Small area estimation of labour force indicators under a multinomial

Transcription

Small area estimation of labour force indicators
under a multinomial mixed model with correlated
time and area effects
M.E. López-Vizcaíno1 , M.J. Lombardía2 and D. Morales 3
1
3
Instituto Galego de Estatística
2
Universidade da Coruña
Universidad Miguel Hernández de Elche
Introduction
The impact of the crisis on the Spanish labour market has been very
intense.
Spanish unemployment rate was 27.16% in the firth quarter of 2013
(around 17 points higher than in 2008).
Unemployment in Galicia was also a major problem. The unemployment
rate was 22,35% in the firth quarter of 2013.
The availability of a hight quality set of labour force indicators at the
county level is very important for police makers.
As the Spanish Labour Force Survey (SLFS) is designed to obtain
estimates at the province level, estimating indicators at a low level of
aggregation (like counties) is a small area estimation problem.
M.E. López-Vizcaíno, M. J. Lombardía and D. Morales (Universities
EstimationofofSomewhere
labour forceand
indicators
Elsewhere)
June 2013
2 / 47
Objective
Estimation of labour force indicators: totals of employed and unemployed
people and unemploymet rates in the counties of Galicia.
We use a multinomial logit mixed model with two random effects, one
associated with the category of employed people and the other
associated with the category of unemployed people.
We model the time dependency with AR(1)-correlated time effects.
We propose mean squared error (MSE) estimators based on analytical
and bootstrap methods.
The new models generalized the model of Molina et al. (2007) and
López-Vizcaíno et al. (2013).
labour forceand
indicators
Elsewhere)
June 2013
3 / 47
Data description
Sample information:
SLFS of Galicia from the third quarter of 2009 to the fourth quarter of
2011 (10 time periods). Information available on individual level.
Sample design:
Quarterly survey that uses a two-stage stratified random sampling design to
extract samples from each Spanish province.
The Primary Sampling units are Census Sections (geographical areas with a
maximum of 500 dwellings-approximately 3000 people).
The Secondary Sampling Units are dwellings, around 8000 in Galicia.
All individuals aged 16 or more in the selected dwelling are interviewed.
Domains of interest:
Domains are the 51 counties of Galicia crossed with sex. There are
D = 102 domains Pdt partitioned in the subsets
Pd1t
Pd2t
Pdt3
employed
unemployed
inactive
labour forceand
indicators
Elsewhere)
June 2013
4 / 47
Data description
Auxiliary variables
0.8
SEXAGE: Combinations of sex and age groups, with 6 values. SEX is coded
1 for men and 2 for women and AGE is categorized in 3 groups with codes 1
for 16-24, 2 for 25-54 and 3 for ≥55. The codes 1, 2, . . . , 6 are used for the
pairs of sex-age (1, 1), (1, 2), . . . , (2, 3).
2
1
2
2
2
2
2
2
2
1
5
5
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1
4
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6
1
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1
1
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0.2
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1
1
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5
1
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0.6
5
0.4
Proportion of employed
5
1
1
0.10
5
Proportion of unemployed
5
1
1
2
0.15
2
TIME
labour forceand
indicators
Elsewhere)
3
TIME
June 2013
5 / 47
Data description
Auxiliary variables
STUD: This variable describes the achieved education level, with values 1-3
for the illiterate and the primary, the secondary and the higher education level
respectively.
3
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3
2
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3
2
0.10
2
2
Proportion of unemployed
2
0.4
Proportion of employed
0.6
2
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0.05
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1
2
TIME
labour forceand
indicators
Elsewhere)
5
6
7
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9
10
TIME
June 2013
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Data description
Auxiliary variables
0
SS: This variable indicates if an individual is registered or not in the social
security system.
REG: This variable indicates if an individual is registered or not as
unemployed in the administrative register of employment claimants.
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log(Employed/Inactive)
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0.35
0.40
0.45
0.50
Proportion of registered in social security system
0.04
0.06
0.08
0.10
0.12
0.14
Proportion of registered unemployed
labour forceand
indicators
Elsewhere)
June 2013
7 / 47
Design-based estimators
Some notation:
k = 1, 2. The categories of the target variable:
1 for employed, and
2 for unemployed.
d = 1, . . . , D. The domains.
t = 1, . . . , T . The time periods.
Pdt . The population in domain d and period t.
Sdt . The sample in domain d and period t.
labour forceand
indicators
Elsewhere)
June 2013
8 / 47
Target population parameters:
The totals of employed and unemployed people and the unemployed rate.
Ydkt =
X
ydtkj ,
Rdt =
j∈Pdt
Yd1
,
Yd1t + Yd2t
k = 1, 2.
