On the Identification of Misspecified Propensity Scores - School of ...

Ichimura, and Todd [6]. In addition, the expression in (15) can be further broken into
n √ hn
n − 1
[Q(Xj, θ) − Q(Xj, θ0)][Q(Xi, θ) − Q(Xi, θ0)] Q(Xi, θ) − Q(Xj, θ)
E
K
|Dj, Xj
j
hn
[Q(Xj, θ) − Q(Xj, θ0)][Q(Xi, θ) − Q(Xi, θ0)] Q(Xi, θ) − Q(Xj, θ)
−E
K
hn
(16)
+n
[Q(Xj, θ) − Q(Xj, θ0)][Q(Xi, θ) − Q(Xi, θ0)] Q(Xi, θ) − Q(Xj, θ)
hnE
K
.
hn
hn
(17)
(16) is a degenerate order 1 process, and hence could be analyzed using the same lemma. We
study (17) at the end.
Our first step is verifying the conditions of the equicontinuity lemma for {ψn(i, j) −
2
E[ψn(i, j)|j]}. Note that since |D − Q(x, θ)| ≤ 1 for all x, θ,
(n−1) √
Q(Xi,θ)−Q(Xj,θ)
K
is
hn
hn
an envelope for this process. Moreover,
lim sup
4
E K
n→∞
2
Q(Xi, θ)− Q(Xj, θ)
= lim sup
Q(Xi, θ)− Q(Xj, θ)
.
n→∞
i
j
(n − 1) 2 hn
In addition,
1
E K
hn
2
Q(Xi, θ)− Q(Xj, θ)
=
hn
1
E K
hn
2
Q(Xi, θ)− Q(Xj, θ)
hn
1 Q(Xi, θ0)−Q(Xj, θ0)
+E
.
K
hn
2
hn
hn
hn
4
E K
hn
2
−K 2
hn
hn
Q(Xi, θ0)−Q(Xj, θ0)
Since K and Q are Lipschitz, and K is bounded, for some constant C, the first expression is
bounded by C/(anh 2 n ), which has a finite limit. Moreover, since Q(X, θ0) is assumed to have
a Lebesgue density, denoted by fQ, we have
Q(Xi, θ0)− Q(Xj, θ0)
=
1
E K
hn
2
hn
hn
K 2 (u)fQ(Q(Xj, θ0) + uhn)dufQ(Q(Xj, θ0))dQ.
But boundedness of K implies that this last expression has a finite limit as well. Thus, the first
condition of the equicontinuity lemma is satisfied for our process. To
verify the second condi-
1
tion, we note that for each δ > 0, if the kernel function is bounded, 1
(n−1) √ hn K hn
equals 1 for only finitely many n because n √ hn → ∞. On the other hand, by the dominated
convergence theorem, we have
limn→∞ E 1
= E
limn→∞
(n−1) 2
i
i
j
j 1
hn K2
1
(n−1) 2 hn K2
Q(Xi,θ)−Q(Xj,θ)
hn
Q(Xi,θ)−Q(Xj,θ)
hn
15
1 1
(n−1) √ hn K
1 1
(n−1) √ hn K
Q(Xi,θ)−Q(Xj,θ)
hn
Q(Xi,θ)−Q(Xj,θ)
hn
Q(Xi,θ)−Q(Xj,θ)
> δ
> δ
= 0.
> δ