Small Area Estimation - Sampling Spatial Units for Agricultural Surveys - page 292

11.5 The Spatially Augmented Approach to Small Area
Estimation

The EBLUP does not consider spatial information in its estimation process. The
EBLUP method can possibly be improved by including a spatial structure to the
random area effects (Cressie 1991 ; Petrucci et al. 2005 ; Petrucci and Salvati 2006 ).
Spatial information can be added to the basic area level model in different ways.
The first possibility is to use GIS (see Chap. 3 ) to add some geographical covariates
for each SA, regarding the centroid coordinates and/or other auxiliary geographical
variables. The idea is that these geographical variables may capture spatial effects
of the phenomenon under investigation. In this case, it seems plausible to assume
that the random SA effects are independent, and that all spatial dependence can be
explained using these additional variables. Following this hypothesis, the tradi-
tional EBLUP is still considered an appropriate predictor.
The second approach adds the geographic information directly to the random
part of the Fay-Herriot model ( 11.28 ). The geographical coordinates of area
centroids may be incorporated into the random part of the model by defining a
d 2 vector Z, where the first column represents the latitude of each area and the
second column is the longitude of each area. For further details see Petrucci
et al. ( 2005 ).
When the two approaches outlined above are unfeasible, we must directly model
the spatial dependence. The Fay-Herriot model in Eq. ( 11.28 ) typically assumes
that

, but in some circumstances it may be more appropriate to
consider a model that allows for spatial correlations among the

iid N 0

˅ d

2
˅

; ˃

e

˅ d s. Spatially

augmented models for the area-specific random effect ˅ d are appropriate when we

have information about neighboring areas. There are two different approaches for
describing spatial information: SAR and CAR (see Sect. 1.4.3.2 ).
Let N ( d ) be the set of neighborhoods of SA d (see Sect. 1.4.3 ). Then, for the
random effect b d ˅ d , it is possible to consider a CAR spatial model as

!

N X
l6 ¼ d

2
˅

b d ˅ d j f ˅ l , l 6 ¼ d g c ld b l ˅ l , b d ˃ ; ð 11 : 40 Þ

e

where c ld denotes spatial dependence parameters that are non-zero only if l 2 NðÞ .
Cressie ( 1991 ) used a CAR model in an SAE framework in the context of US census
undercounts.
Now, consider the Fay-Herriot model defined using matrix notation in
Eq. ( 11.29 ), i.e., ʸ ¼x βþ B ˅ þ e. The error term ˅ can be defined using a SAR
process with the spatial autoregressive coefficient ˁ , and a d d proximity matrix

W (Anselin 1988 )