On Spatial Processes and Asymptotic Inference under Near$Epoch ...

3.3 Cli¤-Ord Type Spatial Processes
Consider the following Cli¤-Ord type model:
Yn = WnYn + Xn + un (9)
un = Mnun + vn
where Yn = (Y1;n; :::; Yn;n) 0 is an n 1 vector of endogenous variables, Xn =
(Xij;n) is n k matrix of regressors, Wn = (wij;n) and Mn = (mij;n) are n n
nonstochastic weight matrices, vn = (v1;n; : : : ; vn;n) 0 is an n 1 vector of disturbances,
and are scalar parameters typically referred to as spatial autoregressive
parameters, and is a k 1 parameter vector. To simplify presentation, we
assume in the following that k = 1 and use the notation Xn = (X1;n; :::; Xn;n) 0 .
The above model is frequently referred to as a spatial ARAR(1,1) model, and
WnYn and Mnun are referred to as spatial lags. For recent contributions on
estimation strategies for this model see, e.g., Robinson (2009, 2007), Kelejian
and Prucha (2007a,b, 2004), and Lee (2007a,b, 2004).
The spatial weights mij;n and wij;n are typically thought of as to depend
on some measure of distance, say dij, between units i and j, where the weights
decline as the distance increases. Thus although in Cli¤-Ord models observations
are indexed by natural numbers, i.e., i = 1; : : : ; n, a leading case would
be where i correspond to some location, say si 2 R d , and the distance between
i and j is given by dij = (si; sj). In the following, we assume that
Dn = fs1; : : : ; sng D, where the lattice D satis…es Assumption 1. However,
to simplify notation we will use (i; j) instead of (si; sj). We assume
furthermore that for all n
In Wn and In Mn are nonsingular. (10)
The reduced form of the model is then given by Yn = AnXn + Bnvn:with
An = (aij;n) = (In Wn) 1 and Bn = (bij;n) = (In Wn) 1 (In Mn) 1 .
In scalar notation we have for i = 1; :::; n:
Yi;n =
nX
j=1
aij;nXj;n +
nX
j=1
bij;nvj;n;
Although for …xed n; the output process Yi;n depends on only a …nite number of
elements of the input process "i;n = (Xi;n; vi;n) 0 , the mixing property of "i;n may
not carry over to Yi;n: The reason is that the number of elements composing the
spatial lags grows unboundedly with the sample size so that the mixing property
can break down in the limit. This is especially important when analyzing the
asymptotic properties of Cli¤-Ord type processes.
Towards establishing that Y = fYi;n; si 2 Dn; n 1g is NED on " =
f"i;n; si 2 Dn; n 1g, we maintain the following assumptions:
lim
s!1 sup sup
n
X
jaij;nj + jbij;nj = 0 (11)
1 i n
1 j n: (i;j)>s
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