Direct estimators:
The direct estimators of the total Ydkt , the size Ndt and the rate Rdt are
dir
Ŷdkt
=
X
j∈Sdt
dir
wdj ydtkj , N̂dt
=
X
dir
wdtj , R̂dt
=
j∈Sdt
dir
Ŷd2t
dir
dir
Ŷd1t
+ Ŷd2t
.
where the wdtj ’s are the official calibrated sampling weights which take
into account for non response.
labour forceand
indicators
Elsewhere)
June 2013
9 / 47
Variance of direct estimators:
dir
dir
cd
ovπ (Ŷdk
, Ŷdk
)=
1t
2t
X
ˆ dir )(y
ˆ dir
wdtj (wdtj − 1)(ydtk1 j − Ȳ
dtk2 j − Ȳdk2 t )
dk1 t
j∈Sdt
This formula is obtained from Särndal et al. (1992), pp. 43, 185 and 391,
with the simplifications wdtj = 1/πdtj , πdtj,dtj = πdtj and πdti,dtj = πdti πdtj ,
i 6= j in the second order inclusion probabilities.
Variance of unemployment rate:
The design-based variance of R̂ddir approximated by Taylor linearisation
dir
V̂π (R̂dt
)=
dir 2
(Ŷd1t
)
dir
(Ŷd1t
+
dir 4
Ŷd2t
)
dir
V̂π (Ŷd2t
)+
2
Ŷd2t
dir
(Ŷd2t
+
dir 4
Ŷd1t
)
dir
V̂π (Ŷd1t
)−
labour forceand
indicators
Elsewhere)
dir
dir
2Ŷd1t
Ŷd2t
dir
dir 4
(Ŷd1t
+ Ŷd2t
)
June 2013
cd
ovπ
10 / 47
In the fourth quarter of 2008 the distribution of the sample sizes per domains
in the SLFS of Galicia has the quantiles
qmin = 13, q1 = 54, q2 = 97, q3 = 153, qmax = 1554.
This means that many of the direct estimators are not reliable.
labour forceand
indicators
Elsewhere)
June 2013
11 / 47
The multinomial logit mixed model
We use indexes k = 1, . . . , (q − 1), d = 1, . . . , D and t = 1, . . . , T for the
categories of the target variable, the domains and the time instants.
In the main model (Model 3), we write the random effects in the vector
form
u1
=
u 2,d
=
col (u 1,d ),
1≤d≤D
col
1≤k ≤q−1
u 1,d =
(u 2,dk ),
col
(u1,dk ),
1≤k ≤q−1
u 2,dk = col (u2,dkt ),
1≤t≤T
u 2 = col (u 2,d )
1≤d≤D
u 2,dt =
col
1≤k ≤q−1
(u2,dkt ),
and we suppose that
1
u 1 and u 2 are independent,
2
u 1 ∼ N(0, V u1 ), where V u1 = diag ( diag (ϕ1k )), k = 1, . . . , q − 1.
1≤d≤D 1≤k ≤q−1
3
u 2,dk ∼ N(0, V u2,dk ), d = 1, . . . , D, k = 1, . . . , q − 1, are independent with
covariance matrix AR(1), i.e. V u2,dk = ϕ2k Ωd (φk ) and
labour forceand
indicators
Elsewhere)
June 2013
12 / 47

Ωd (φk ) = Ωd,k
1

 φk
1 
 ..
=

1 − φ2k  .

 φT −2
k
φTk −1
φk
1
..
.
φTk −2
...
..
.
..
.
..
.
...
φkT −2
φTk −1

..
φTk −2
..
.








.
1
φk
φk
1
.
T ×T
It holds that V u = var(u) = diag(V u1 , V u2 ), where
V u2 = var(u 2 ) = diag ( diag (V u2,dk )).
1≤d≤D 1≤k ≤q−1
We assume that the response vectors y dt conditioned to u 1,d and u 2,dt ,
are independent with multinomial distributions
y dt |u 1,d ,u 2,dt ∼ M(νdt , p1,dt , . . . , pq−1,dt ), d = 1, . . . , D, t = 1, . . . , T .
labour forceand
indicators
Elsewhere)
June 2013
13 / 47
The model
ηdkt = log
pdkt
= x dkt β k + u1,dk + u2,dkt
pdqt
where
d = 1, . . . , D, k = 1, . . . , q − 1, t=1,. . . ,T,
x dkt = col0 (xdktr ),
1≤r ≤lk
β k = col (βkr ),
1≤r ≤lk
l=
Pq−1
k =1 lk .
pdkt =
exp{ηdkt }
Pq−1
1 + `=1 exp{ηd`t }
is the probability of the category k in the domain d and the time instant t.
labour forceand
indicators
Elsewhere)
June 2013
14 / 47
Other simpler models
Along this work we also consider three simpler models.
