Agriculture Reference
In-Depth Information
11.3 Small Area Models
The traditional methods described in the previous section assume implicit models
that specify a link to some SAs using observed supplementary data. In this section,
we present some methods that use explicit models to take into account specific
variations between different areas. We refer to this group of techniques as small
area models (SAMs).
SAMs are model-based methods that assume a model for the sample data, and
use the optimal predictor of the area variable under investigation. The MSE of the
prediction is obviously defined and estimated with respect to the model (see
Sect.
11.1
).
The explicit models used in SAE are a special case of the mixed effects models.
They are very flexible when handling complex problems in SAE (Fay and Herriot
1979
; Battese et al.
1988
).
These approaches can be mostly classified into area level and unit level models
(Rao
1999
,
2003
). In area level models, information on the response variable is only
available at the SA level. In unit level models, data exist at the unit or respondent
level. A description of these two types of SAMs is given below.
11.3.1 Area Level Models
The area level model has two main components: the linking model and the sampling
model. This approach is used when the covariate information is only available at the
area level.
First, let x
d
¼
t
denote the covariates vector for each area
d
. Assume that the parameters of interest
ð
x
d
1
x
d
2
...
x
dq
Þ
ʸ
d
¼
g
(
t
d
), for some function
g
(.) are
related to x
d
by a linear (typical, but not necessary) model
ʸ
d
¼ x
d
β þ b
d
˅
d
,
d
¼ 1, 2,
...
,
D
;
ð
11
:
26
Þ
where
β
is the
q
1 regression parameters vector, the
b
d
s are known positive
constants, and
:
2
˅
Model (
11.26
) is denoted as the link function. The
˅
d
s are area-specific random effects that represent the homogeneity of the areas
after accounting for the covariates x
d
. It is common to assume that the random
effects
˅
d
iid
0
; ˃
e
˅
d
are normally distributed, but it is possible to make robust inferences also
when this assumption is relaxed (Rao
2003
). One possible solution to this problem
is the quantile regression approach (Chambers and Tzavidis
2006
).
Second, to apply inferential procedures, we need to suppose that the direct
estimator
t
d
(or its transformation
ʸ
d
¼
ʸ tðÞ
) is available and defined as
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