poverty mapping at local level with suitable modelling of ... s16bp1.… · poverty indicators e di...
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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)
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POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
POPULATION AT RISK OF POVERTY
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POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
POVERTY INDICATORS
DIRECT ESTIMATORS
EB METHOD
APPLICATION
SIMULATIONS WITH OUTLIERS
PROPOSAL
RESULTS
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POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
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.
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POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
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
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POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
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 .
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POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
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
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POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
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
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POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
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).
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POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
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
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POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
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
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POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
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
![Page 13: Poverty mapping at local level with suitable modelling of ... S16BP1.… · POVERTY INDICATORS E di welfare measure for indiv. i in domain d. z = poverty line. Poverty indicator of](https://reader034.vdocuments.us/reader034/viewer/2022051911/6000f82d76ec204d8b1d60c0/html5/thumbnails/13.jpg)
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
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
![Page 14: Poverty mapping at local level with suitable modelling of ... S16BP1.… · POVERTY INDICATORS E di welfare measure for indiv. i in domain d. z = poverty line. Poverty indicator of](https://reader034.vdocuments.us/reader034/viewer/2022051911/6000f82d76ec204d8b1d60c0/html5/thumbnails/14.jpg)
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
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
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POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
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
●●●●
●●●●●
●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●
●●●
●
●
● True ELL
15
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POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
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
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POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
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
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POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
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
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POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
MULTIVARIATE GB2 MODEL
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.
X Graf, Marın and Molina (2014), Unpublished 19
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POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
MULTIVARIATE GB2 MODEL
• 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
X Graf, Marın and Molina (2014), Unpublished 20
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POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
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
●
●
●
●
●
●●●
●●
●
●
●
●
●
●
●
●
●●●
●●
●
●●●●●
●
●
●●●●
●
●
●●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●●
●
●●
●●●●
●●
●
●
●
●●
●●
● 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
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POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
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
●
●●
●
●●
●●
●●
●
●
●●
●
●
●
●
●●
●
●●
●
●●●●
●
●●
●
●●
●
●
●
●
●
●●●
●●
●
●
●
●
●●●
●
●
●
●
●
●●
●
●
●●
●●●
●
●●●
●
●
●●
●
●
●●●●●
● 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
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POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
DOMO ARIGATO!!EZKERRIK ASKO!!GRAZIE MILLE!!MERCI BEAUCOUP!!¡¡MOITAS GRAZAS!!¡¡MOLTES GRACIES!!¡¡MUCHAS GRACIAS!!THANK YOU VERY MUCH!!VIELEN DANK!!
23