Model 2 is obtained by restricting Model 3 to φ1 = . . . = φq−1 = 0.
Model 1 is obtained by restricting Model 2 to one time period (T = 1) and
by considering only the random effect u 1 . This is the model studied by
López-Vizcaíno et al. (2013).
Model 0 is obtained by making u1,d1 = . . . = u1,dq−1 in Model 1. This is
the model studied by Molina et al. (2007).
Nevertheless, in the application to real data we apply the non-temporal
Models 1 and 0 to all the considered periods.
labour forceand
indicators
Elsewhere)
June 2013
15 / 47
Fit the model
We combine the penalized quasi-likelihood (PQL) method, described by
Breslow and Clayton (1996) for estimating and predicting the β k ’s, the
u 1d ’s and the u 2d ’s, with the residual maximum likelihood (REML) method
for estimating the variance components ϕ1 and ϕ2 .
The presented method is based on a normal approximation to the joint
probability distribution of the vector (y, u)
The combined algorithm was first introduced by Schall (1991) and later
used by Saei and Chambers (2003) and Molina et al. (2007). We adapt it
to the presented multinomial logit mixed model.
Other alternative fitting algorithms producing consistent estimators of
model parameters are the method of simulated moments of Jiang (1988)
and the EM algorithm.
labour forceand
indicators
Elsewhere)
June 2013
16 / 47
Model-based small area estimation
The problem is to estimate the domain total
mdt = N̂dt p dt ,
d=1,. . . ,D; t=1,. . . ,T.
N̂d is an estimated population size that can be obtained from the
dir
unit-level survey data. In the application to real data, we take N̂dt = N̂dt
We estimate mdt by means of
m̂dt = N̂dt p̂ dt
labour forceand
indicators
Elsewhere)
June 2013
17 / 47
MSE estimation
Estimator 1: Analytical estimation
The employed fitting method requires two approximations:
1
The linearization of the GLMM link function.
2
The approximation of the distribution of the resulting linearized parameter
by a normal distribution.
The consequence of 1+2 is a LMM, where the G1 − G3 formulas of (Prasad and
Rao (1990)) can be applied. For the sake of brevity, we skip these formulas.
Therefore we approximate the mean square error of m̂d by means of
MSE(m̂dt ) = G1dt (σ) + G2dt (σ) + G3dt (σ),
where σ = (ϕ11 , . . . , ϕ1q−1 , ϕ21 , . . . , ϕ2q−1 , φ1 , . . . , φq−1 ) is the vector of
variance components.
The proposed analytic mean square error estimator is
mse(m̂dt ) = G1dt (σ̂) + G2dt (σ̂) + 2G3dt (σ̂).
labour forceand
indicators
Elsewhere)
June 2013
18 / 47
MSE estimation
Estimator 2: Parametric bootstrap (González-Manteiga et al. (2007))
b , k = 1, . . . , q − 1.
1
Fit the model and calculate ϕ̂1k , ϕ̂2k , φ̂k and β
k
2
Generate u ∗1,d ∼ N(0, diag (ϕ̂1k )), u ∗2,dk ∼ N(0, ϕ̂2k Ωd (φ̂k )), and
1≤k ≤q−1
∗
∗
), where
, . . . , pdqt−1
y ∗dt ∼ M(νdt , pd1t
∗
pdkt
=
3
b ∗ and
Calculate ϕ̂∗1k , ϕ̂∗2k , φ̂∗k , β
k
∗
p̂dkt
=
4
∗
}
exp{ηdkt
∗
b xdkt + u ∗ + u ∗ , m∗ = N̂dt p∗ .
, ηdkt
=β
Pq−1
k
1,dk
2,dkt
dkt
dkt
∗
1 + `=1 exp{ηd`t }
∗
exp{η̂dkt
}
∗
b ∗ xdkt + u ∗ + u ∗ , m̂∗ = N̂dt p̂∗ .
, η̂dkt
=β
Pq−1
k
1,dk
2,dkt
dkt
dkt
∗
1 + `=1 exp{η̂d`t }
Repeat B times steps 2-3 and calculate the bootstrap mean square error
estimator
B
1X ∗
∗
∗ 2
msedkt
=
(m̂dkt − mdkt
) .
B
b=1
labour forceand
indicators
Elsewhere)
June 2013
19 / 47
Simulation
We simulate a multinomial logit mixed model with three categories (q = 3).
