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

with cin = 1. (Note that Assumption 3(d) is satis…ed automatically since the
qi;n(Zi;n; o n) are uniformly L2-NED.) Hence,
1=2
n jDnj Rn( o X
1=2
n) = n qi;n(Zi;n; o n) =) N(0; Ipq ); (C.11)
with n = V ar P
i2Dn
i2Dn qi;n(Zi;n; o n) and sup n max
h
jDnj 1 i
n < 1.
Step 3. By Assumptions 7(c),(d),(e) the functions r qi;n(Zi;n; ) satisfy for
each 2 the LLN given as Theorem 2 in the text with ci;n = 1, observing
that Assumption 4(b) is implied by 2. By argumentation analogous as used in
the proof of consistency we have
jDnj
1 X
i2Dn
(r qi;n(Zi;n; ) Er qi;n(Zi;n; )) p ! 0:
By Proposition 1 of Jenish and Prucha (2009), Assumption 7(f) implies that the
r qi;n(Zi;n; ) are uniformly L0-equicontinuous on . Given L0-equicontinuity
and Assumption 7(e), we have by the ULLN of Jenish and Prucha (2009), Theorem
2,
sup jr Rn( )
2
Er Rn( )j p ! 0: (C.12)
and furthermore, the Er Rn( ) are uniformly equicontinuous on
that
in the sense
sup jEr Rn( ) Er Rn( )j ! 0 (C.13)
lim sup sup
n!1 02 j
as ! 0. In light of (C.12) and (C.13), and given that b n
hence e o
n n = op(1), if follows further that
0 j<
o
n = op(1) and
r Rn( b n) Er Rn( o n) p ! 0; and r Rn( e n) Er Rn( o n) p ! 0:
Hence,
bAn An
where b An and b Bn are as de…ned above, and
p
p
! 0 and Bn
b Bn ! 0; (C.14)
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 :
Step 4. Given Assumptions 7(e),(f), and since P is positive de…nite, we
have jAnj = O(1) and A 1
n = O(1), respectively. Hence by, e.g., Lemma
F1 in Pötscher and Prucha (1997) we have b An = Op(1), b A + n = Op(1), b An is
nonsingular with probability tending to one, and b A + n A 1 p
n ! 0. In light of the
above it follows from (C.10) that
jDnj 1=2 b
n
o
n =
h
A b+ b 1=2
n Bn n jDnj Rn( o =
i
n) + op(1)
A 1
h
1=2
n Bn n jDnj Rn( o i
n) + op(1)
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