Estimation of Poverty Measures in Small Areasec.europa.eu/.../4398377/S3P3-ESTIMATION-OF-POVERTY-MEASURE… · FGT POVERTY MEASURES X Complex non-linear quantities ... Estimation

INTRODUCTION SMALL AREA ESTIMATION EBLUP METHOD EB METHOD APPLICATION SIMULATIONS

Estimation of Poverty Measures in SmallAreas

Isabel MolinaDepartamento de Estadıstica, Universidad Carlos III de Madrid

J.N.K. RaoSchool of Mathematics and Statistics, Carleton University Página 1 de 1

04/02/2009http://www.sample-project.eu/images/logo_sample_trasp.png



FGT POVERTY MEASURES

• Finite population of size N.

• Ej welfare measure for indiv. j : for instance, normalized annualnet income.

• z poverty line: in Spain, INE uses:

z = 0.6×Median(Ej ).



FGT Family (Foster, Greer & Thornbecke, 1984):

Fα =1

N

N∑j=1

(z − Ej

z

)αI (Ej < z), α = 0, 1, 2.

• α = 0⇒ Poverty incidence

• α = 1⇒ Poverty gap

• α = 2⇒ Poverty severity



X Complex non-linear quantities (non continuous).

X Not easy to obtain estimators with good bias and MSEproperties.

X A method valid to estimate poverty measures for any α andfor other poverty or inequality measures would be desirable.


SMALL AREA ESTIMATION

• Due to the relative nature of the poverty line, poverty hasusually low frequency.

X In Spain, the poverty line for 2006 is 6557 euros, approx.20 % population under the line.

• Survey on Income and Living Conditions (EU-SILC) haslimited sample size.

X In the Spanish SILC 2006, n = 34,389 out ofN = 43,162,384 (8 out 10,000).


SAMPLE SIZES OF PROVINCES BY GENDER

• Direct estimators at Spanish provinces are not very precise.

• Provinces × Gender → Small areas.

• Descriptives of sample sizes and CVs of poverty incidences forthe 52 × 2 areas:

nd Obs. Pour CV Dir. CV EB

Min 17.0 4.0 6.3 3.6Q1 128.5 27.0 9.3 5.9

Median 231.5 58.5 11.8 7.4Mean 330.7 67.9 13.6 8.8

Q3 475.3 96.3 15.7 10.5Max 1483.0 267.0 40.4 20.3


MODELS FOR SMALL AREA ESTIMATION

• Necessary to apply Small Area Estimation Techniques toobtain estimators with lower error.

• Regression models assume that the relation between theresponse variable and the explanatory variables is constantacross all areas.

• They allow to estimate within an area using the observed datafrom the whole population.


NESTED ERROR MODEL

X Population with D areas of sizes N1, . . . ,ND .

X The variable of interest Y has Normal distribution.

X Usual target quantities are the means of Y in the areas,

Yd =1

Nd

Nd∑j=1

Ydj , d = 1, . . . ,D.

X We have data from p auxiliary variables X = (X1, . . . ,Xp)′ atindividual level.

X A sample is drawn from the population,sd sample units from area drd non-sample units from area d


NESTED ERROR MODEL

X Model:

Ydj = x′djβ + ud + edj , j = 1, . . . ,Nd , d = 1, . . . ,D.

udiid∼ N(0, σ2

u), edjiid∼ N(0, σ2

e ).

X Model fitting: → β, σ2u, σ2

e , ud , d = 1, . . . ,D.

X Best Linear Unbiased Predictor (BLUP) of Yd :

ˆYd =1

Nd

∑j∈sd

Ydj +∑j∈rd

Ydj

, d = 1, . . . ,D,

where Ydj = x′dj β + ud .


PROBLEMS

X Even if FGT poverty measures are also means

Fαd =1

Nd

Nd∑j=1

Fαdj , Fαdj =

(z − Edj

z

)αI (Edj < z),

we cannot assume normality for the Fαdj .

X We would like to have a method suitable to estimate FGTmeasures for different α.

X We would like to estimate other poverty or inequalitymeasures.


EB METHOD (EMPIRICAL BEST/BAYES)

X Variable of interest for all population unitsy = (Y1, . . . ,YN)′ = (y′s , y

′r ).

X We want to estimate δ = h(y).

X The estimator δ that minimizes the MSE is

δEB = Eyr (δ|ys).


EB METHOD FOR POVERTY ESTIMATION

X We assume that there exists a transformation Ydj of thewelfare variables Edj with Normal distribution

y ∼ N(µ,V).

X The distribution of yr given ys is

yr |ys ∼ N(µr |s ,Vr |s),

where

µr |s = µr + VrsV−1s (ys − µs), Vr |s = Vr − VrsV−1

s Vsr .


EB METHOD FOR POVERTY ESTIMATION

X The EB estimator of Fαd is

F EBαd = Eyr

1

Nd

Nd∑j=1

Fαdj | ys

=1

Nd

Nd∑j=1

Eyr [Fαdj | ys ]

=1

Nd

∑j∈sd

Fαdj +∑j∈rd

Eyr [Fαdj |ys ]

,

X Observe that

Fαdj =

(z − Edj

z

)αI (Edj < z) = hα(Ydj ).


MONTE CARLO APPROXIMATION

• The conditional expectation can be approximated by MonteCarlo.