For d = 1, . . . , D, k = 1, 2 and t = 1, . . . , T , generate:
the explanatory variables
1 d −D
k
t
Udkt =
+
+
,
3
D
q−1 T
xd1t
q
1/2
1/2 = µ1 + σx11 Ud1t , xd2t = µ2 + σx22 ρx Ud1 + 1 − ρ2x Ud2t ,
where µ1 = µ2 = 1, σx11 = 1, σx22 = 2 and ρx = 0.
the random effects u1,dk ∼ N(0, ϕ1k ) and u 2,dk ∼ N(0, V u2,dk ), with
ϕ11 = 1, ϕ12 = 2, ϕ21 = 0.25, ϕ22 = 0.5, φ1 = 0.5 and φ2 = 0.75.
the target variable y dt = (yd1t , yd2t )0 ∼ M(νdt , pd1t , pd2t ), where
pdkt =
exp{ηdkt }
, ηdkt = β0k + β1k xdkt + u1,dk + u2,dkt .
1 + exp{ηd1t } + exp{ηd2t }
νdt = 100, β01 = 1.3, β02 = −1, β11 = −1.6 and β12 = 1.
labour forceand
indicators
Elsewhere)
June 2013
20 / 47
Simulation I
The target of the first simulation is to analyze the behaviour of the
estimators β k , ϕ1k , ϕ2k , φk and mdkt (with N̂dt = 1000).
As efficiency measures, we consider the relative empirical bias (RBIAS)
and relative mean squared error (RMSE).
Number of iterations is I = 1000.
labour forceand
indicators
Elsewhere)
June 2013
21 / 47
Simulation I
RMSE and RBIAS for D = 100
T
RMSE(βb01 )
RMSE(βb02 )
RMSE(βb11 )
RMSE(βb12 )
RMSE(ϕ
b11 )
RMSE(ϕ
b12 )
RMSE(ϕ
b21 )
RMSE(ϕ
b22 )
RMSE(φb11 )
RMSE(φb12 )
4
0.33
0.34
0.34
0.34
0.17
0.25
0.39
0.39
0.94
0.79
8
0.28
0.28
0.36
0.35
0.17
0.18
0.19
0.18
0.50
0.39
12
0.22
0.23
0.31
0.32
0.15
0.18
0.13
0.12
0.30
0.24
T
RBIAS(βb01 )
RBIAS(βb02 )
RBIAS(βb11 )
RBIAS(βb12 )
RBIAS(ϕ
b11 )
RBIAS(ϕ
b12 )
RBIAS(ϕ
b21 )
RBIAS(ϕ
b22 )
RBIAS(φb11 )
RBIAS(φb12 )
labour forceand
indicators
Elsewhere)
4
-0.04
-0.05
-0.04
-0.03
0.02
0.15
-0.39
-0.39
-0.57
-0.38
8
-0.03
-0.02
-0.06
-0.05
-0.04
0.07
-0.18
-0.17
-0.50
-0.33
12
0.01
0.01
-0.06
-0.05
-0.05
-0.05
-0.11
-0.11
-0.29
-0.19
June 2013
22 / 47
Simulation I
RMSEdkt
RMSEd1t
RMSEd2t
RBIASd1t
RBIASd2t
T
1
D/2
D
1
D/2
D
1
D/2
D
1
D/2
D
4
0.09
0.11
0.13
0.12
0.09
0.09
0.004
-0.008
0.002
0.004
0.005
-0.002
8
0.09
0.11
0.14
0.12
0.10
0.08
-0.018
0.001
0.015
0.017
0.009
0.005
labour forceand
indicators
Elsewhere)
12
0.09
0.11
0.12
0.13
0.10
0.08
0.004
0.005
0.010
0.012
-0.011
-0.003
June 2013
23 / 47
Simulation II
The target of the second simulation is to study the behaviour of the two
mean square error estimators (analytic and bootstrap).
We consider D = 50 and T = 2, 4, 6.
Numbers of iterations are I = 500 and B = 500.
We present boxplots of MSE estimates for d = 25.
labour forceand
indicators
Elsewhere)
June 2013
24 / 47
Simulation II
Boxplots for (d, t) = (25, 2), (25, 4), (25, 6).
