poverty mapping at local level with suitable modelling of ... s16bp1.… · poverty indicators e di...

POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS

Poverty mapping at local level with suitablemodelling of income

Isabel MolinaDept. of Statistics, Univ. Carlos III de Madrid

(Coauthors: M. Graf and J.M. Marın)


POPULATION AT RISK OF POVERTY

2


POVERTY INDICATORS

DIRECT ESTIMATORS

EB METHOD

APPLICATION

SIMULATIONS WITH OUTLIERS

PROPOSAL

RESULTS

3


NOTATION

• U finite population of size N.

• U partitioned into D subsets U1, . . . ,UD of sizes N1, . . . ,ND ,called domains or areas.

• s sample of size n drawn from the population U.

• sd = s ∩ Ud sub-sample from domain d of size nd .

• rd = Ud − sd out-of-sample elements from domain d .

• Small area problem: nd too small for some domains.

4


POVERTY INDICATORS

• Edi welfare measure for indiv. i in domain d .

• z = poverty line.

• Poverty indicator of order α ≥ 0 for domain d :

Fαd =1

Nd

Nd∑i=1

(z − Edi

z

)αI (Edi < z).

• When α = 0⇒ Poverty incidence

• When α = 1⇒ Poverty gap

• Other: Quintile share ratio, Gini coef., Sen index, Theil index,Generalized entropy, Fuzzy monetary/supplementary index.

X Foster, Greer & Thornbecke (1984), Econom.X Neri, Ballini & Betti (2005), Stat. in Transition 5


DIRECT ESTIMATORS

• Direct estimator for a domain d : Estimator that uses onlythe sample data from that domain.

• Example: For linear parameters such as means or totals, theHorwitz-Thompson estimator.

• FGT poverty indicators as means:

Fαd =1

Nd

Nd∑i=1

Fαdi , Fαdi =

(z − Edi

z

)αI (Edi < z).

• πdi inclusion probability of unit i in sd .• wdi = 1/πdi sampling weight.• Horwitz-Thompson estimator (random sampling without

replacement):

FDIRαd =

1

Nd

∑i∈sd

wdiFαdi .

6


EB METHOD

• Assumption: For a given one-to-one transformation of thewelfare variables, Ydi = T (Edi ), it holds

Ydi = x′diβ + ud + edi , udiid∼ N(0, σ2u), edi

iid∼ N(0, σ2e ).

• FGT poverty indicator in terms of transformed variables Ydi :

Fαd =1

Nd

Nd∑i=1

{z − T−1(Ydi )

z

}αI{T−1(Ydi ) < z

}= hα(Yd),

where Yd = (Yd1, . . . ,YdNd)′.

X Molina and Rao (2010), CJS 7


EB METHOD

• Partition Yd into sample and out-of-sample: Yd = (Y′ds ,Y′dr )′

• General parameter: δd = h(Yd), h(·) real measurable.

• Best estimator: δd with minimum MSE (δd) = E (δd − δd)2

• The best estimator is given by

δBd = EYdr[δd |yds ]

• For δd = hα(Yd) = Fαd , the best estimator is

FBαd = EYdr

[Fαd |yds ] = FBαd(β, σ2u, σ

2e )

• Empirical best (EB) estimator: FEBαd = FB

αd(β, σ2u, σ2e ).

X Molina and Rao (2010), CJS 8


POVERTY MAPPING IN SPAIN

• Data source: Spanish Survey on Income and LivingConditions (EU-SILC) of 2006.

• Areas: D = 52 provinces for each gender. We fit a separatemodel for each gender.

• Transformation: We consider the nested-error model for thelog-equivalized disposable income: Ydi = T (Edi ) = log Edi .

• Explanatory variables: indicators of 5 age groups, of havingSpanish nationality, of 3 education levels and of labor forcestatus (unemployed, employed or inactive).

9


MODEL DIAGNOSTICS

Residuals

−10 −8 −6 −4 −2 0 2

0.0

0.2

0.4

0.6

0.8

−4 −2 0 2 4−1

0−8

−6−4

−20

2

Theoretical Quantiles

Sam

ple

Qua

ntile

s

10


MODEL DIAGNOSTICS

0 5000 10000 15000

−10

−8−6

−4−2

02

Index

Res

idua

ls

9.2 9.4 9.6 9.8 10.0 10.2−1

0−8

−6−4

−20

2

Fitted values

Res

idua

ls

11


SIMULATION: NO OUTLIERS

y

De

nsity

−2 0 2 4 6

0.0

00

.05

0.1

00

.15

0.2

00

.25

0.3

00

.35

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−1 0 1 2

−2

02

4

x1s

ys

12


POV. INCIDENCE: NO OUTLIERS

0 20 40 60 80

0.1

0.2

0.3

0.4

0.5

0.6

Area

Dire

ct, T

rue

●●●●

●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●

●

● True Direct

0 20 40 60 80

0.1

0.2

0.3

0.4

0.5

0.6

Area

EB

, Tru

e

●●●●

●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●

●

● True EB

0 20 40 60 80

0.1

0.2

0.3

0.4

0.5

0.6

Area

FH

, Tru

e

●●●●

●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●

●

● True FH

0 20 40 60 80

0.1

0.2

0.3

0.4

0.5

0.6

Area

EL

L, Tru

e

●●●●

●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●

●

● True ELL

13


SIMULATION: 2 % OUTLIERS

y

De

nsity

−10 −5 0 5

0.0

00

.05

0.1

00

.15

0.2

00

.25

0.3

0

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−1 0 1 2

−1

0−

50

5

x1s

ys

14


POV. INCIDENCE: 2 % OUTLIERS

0 20 40 60 80

0.1

0.2

0.3

0.4

0.5

0.6

Area

Dire

ct, T

rue

●●●●

●●●●●

●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●

●●●

●

●

● True Direct

0 20 40 60 80

0.1

0.2

0.3

0.4

0.5

0.6

Area

EB

, Tru

e

●●●●

●●●●●

●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●

●●●

●

●

● True EB

0 20 40 60 80

0.1

0.2

0.3

0.4

0.5

0.6

Area

FH

, Tru

e

●●●●

●●●●●

●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●

●●●

●

●

● True FH

0 20 40 60 80

0.1

0.2

0.3

0.4

0.5

0.6

Area

EL

L, Tru

e

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●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●

