Spatial Regression Models
Spatial Independent Variables
Y = Xb + WXg + u
where W is the spatial weights matrix, and u ~ iid(0,s2I).
This is a straightforward extension of the classical regression model. The parameter vector
g is interpreted as spatial externalities or spillover.
Spatial Lag Model
Y = lWY + Xb + u
where W is the spatial weights matrix, and u ~ iid(0,s2I).
u = (I-lW)Y - Xb
Therefore, Var(Y) = s2[(I-lW)-1(I-lW)-1']
Spatial Error Structure
Spatial AR(1)
Y = Xb + e
e = rWe + u
where W is the spatial weights matrix, and u ~ iid(0,s2I).
u = (I-rW)e = (I-rW)(Y - Xb)
Therefore, Var(Y) =
s2[(I-rW)-1(I-rW)-1']
Spatial MA(1)
Y = Xb + e
e = u - qWu
where W is the spatial weights matrix, and u ~ iid(0,s2I).
e = (I-qW)u, or
u = (I-qW)-1e = (I-qW)-1(Y - Xb)
Therefore, Var(Y) =
s2[(I-qW)(I-qW)']
Spatial ARMA(1,1)
Y = Xb + e
e = rWe - qWu + u
where W is the spatial weights matrix, and u ~ iid(0,s2I).
(I-rW)e = (I-qW)u, or
u = (I-qW)-1(I-rW)e =
(I-qW)-1(I-rW)(Y - Xb)
Therefore, Var(Y) =
s2[(I-rW)-1(I-qW)(I-qW)'(I-rW)-1']