0.003
0.002
0.004
0.003
●
0.002
0.003
0.002
k=1
(d,t)=(25,6)
0.004
(d,t)=(25,4)
0.004
(d,t)=(25,2)
●
●
0.000
mse
mse*
0.003
0.002
0.003
0.004
mse*
0.002
0.003
0.002
k=2
mse
0.004
mse*
0.004
mse
0.001
0.001
0.000
0.000
0.001
●
●
●
●
0.001
0.001
0.001
●
●
●
●
mse*
T=2
0.000
0.000
0.000
●
●
mse
mse
mse*
T=4
labour forceand
indicators
Elsewhere)
mse
mse*
T=6
June 2013
25 / 47
Application to real data
Unemployment rate − women − IV quarter 2008
Objective
Estimate the total of employed and
unemployed people and the
unemployment rates per sex in the
counties of Galicia.
No data (5)
<=5 (0)
5 − 10 (24)
10 − 15 (20)
>15 (4)
labour forceand
indicators
Elsewhere)
June 2013
26 / 47
Model 0: parameter estimates for the fourth quarter of 2011.
Variable
CONSTANT
SEXAGE=1
SEXAGE=2
SEXAGE=3
SEXAGE=4
SEXAGE=5
STUD=1
SS
ϕ
Employed
Estimate
-1.85
-1.36
1.81
0.18
0.31
1.26
-0.39
2.95
Estimate
0.06
p-value
0.00
0.14
0.00
0.48
0.77
0.01
0.23
0.00
Std.Dev.
0.0055
Variable
CONSTANT
SEXAGE=1
SEXAGE=2
SEXAGE=3
SEXAGE=4
SEXAGE=5
STUD=1
REG
labour forceand
indicators
Elsewhere)
Unemployed
Estimate
-4.87
0.61
3.72
0.18
3.43
2.86
0.54
10.94
p-value
0.00
0.72
0.00
0.71
0.08
0.00
0.34
0.00
June 2013
27 / 47
Model 1: parameter estimates for the fourth quarter of 2011.
Variable
CONSTANT
SEXAGE=1
SEXAGE=2
SEXAGE=3
SEXAGE=4
SEXAGE=5
STUD=1
SS
ϕ11
Employed
Estimate
-1.26
-0.76
1.58
0.18
-1.36
1.41
-0.90
2.01
Estimate
0.011
p-value
0.00
0.42
0.00
0.49
0.21
0.00
0.01
0.00
Std.Dev.
0.0055
Variable
CONSTANT
SEXAGE=1
SEXAGE=2
SEXAGE=3
SEXAGE=4
SEXAGE=5
STUD=1
REG
ϕ12
labour forceand
indicators
Elsewhere)
Unemployed
Estimate
-4.90
1.46
3.72
0.42
2.34
3.46
0.58
8.99
Estimate
0.09
p-value
0.00
0.43
0.00
0.45
0.30
0.00
0.39
0.00
Std.Dev.
0.0295
June 2013
28 / 47
Model 2: parameter estimates.
Variable
CONSTANT
SEXAGE=1
SEXAGE=2
SEXAGE=3
SEXAGE=4
SEXAGE=5
STUD=1
SS
ϕ11
ϕ21
Employed
Estimate
-1.43
0.92
2.05
0.15
0.48
1.68
-0.82
1.49
Estimate
0.031
0.013
p-value
0.00
0.02
0.00
0.38
0.28
0.00
0.00
0.00
Std.Dev.
0.0032
0.0002
Variable
CONSTANT
SEXAGE=1
SEXAGE=2
SEXAGE=3
SEXAGE=4
SEXAGE=5
STUD=1
REG
ϕ12
ϕ22
labour forceand
indicators
Elsewhere)
Unemployed
Estimate
-3.87
1.88
2.35
-0.48
1.57
1.51
-0.49
11.84
Estimate
0.086
0.104
p-value
0.00
0.01
0.00
0.11
0.07
0.00
0.12
0.00
Std.Dev.
0.0186
0.011
June 2013
29 / 47
Model 3: parameter estimates.
Variable
CONSTANT
SEXAGE=1
SEXAGE=2
SEXAGE=3
SEXAGE=4
SEXAGE=5
STUD=1
SS
ϕ11
ϕ21
φ1
Employed
Estimate
-1.47
0.69
2.14
0.16
0.56
1.71
-0.78
1.50
Estimate
0.024
0.013
0.58
p-value
0.00
0.07
0.00
0.33
0.18
0.00
0.00
0.00
Std.Dev.
0.017
0.017
0.099
Variable
CONSTANT
SEXAGE=1
SEXAGE=2
SEXAGE=3
SEXAGE=4
SEXAGE=5
STUD=1
REG
ϕ12
ϕ22
φ2
labour forceand
indicators
Elsewhere)
Unemployed
Estimate
-4.40
2.26
2.98
-0.45
2.62
2.13
-0.27
12.40
Estimate
0.081
0.098
0.29
p-value
0.00
0.00
0.00
0.06
0.00
0.00
0.29
0.00
Std.Dev.
