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
that under Assumptions A1–A6 ϒ
u =ϒu +o
P
1. Alterna- tively, we can obtain the estimates
α u and thus
g
j
u for u = U
i
, i = 1, . . . , n, and then we can define the global resid- ual
ε
i
= Y
i
−
d j =1
g
j
U
i
X
i,j
. Replacing
ε
i
u in the defini- tion of
ϒ u by
ε
i
also yields a consistent estimate of ϒu, but
this needs preliminary estimation of the functional coefficients at all data points and thus is much more computationally ex-
pensive. By choosing
n
u = ϒ
u, we denote the resulting local linear GMM estimator of αu as
α
ϒ
u; h, λ. We summa-
rize the asymptotic properties of this estimator in the following corollary, whose proof is straightforward.
Corollary 3.2. Suppose that Assumptions A1–A4i and
A5 and A6 hold. Then √
nh {H[
α
ϒ
u; h, λ − αu] −
′
ϒ
−1 −1
′
ϒ
−1
B u; h, λ}
d
→ N0,
′
ϒ
−1 −1
. In particular, √
nh {
g
ϒ
u; h, λ − gu − f
U
u
−1
[
′ 1
u
2
u
−1 1
u]
−1 ′
1
u
2
u
−1
B u; h, λ}
d
→ N0, μ
p
c
0,2
f
U
u
−1
[
′ 1
u
2
u
−1 1
u]
−1
, where
g
ϒ
u; h, λ and B u; h, λ denote the
first d elements of α
ϒ
u; h, λ and Bu; h, λ, respectively.
Remark 4 Asymptotic independence between estimates of functional coefficients and their first-order derivatives.
Theo- rem 3.1 indicates that, for the general choice of
n
that may not be block diagonal, the estimators of the functional coef-
ficients and those of their first-order derivatives may not be asymptotically independent. Nevertheless, if one chooses
n
as an asymptotically block diagonal matrix i.e., the limit of
n
is block diagonal as in Corollary 1, then we have asymptotic
independence between the estimates of gu and
·
g u. If further
k = d, then the formulas for the asymptotic bias and variance of
g
ϒ
u can be simplified to
1
u
−1
B u; h, λf
U
u and
μ
p
c
0,2 1
u
−1 2
u
1
u
−1 ′
f
U
u, respectively.
3.3 Optimal Choice of Global Instruments
To derive the optimal global instruments for the estimation of α
u based on the conditional moment restriction given in 1
, define
Q
∗
V
i
= CE X
i
|V
i
σ
2
V
i
, 3.4
where C is any nonsingular nonrandom d × d matrix. As Q
∗
V
i
is a d × 1 vector, the weight matrix
n
does not play a role. It is easy to verify that the local linear GMM estimator corresponding
to this choice of IV has the following AVC matrix:
∗
u = f
−1 U
u
μ
p
c
0,2 ∗
u
−1 d×dp
c
dp
c
×d
μ
2,2
μ
2 2,1
∗
u
−1
⊗ I
p
c
= f
−1 U
u K
μ
p
c
0,2 ∗
u
d×dp
c
dp
c
×d
μ
2,2 ∗
u ⊗ I
p
c
−1
K, where
∗
u ≡ E[E X
i
|V
i
E X
i
|V
i ′
σ
−2
V
i
|U
i
= u]
and K ≡
μ
p
c
0,2
I
d d×dp
c
dp
c
×d
μ
2,2
μ
2,1
I
dp
c
. Noting that
∗
u is free of the choice of C, hereafter we simply take C = I
d
and continue to use Q
∗
V
i
to denote
Downloaded by [Universitas Maritim Raja Ali Haji] at 22:05 11 January 2016
E X
i
|V
i
σ
2
V
i
. We now follow Newey 1993
and argue that
Q
