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

5 Large Sample Properties of Spatial GMM Estimators
In this section, we apply the above developed limit theorems to establish the
large sample properties of spatial GMM estimators under a reasonably general
set of assumptions that should cover a wide range of empirical problems. More
speci…cally, our consistency and asymptotic normality results (i) maintain only
that the spatial data process is NED on an -mixing basis process to accommodate
spatial lags in the data process as discussed above, (ii) allow for the data
process to be located on an unevenly spaced grid, and (iii) allow for the data
process to be non-stationary, which will frequently be the case in empirical applications.
We also give our results under a set of primitive su¢ cient conditions
for easier interpretation by the applied researcher. 7
We continue with the basic set-up of Section 2. Consider the moment function
qi;n : R pz ! R pq , where denotes the parameter space, and let 0 n 2
denote the parameter vector of interest (which we allow to depend on n for reasons
of generality). Suppose the following moment conditions hold
Eqi;n(Zi;n; 0 n) = 0; (22)
then the corresponding spatial GMM estimator is de…ned as
where Qn: ! R,
b n = arg min
2 Qn(!; ); (23)
Qn(!; ) = Rn( ) 0 PnRn( );
Rn( ) = jDnj
X
1
qi;n(Zi;n; );
i2Dn
where the Pn are a positive de…nite weighting matrices. To prove that b n is
0
a consistent estimator for n consider the following non-stochastic analogue of
Qn, say
Qn( ) = [ERn( )] 0 P [ERn( )] ; (24)
where P denotes the probability limit of Pn. Then given the moment condition
(22) clearly Rn( o o
n) = 0, and the functions Qn are minimized at n with
Qn( o n) = 0. In proving consistency, we follow the classical approach; see, e.g.,
7 In an important contribution, Conley (1999) gives a …rst set of results regarding the
asymptotic properties of GMM estimators under the assumption that the data process is stationary
and -mixing. Conley also maintains some high level assumption such as …rst moment
continuity of the moment function, which in turn immediately implies uniform convergence
- see, e.g., Pötscher and Prucha (1989) for a discussion. Our results extend Conley (1999)
in several important directions, as indicated above. We establish uniform convergence from
primitive su¢ cient conditions via the generic uniform law of large numbers given in Jenish
and Prucha (2009) and the law of large numbers given as Theorem 2 above.
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