadian Journal of Statistics 38, 369-385.

• IBGE (2010), Pesquisa de Orcamentos Familiares 2008-2009 Despesas,

Rendimentos e Condições de Vida, Rio de Janeiro: IBGE.

• J. Foster, J. Greer and E. Thorbecke (1984) A class of decomposable poverty

measures, Econometrica 52, 761-766.

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Additional Information

• Some examples of explanatory variables used in the models:

– Years of schooling, ethnic group and age of the head of the household – Indicator of presence in the household of durable goods, indoor toilet, water supply – Aggregated information at the Census enumeration area level

16

Results

• Poverty Incidence – Box-plots of the ratios FH/POP, ELL/POP, MR/POP

17

Results

• Poverty Gap – Box-plots of the ratios of the estimates: FH/POP,

ELL/POP, MR/POP