Luc Anselin and Nancy Lozano-Gracia 1227A simpler spatio-temporal approach is introduced in Huanget al. (2006). Here, an
expanded version of a spatial lag model is estimated using ML, under the assump-
tion that spatial correlation remains constant through time. Furthermore, after the
inclusion of the spatially lagged dependent variable, the error terms are assumed
to be correlated only in time. This spatio-temporal version of the spatial lag model
takes the following form:
y∗=ρ(W⊗IT)y∗+X∗β+u∗,u∗∼N(0,σu^2 ), (26.29)withy∗=(IN⊗Q)y,X∗=(IN⊗Q)X, andu∗=(IN⊗Q)u.Qis a transformation
matrix that “removes the effect” of theAR( 1 )process in the residuals following
a method suggested in Judgeet al. (1988) and Hsiehet al. (2001). If theAR( 1 )
correlated residuals are defined as:
uit=λui,t− 1 +υit, (26.30)withυit∼N(0,συ^2 ),E[uitu′is]=συ^2 (λ), andt =s, then:
(λ)=In⊗
1
1 −λ^2⎡
⎢
⎢⎢
⎢
⎢⎢
⎢⎣1 λλ^2 ··· λT−^1
λ 1 λ ··· λT−^2
λ^2 λ 1 ··· λT−^3
..
...
...
...
...
.
λT−^1 λT−^2 λT−^3 ··· 1⎤
⎥
⎥⎥
⎥
⎥⎥
⎥⎦, (26.31)andQis the transformation matrix such that−^1 =I⊗(Q(λ)′Q(λ)). In particular,
Qtakes the following form:
Q(λ)=⎡
⎢
⎢⎢
⎢
⎢⎢
⎣√
1 −λ^200 ··· 00
−λ 10 ··· 00
0 −λ 1 ··· 00
..
...
...
...
...
...
.
000 ··· −λ 1⎤
⎥⎥
⎥⎥
⎥⎥
⎦. (26.32)
ThisQis used to define the transformed model in equation (26.29), which can
then be estimated using ML. Huanget al. (2006) conclude that introducing these
spatial and temporal correlations improves the goodness-of-fit of the model.
26.3.2 Spatial heterogeneity
26.3.2.1 Discrete spatial heterogeneity
Discrete spatial heterogeneity is a special case of structural instability in the model
specification (functional form and/or parameters), where the form of the instabil-
ity follows a spatial structure, referred to asspatial regimes(Anselin, 1988, 1990).
In practice, this often occurs when structural breaks can be observed between dif-
ferent sub-regions, such as core and peripheral regions or urban and rural areas. In