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

The …rst part of Assumption 7(a) is needed to ensure that the estimator
b n lies in the interior of with probability tending to one, and facilitates the
application of the mean value theorem to Rn( b n) around o n. The second part
states in essence that the moment conditions are correctly speci…ed. Its violation
will generally invalidate the limiting distribution result.
Assumptions 7(c),(d),(g) enable us to apply the CLT for vector-valued NED
processes given above as Corollary 1 to Rn( o n). Some low level su¢ cient conditions
for Assumption 7(c) are given below. To establish asymptotic normality,
we also need uniform convergence of r Rn on , which is implied via Assumptions
7(c),(d),(e),(f).
Given the above assumptions, we have the following asymptotic normality
result for the spatial GMM estimator de…ned by(23).
Theorem 4 Suppose fDng is a sequence of …nite sets of D such that jDnj ! 1
as n ! 1; where D R d ; d 1 is as in Assumption 1. Suppose further that
Assumptions 5- 7 hold. Then
where
A 1
n BnB 0 nA 10
n
1=2 jDnj 1=2 b n
o
n =) N(0; Ik);
An = [Er Rn( o n)] 0 P [Er Rn( o n)] and Bn = [Er Rn( o n)] 0 P
h
jDnj 1 i1=2 n :
Moreover, jAnj = O(1); A 1
n = O(1); jBnj = O(1); (BnB 0 n) 1 = O(1) and
hence, b n is jDnj 1=2 -consistent for
o
n.
As remarked above, relative to the existing literature Theorem 4 allows for
nonstationary processes and only assumes that qi;n and r qi;n are NED on
an -mixing input process, rather than postulating that qi;n and r qi;n are -
mixing. The latter assumption is restrictive in that the -mixing property is not
preserved under transformation. Since the NED property is preserved under a
wide range of transformations it accommodates a much larger class of dependent
spatial processes, including in…nite moving average and spatial autoregressive
models. As such, Theorem 4 should provide a basis for constructing con…dence
intervals and hypothesis testing in a wider range of spatial models.
Using Proposition 2, we now give some su¢ cient conditions for Assumption
7(c).
Assumption 8 The process fZi;n; i 2 Dn Tn; n 1g is uniformly L2-NED
on f"i;n; i 2 Tn; n 1g of size 2d(r 1)=(r 2) for some r > 2:
Assumption 9 For every sequence f ng on , the functions qi;n(Zi;n; n) and
r qi;n(Zi;n; n) satisfy Lipschitz condition (2) in z, that is, for gi;n = qi;n or
r qi;n:
jgi;n(z; n) gi;n(z ; n)j Bi;n(z; z ) jz z j :
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