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

where ˆεi := Yi − Q(Xi, ˆ θn), ˆ θn is a √ n-consistent estimator of arg minθ∈Θ E[Y − Q(X, θ)] 2 , h
is a smoothing parameter, and K(·) is a kernel.
Zheng shows that under certain assumptions
where Σ can be consistently estimated by
ˆΣn :=
2
n(n − 1)
nh m/2 ˆVn n
d → N(0, Σ)
n
1
h
i=1 j=i
m n
K 2
Xi − Xj
hn
ˆε 2 i ˆε 2 j .
We would like to test whether E[D|Q] = Q, where Q(X) = Q(X, θ0). Our null and
alternative hypotheses are therefore given by
H0: ∃θ0 ∈ Θ s.t. Pr[E[D|Q(X, θ0)] = Q(X, θ0)] = 1
H1: Pr[E[D|Q(X, θ)] = Q(X, θ)] < 1 ∀θ ∈ Θ
Note that this differs from the framework of Zheng [18] in that the variable that is conditioned
on is not observed but has to be estimated. