Essential Statistical Concepts, Definitions, and Terminology - Sampling Spatial Units for Agricultural Surveys - page 25

h i .SoY is multivariate normal such

Þ 1 Þ 1 I B t

2 I, 2

Var ðÞ ¼ ˃ ʣ SAR ¼ ˃ ð I B ð

that

h i

Þ 1 Þ 1

2

I B t

y / MVN ʼ , ˃ ð I B ð : ð 1 : 51 Þ

The CAR and the SAR models, as defined above, have too many unknown
parameters to be useful in practice. Typically, these models are parameterized in
terms of a single parameter and a given neighborhood structure. Thus, we set
C¼ ʳ W for the CAR model, and B¼ ˁ W for the SAR model, where W is a

connectivity weight matrix, and ʳ and ˁ are spatial dependence parameters that

must be estimated.
The SAR random field is widely used in spatial econometrics (Anselin 1988 ).
Assume that E y ðÞ ð Þ ¼ ʼ ¼ X ʲ and B¼ ˁ W. Then Eq. ( 1.50 ) can be re-written as

:

Þ 1

2 I

y ¼ X ʲ þ I ˁ ð W ε ε / N 0, ˃ ð 1 : 52 Þ

In spatial econometrics literature, this model is called the spatial error model
(SEM, Anselin 1988 ).
There are some alternatives for estimating SEM. For example, the maximum
likelihood (ML) and generalized method of moments (GMM) approaches (Anselin
1988 ; Kelejian and Prucha 1999 ). In spdep the two methods are implemented with
two different commands: errorsarlm and GMerrorsar.
In the example below, we have only presented ML estimation. For this exercise,
we have used the dataset LasRosas, which can be downloaded at https://
geodacenter.asu.edu/sdata . LasRosas contains yield data and nitrogen fertilizer
treatment with field characteristics for the Las Rosas farm at Rio Cuarto (for 1999
and 2001). For 1999, the data set contains 32 variables at 1738 observation points.
The variables considered here are YIELD (corn yield in quintals per hectare), N
(Nitrogen fertilizer in kg per hectare), and N2 (nitrogen squared).
Note that the LasRosas data are in the form of shape-files, and so some initial
transformations were required. For further details about this study see Anselin
et al. ( 2004 ).

> library(maptools)
> LasRosas < - readShapePoly(("etc/rosas1999"))
> library(spdep)
> attach(LasRosas@data)
> options(scipen¼10)
> LasRosas_nb < -poly2nb(LasRosas)
> LasRosas_listw < -nb2listw(LasRosas_nb,style¼"W", zero.policy¼FALSE)

The reference manual can be downloaded from http://cran.r-project.org/web/
packages/maptools/maptools.pdf .