0.010
0.010
0.084
June 2013
30 / 47
Domain relative residuals versus predicted values
ydkt − ŷdkt
,
ŷdkt
rdkt =
d = 1, . . . , 96,
k = 1, 2,
t = 10.
●
Model 0
Model 1
Model 2
Model 3
●
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0.0
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0.4
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0.2
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0.0
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Residuals
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0.0
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Residuals
0.2
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−0.2
0.0
−0.2
Residuals
0.2
●
−0.2
0.4
0.4
●
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0
100
●
200
300
400
500
600
700
0
100
200
300
400
500
600
0
100
200
300
400
500
0 ●●
600
●
●
Predicted employed people
●
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100
150
Predicted unemployed people
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0.4
0.2
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500
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200
0
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100
Model 3
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Residuals
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●● ●
200
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Model 2
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Residuals
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−0.4
0.0
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Residuals
0.2
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Model 0
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100
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50
100
150
200
0
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50
100
150
200
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labour forceand
indicators
Elsewhere)
June 2013
31 / 47
Direct estimator versus model-based estimator.
Total of employed − model 1
80
80
●
●
●
●
● ●
0
20
40
60
0
Total of unemployed − model 1
20
40
60
10
15
Direct estimate
20
25
80
●
●
20
20
5
10
15
Direct estimate
20
25
model estimate
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●
●
●
●●
●
●
●
●
●
●
●
●●●
0
●
5
●● ●
●● ●
● ● ●●
●
15
model estimate
●
●
●
0
5
60
●
20
0
●
5
5
●
●● ●
40
Direct estimate
●
0
●●
●●
●●
●
● ●
●
● ● ●●● ●
●
● ●
●●●
●●●●
●
●
●●
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●●●
●
●
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●
●
●
●●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
20
●
15
model estimate
10
●
●●
0
10
20
15
●
60
80
●
●
40
model estimate
20
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
Direct estimate
●
model estimate
60
80
●
●
●
●●
●
●
●●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
Direct estimate
0
40
0
●
●
●●
●●
●●
15
80
●
●●
Direct estimate
●
10
60
● ●
●●
●●
●●
10
0
0
40
●
●
0
●
● ●
●
●●●
●
●
●●●
●●
●●●
●
●
●
●●
●
●●
●●
●
●
●●
●
●
●
●●
●●
●
●
●●
●●
●
20
20
●
●●
●
●
●
●●
●●
●
●
●●●
●
●●●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●●
●
model estimate
60
●
40
model estimate
60
40
20
model estimate
●
●
20
●
●
●
0
●
● ●
●
●
●●●
●
●
●
●
80
●
●
80
10
15
Direct estimate
labour forceand
indicators
Elsewhere)
20
25
●●
●
● ●
●
● ●
●
●
●
● ●
● ●
●●●●
●
●●●
●●●●
●
●●
●●●
●
●
●●
●
●
●
●
●●●
●
●
●
●●
●
●●
●
●
●
●●
●
●●
●●
●
●
●
●
●
●
●
●
0
5
10
15
20
25
Direct estimate
June 2013
32 / 47
Men and women employment totals for the fourth quarter of 2011.
Employed men (thousands)− IV/2011
Employed women (thousands)− IV/2011
●
Direct
●
Model 0
Model 1
40
●
Direct
●
Model 0
Model 1
40
Model 2
Model 2
Model 3
Model 3
●
30
●
30
●
●
●
●
●
20
20
●
●
●
●
●
●
10
●
●
●
●
●●
●
●
0
●●
13
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
105
172
●
●●
13
●
●●
●
●
43
Sample size
labour forceand
indicators
Elsewhere)
●
●
●
0
63
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
38
●
●
●
●
10
●
●
●
●
●
●●●
71
107
189
Sample size
June 2013
33 / 47
Men and women unemployment rates for the fourth quarter of 2011.
●
Unemployment rate − men − IV/2011
Unemployment rate − women − IV/2011
●
Direct
●
Direct
●
Model 0
Model 0
Model 1
Model 1
Model 2
40
Model 2
40
Model 3
Model 3
●
●
●
●
30
30
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
20
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
10
●
●●
●
10
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
20
●
●
●
●
●
●
●
●
●●
●
●
0
●
13
●
0
●
38
63
105
172
●
13
●
●
43
Sample size
labour forceand
indicators
Elsewhere)
71
107
189
Sample size
June 2013
34 / 47
RMSEs of model-based estimator of unemployment rates in the fourth
quarter of 2011.
