3.2 Asymptotic Theory for the Local Linear Estimator
Let μ
s,t
=
R
v
s
w v
t
dv, s, t = 0, 1, 2. Define
u = f
U
u
1
u
k×dp
c
kp
c
×d
μ
2,1 1
u ⊗ I
p
c
, 3.1
and ϒ
u = f
U
u
μ
p
c
0,2 2
u
k×kp
c
kp
c
×k
μ
2,2 2
u ⊗ I
p
c
. 3.2
Clearly, u is a k1 + p
c
× d1 + p
c
matrix and ϒu is
k 1 + p
c
× k1 + p
c
matrix. To describe the leading bias term associated with the discrete
random variables, we define an indicator function I
s
·, · by I
s
u
d
, u
d
= 1{u
d
= u
d s
}
p
d
t =s
1{u
d
= u
d t
}. That is, I
s
u
d
, u
d
is one if and only if u
d
and u
d
differ only in the sth component and is zero otherwise. Let
B u; h, λ =
1 2
μ
2,1
f
U
u
1
uA u
; h
kp
c
×1
+
u
d
∈U
d
p
d
s=1
λ
s
I
s
u
d
, u
d
f
U
u
c
, u
d
×
1
u
c
, u
d
gu
c
, u
d
− gu
c
, u
d
−μ
2,1 1
u
c
, u
d
⊗ I
p
c
·
g u
c
, u
d
, 3.3
where A
u; h =
p
c
s=1
h
2 s
g
1,ss
u,
. . . ,
p
c
s=1
h
2 s
g
d,ss
u
′
,
g u = g
1
u, . . . , g
d
u
′
, and
·
g u =
.
g
1
u
′
, . . . ,
.
g
d
u
′ ′
. Now we state our first main theorem.
Theorem 3.1. Suppose that Assumptions A1–A6 hold.
Then √
nh {H[
α
n
u; h, λ − αu] −
′ −1
−1 ′
−1
B u; h, λ}
d
→ N0,
′ −1
−1 ′
−1
ϒ
−1 ′
−1 −1
,
where we have suppressed the dependence of ,, and ϒ on u, and H = diag 1, . . . , 1, h
′
, . . . , h
′
is a dp
c
+ 1 × dp
c
+ 1
diagonal matrix with both 1 and h appearing d times.
Remark 3 Optimal choice of the weight matrix. To mini-
mize the AVC matrix of α
n
, we can choose
n
u as a con- sistent estimate of ϒu, say
ϒ
u. Then the AVC matrix of
α
ϒ
u; h, λ is given by u = [u
′
ϒ u
−1
u]
−1
, which
is the minimum AVC matrix conditional on the choice of the global instruments QV
i
. Let α
u be a preliminary estimate of αu by setting
n
u = I
k p
c
+1
. Define the local residual
ε
i
u = Y
i
−
d j =1
g
j
uX
i,j
, where
g
j
u is the jth component
of α
u. Let
ϒ u =
h
n
n i=1
× Q
i
Q
′ i
ε
i
u
2
Q
i
Q
′ i
⊗ η
i
u
c ′
ε
i
u
2
Q
i
Q
′ i
⊗ η
i
u
c
ε
i
u
2
Q
i
Q
′ i
⊗ [η
i
u
c
η
i
u
c ′
] ε
i
u
2
× K
2
hλ,iu
,
where Q
i
≡ QV
i
and η
i
u
c
≡ U
c i
− u
c
h. It is easy to show