• Generate L non-sample vectors y(`)r , ` = 1, . . . , L from the

conditional distribution of yr |ys .

• Calculate F(`)αdj = hα(Y

(`)dj ), ` = 1, . . . , L and take

Eyr [Fαdj |ys ] ∼=1

L

L∑`=1

F(`)αdj .


APPLICATION

• We assume the nested error model for the log-normalizedincome.

• We take as explanatory variables, the indicators of 5 agegroups, of having Spanish nationality, of 3 education levelsand of the situation against activity.

• We estimate the MSE by parametric bootstrap for finitepopulations (Gonzalez-Manteiga, Lombardıa, Molina, Moralesand Santamarıa, 2008).


RESULTS

−15 −10 −5 0 5

30

35

40

45

Poverty incidence − Men

under 1515 − 2020 − 2525 − 30over 30

−15 −10 −5 0 5

30

35

40

45

Poverty incidence − Women

under 1515 − 2020 − 2525 − 30over 30


RESULTS

−15 −10 −5 0 5

30

35

40

45

Poverty gap − Men

under 55 − 7.57.5 − 1010 − 12.5over 12.5

−15 −10 −5 0 5

30

35

40

45

Poverty gap − Women

under 55 − 7.57.5 − 1010 − 12.5over 12.5


MODEL-BASED EXPERIMENT

Sizes:

N = 20000D = 80Nd = 250, d = 1, . . . ,D.

Variance components:

σ2e = (0,5)2

σ2u = (0,15)2



Explanatory variables:

X1 ∈ {0, 1}, p1d = 0.3 + 0.5d/20, d = 1, . . . ,D.

X2 ∈ {0, 1}, p2d = 0.2, d = 1, . . . ,D.

Coefficients:β = (3, 0.03,−0.04)′.

X The response increases when moving from X1 = 0 to X1 = 1,and decreases when moving from X2 = 0 to X2 = 1.

X The “richest” people are those with X1 = 1 and X2 = 0.

X The last areas have “richer” individuals than the first areas,i.e., poverty decreases with the area index.



X We simulated I = 1000 populations from the nested errormodel;

X For each population, we computed the true domain povertymeasures.

X We computed the MSE of the EB estimators as

MSE(F EBαd ) =

1

I

I∑i=1

(F

EB(i)αd − F

(i)αd

)2, d = 1, . . . ,D.

X Idem for direct estimators and for WB estimators (Elbers,Lanjouw & Lanjouw, 2003).


POVERTY INCIDENCE

a) Means (×100)

0 20 40 60 80

15.0

15.5

16.0

16.5

Area

Pov

erty

inci

denc

e (x

100)

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b) MSEs (×104)

0 20 40 60 8010

2030

4050

6070

Area

MS

E p

over

ty in

cide

nce

(x10

000)

EB Sample ELL

Figure 1. a) Means and b) MSEs of true values and of EB, direct and

ELL estimators of poverty incidences F0d for each area d .


POVERTY GAP

a) Means (×100)

0 20 40 60 80

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Area

Pov

erty

gap

(x1

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b) MSEs (×104)

0 20 40 60 801

23

45

6

Area

MS

E p

over

ty g

ap (

x100

00)

EB Sample ELL

Figure 2. a) Means and b) MSEs of true values and of EB, direct and

ELL estimators of poverty gaps F1d for each area d .


BOOTSTRAP

a) MSE of poverty incidence

0 20 40 60 80

10.0

10.2

10.4

10.6

10.8

Area

MS

E P

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ty in

cide

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● True MSEBootstrap MSE

b) MSe of poverty gap

0 20 40 60 800.

780.

800.

820.

840.

86

Area

MS

E P

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ty g

ap (

x100

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● True MSEBootstrap MSE

Figure 3. True MSEs and bootstrap estimators (×104) of EB estimators

with B = 500 for each area d .


DESIGN-BASED EXPERIMENTX Generate only ONE population from the nested error model;

X Draw I = 1000 stratified samples with SRSWR from each prov.

0 20 40 60 80

15.0

15.5

16.0

16.5

Area

Pove

rty in

ciden

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100)

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MEAN SQUARED ERROR: POVERTY INCIDENCE

0 20 40 60 80

1020

3040

5060

70

Area

MSE

pov

erty

incide

nce

(x10

000)

EB Sample ELL


CONCLUSIONS:

• We propose the EB method with Monte Carlo approximationof the estimator.

• Method applicable to unit level data.

• Method applicable to many types of population measures.

• We obtain the estimator with lowest MSE under the model.


REFERENCES

Fay, R. & Herriot, R. (1979).Estimates of income for small places: an application ofJames-Stein procedures to census data.Journal of the American Statistical Association 74:269–277.

Gonzalez-Manteiga, W., Lombardıa, M., Molina, I.,Morales, D. & Santamarıa, L. (2007).Estimation of the mean squared error of predictors of smallarea linear parameters under logistic mixed model.Computational Statistics and Data Analyis 51:2720–2733.

Kackar, R. & Harville, D. (1984).Approximations for standard errors of estimators for fixed andrandom effects in mixed models.Journal of the American Statistical Association 79:853–862.

Documents

Estimation of Poverty Measures in Small Areasec.europa.eu/.../4398377/S3P3-ESTIMATION-OF-POVERTY-MEASURE… · FGT POVERTY MEASURES X Complex non-linear quantities ... Estimation