RMSE unemployment rate men − IV/2011
RMSE unemployment rate women − IV/2011
●
60
30
●
●
●
20
40
25
50
●
●
30
●
5
10
10
20
15
●
Model 0
Model 1
Model 2
Model 3
Model 0
labour forceand
indicators
Elsewhere)
Model 1
Model 2
Model 3
June 2013
35 / 47
Estimated totals of employed men (top) and their estimated
RMSEs (bottom) in the fourth quarter of 2011.
County
27
49
15
34
44
53
27
49
15
34
44
53
n
13
40
65
107
179
1347
13
40
65
107
179
1347
dir
410
2593
4369
4277
9971
90346
40.1
18.6
14.3
13.5
9.7
2.7
mod0
391
2317
4024
4774
9942
87308
20.0
26.9
28.1
13.3
4.2
2.2
mod1
399
2056
4229
4155
10013
84926
25.0
18.0
12.2
11.6
7.5
2.7
labour forceand
indicators
Elsewhere)
mod2
383
2077
4674
4333
10551
83662
17.7
12.6
11.3
8.8
6.8
2.4
mod3
393
2053
4644
4297
10651
84055
19.4
13.3
11.1
9.6
7.9
3.1
June 2013
36 / 47
Estimated totals of employed women (top) and their estimated
County
27
11
10
41
42
53
27
11
10
41
42
53
n
13
45
75
108
193
1554
13
45
75
108
193
1554
dir
292
2209
3301
3594
4616
78128
50.6
20.9
18.3
12.4
12.7
3.2
mod0
358
2332
3417
3614
5184
76854
10.9
9.7
12.2
8.5
9.3
2.8
mod1
318
2119
3226
3505
4983
75033
27.8
16.5
14.5
11.3
8.9
3.0
labour forceand
indicators
Elsewhere)
mod2
330
1974
3143
3593
4789
73316
16.1
10.8
11.1
6.1
6.0
1.8
mod3
336
2019
3179
3566
4830
74199
19.2
13.4
12.6
9.5
8.0
3.2
June 2013
37 / 47
Estimated men unemployment rates (top) and their estimated
County
27
49
15
34
44
53
27
49
15
34
44
53
n
13
40
65
107
179
1347
13
40
65
107
179
1347
dir
18.6
18.1
19.0
28.5
29.3
21.7
146.4
71.9
51.5
26.5
19.1
8.4
mod0
18.0
22.7
19.7
16.4
19.8
18.0
23.5
24.6
28.6
15.1
9.8
6.4
mod1
17.8
21.3
18.7
25.6
24.2
20.7
32.6
20.9
15.8
15.5
12.1
4.6
labour forceand
indicators
Elsewhere)
mod2
15.5
19.0
18.2
19.6
21.0
20.5
25.6
18.1
16.2
12.1
10.7
4.8
mod3
15.3
19.4
18.3
19.9
20.7
20.4
22.1
16.0
12.1
10.7
8.6
3.6
June 2013
38 / 47
Estimated women unemployment rates (top) and their estimated
County
27
11
10
41
42
53
27
11
10
41
42
53
n
13
45
75
108
193
1554
13
45
75
108
193
1554
dir
37.2
23.3
21.1
22.7
27.6
25.4
59.0
55.1
55.7
34.8
25.2
7.7
mod0
17.7
16.5
16.9
20.8
24.5
22.2
16.6
12.9
22.1
7.0
18.4
4.3
mod1
22.7
21.6
20.3
20.9
26.0
24.4
36.2
22.7
18.7
14.4
10.9
4.2
labour forceand
indicators
Elsewhere)
mod2
20.0
17.1
20.7
17.7
25.1
22.5
22.5
17.4
14.5
16.5
12.1
5.3
mod3
19.2
16.7
20.3
17.9
25.0
22.9
28.7
19.2
16.2
11.0
10.2
3.1
June 2013
39 / 47
Unemployment rates in some counties and all periods.
Unemploment rate − women − County 27
●
Direct
●
40
Direct
●
40
Model 0 ●
Direct
●
●
40
Model 0
Model 0
●
●
30
Model 1
Model 1
Model 2
Model 2
30
●
Model 3
Model 1
Model 2
30
Model 3
Model 3
●
●
●
●
●
●
●
20
●
20
20
●
●
●
●
●
●
●
●
●
●
●
●
10
0
●
10
●
●
●
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
Time, n= 13
5
40
40
10
1
3
4
6
7
●
Model 1
Model 2
Model 3
●
●
●
●
10
Model 0
30
●
20
●
9
Direct
●
40
●
●
●
8
Model
3
●
●
●
●
5
Time, n= 66
Model 2
●
30
Model 3
20
●
●
●
●
●
2
Model 1 ●
Model 2
20
9
Model 0
Model 1
●
8
Direct
●
Model 0
30
7
Direct
●
6
Time, n= 47
●
10
0
●
●
●
●
●
10
10
●
10
●
0
0
1
2
3
4
5
6
Time, n= 115
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
Time, n= 191
labour forceand
indicators
Elsewhere)
1
2
3
4
5
6
7
8
9
10
Time, n= 1510
June 2013
40 / 47
Measures for model comparison.