●●●

●

●

● True ELL

15


GENERALIZED GAMMA

• Gamma distribution: Y ∼ G (k , θ), k shape, θ scale

f (y ; k , θ) =(y/θ)k−1

Γ(k)

e−y/θ

θ, y > 0, k, θ > 0.

• Generalized Gamma distribution: Y ∼ GG (a, θ, k),

f (y ; a, θ, k) =a(y/θ)k−1

Γ(k/a)

e−(y/θ)a

θ, y > 0, a, θ, k > 0.

• Reparameterization: Y ∼ GG (a, θ, p), p = k/a⇔ k = ap

f (y ; a, θ, p) =a(y/θ)ap−1

Γ(p)

e−(y/θ)a

θ, y > 0, a, θ, p > 0.

• For p large, GG (a, θ, p) ≈ logN(a, θ).

16


GENERALIZED BETA OF THE SECOND KIND

• It can be obtained as a scale mixture:

Y |u ∼ GG (a, bu1/a, p)U ∼ InvG (1, q)

}⇒ Y ∼ GB2(a, b, p, q).

• Density of Y ∼ GB2(a, b, p, q):

f (y ; a, b, p, q) =a

bB(p, q)

(y/b)ap−1

{1 + (y/b)a}p+q , y > 0, a, b, p, q > 0.

• a shape, b scale, p length of left tail, q length of right tail.

• logY has location parameter log b.

17


MULTIVARIATE GB2 MODEL

• MGB2 mixed regression model:

Ydi |udind∼ GG (a, bdiu

1/ad , p),

ud ∼ InvG (1, q),log bdi = x′diβ, i = 1, . . . ,Nd , d = 1, . . . ,D.

• Equivalent model:

Yd = (Yd1, . . . ,YdNd)′

ind∼ MGB2(a, {bdi}Ndi=1, p, q),

log bdi = x′diβ, i = 1, . . . ,Nd , d = 1, . . . ,D.

• Marginals are univariate GB2:

Ydi ∼ GB2(a, bdi , p, q), i = 1, . . . ,Nd , d = 1, . . . ,D.

X Graf, Marın and Molina (2014), Unpublished 18



Moments of log variables:

• E [log(Ydi )|ud ] = x′diβ +1

a[log ud + ψ(p)]︸︷︷︸

vd

,

• E [log(Ydi )] = x′diβ +1

a[ψ(p)− ψ(q)]︸︷︷︸

c

,

• V [log(Ydi )] =1

a2

[ψ(1)(p) + ψ(1)(q)

],

• Cov [log(Ydi ), log(Ydj)] =1

a2ψ(1)(q), j 6= i ,

where ψ(x) =∂ log Γ(x)

∂xand ψ(1)(x) =

∂ψ(x)

∂x.




• Decomposition into non-sample and sample:

Yd = (Y′dr ,Y′ds)′

• Conditional distribution of non-sample given sample:

Ydr |ydsind∼ MGB2(a, {b∗di}i∈rd , p, q

∗d),

b∗di = bdi

1 +∑i∈sd

(ydibdi

)a1/a

, i ∈ rd ,

q∗d = q + ndp, d = 1, . . . ,D



SIMULATION RESULTS: POV. INCIDENCE• If income simulated from the MGB2 model fitted to realincome, EB estimates based on Log Normal biased upwards!

Mean values, Pov. Inc. ( %)

0 20 40 60 80

1820

2224

2628

3032

Area

Pov

. Inc

iden

ce

●

●

●

●

●

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●

●

●

●

●

●

●

●

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●

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●

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●

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●

●

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●

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●

●

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●

●●

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●●

●

●

●

●●

●●

● TrueEB GB2

EB Log NormalDirect

MSEs, Pov. Inc. ( %)

0 20 40 60 80

1015

2025

3035

40

Area

MS

E P

ov. I

ncid

ence

EB GB2 EB Log Normal Direct

21


SIMULATION RESULTS: POV. INCIDENCE• If income simulated from the Log Normal model fitted to realincome, EB estimates based on MGB2 are good!

Mean values, Pov. Inc. ( %)

0 20 40 60 80

1015

20

Area

Pov

. Inc

iden

ce

●

●●

●

●●

●●

●●

●

●

●●

●

●

●

●

●●

●

●●

●

●●●●

●

●●

●

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●

●

●

●

●

●●●

●●

●

●

●

●

●●●

●

●

●

●

●

●●

●

●

●●

●●●

●

●●●

●

●

●●

●

●

●●●●●

● TrueEB GB2

EB Log NormalDirect

MSEs, Pov. Inc. ( %)

0 20 40 60 80

510

1520

2530

35

Area

MS

E P

ov. I

ncid

ence

EB GB2 EB Log Normal Direct

22


DOMO ARIGATO!!EZKERRIK ASKO!!GRAZIE MILLE!!MERCI BEAUCOUP!!¡¡MOITAS GRAZAS!!¡¡MOLTES GRACIES!!¡¡MUCHAS GRACIAS!!THANK YOU VERY MUCH!!VIELEN DANK!!

23

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