Loglikelihood
BIC
Model 2
-5269,5
10593.1
Model 3
-5037,6
10135.4
labour forceand
indicators
Elsewhere)
June 2013
41 / 47
Model 3 estimates of men unemployment rates in Galician counties in
IV/2011 (left) and their variations between from IV/2009 to IV/2011 (right).
Unemployment rate − men − IV/2011
No data (2)
<=10 (5)
10 − 15 (25)
15 − 20 (14)
>20 (7)
Variation unemployment rate − men − IV/2009−IV/2011
No data (2)
<=1 (8)
1 − 2 (6)
2 − 3 (15)
>3 (22)
labour forceand
indicators
Elsewhere)
June 2013
42 / 47
Model 3 estimates of women unemployment rates in Galician counties in
IV/2011 (left) and their variations between from IV/2009 to IV/2011 (right).
Unemployment rate − women − IV/2011
No data (2)
<=10 (4)
10 − 15 (21)
15 − 20 (15)
>20 (11)
Variation unemployment rate − women − IV/2009−IV/2011
No data (2)
<=1 (24)
1 − 2 (15)
2 − 3 (7)
>3 (5)
labour forceand
indicators
Elsewhere)
June 2013
43 / 47
Conclusions
The model-based estimates have less mean squared error than the direct
ones.
The estimates are consistent in the sense that estimates of domain totals
of employed, unemployed and inactive people sum up to the size of the
domain.
In the simulations we have observed that the bootstrap estimators work
better than the G1 − G3 approximations.
Area-level temporal multinomial mixed models can be recommended for
estimating domain unemployment indicators.
Future research: Introduce alternative fitting algorithm and derive domain
estimates based on Empirical Best Predictors.
labour forceand
indicators
Elsewhere)
June 2013
44 / 47
Acknowledgements
Acknowledgements
Supported by the Instituto Galego de Estatística, by the grants
MTM2009-09473 and MTM2008-03010 of the Spanish “Ministerio de Ciencia
e Innovación” and AAII DE2009-0030 “Grupos de referencia competitiva”
(2007/132) of the “Consellería de Educación e Ordenación Universitaria” and
by the Belgian network IAP-Network P6/03.
Thank you
labour forceand
indicators
Elsewhere)
June 2013
45 / 47
References
Breslow, N. and Clayton, D. (1996). Approximate inference in generalized
linear mixed models. Journal of the American Statistical Association, 88,
9-25.
González-Manteiga, W., Lombardía, M. J., Molina, I., Morales, D. and
Santamaría, L. (2007). Estimation of the mean squared error of
predictors of small area linear parameters under a logistic mixed model.
Computational Statistics and Data Analysis, 51, 2720-2733.
Jiang, J. (1998). Consistent estimators in generalized linear mixed
models. Journal of the American Statistical Association, 93, 720-729.
López-Vizcaíno, E., Lombardía, M.J. and Morales, D. (2013).
Multinomial-based small area estimation of labour force indicators.
Statistical Modelling, 13, 153-178.
labour forceand
indicators
Elsewhere)
June 2013
46 / 47
References
Molina, I., Saei, A. and Lombardía, M. J. (2007). Small area estimates of
labour force participation under multinomial logit mixed model. The
Journal of the Royal Statistical Society, series A, 170, 975-1000.
Prasad, N. G. N. and Rao, J. N. K. (1990). The estimation of the mean
squared error of small area estimators Journal of the American Statistical
Association, 85, 163-171.
Saei, A. and Chambers, R. (2003). Small area estimation under linear an
generalized linear mixed models with time and area effects. S3RI
Methodology Working Paper M03/15, Southampton Statistical Sciences
Research Institute.
Schall, R. (1991). Estimation in generalized linear models with random
effects. Biometrika, 78, 719-727.
Särndal, C. E., Swensson, B. and Wretman, J. (1992). Model Assisted
Survey Sampling, Springer.
labour forceand
indicators
Elsewhere)
June 2013
47 / 47

Small area estimation of labour force indicators under a multinomial

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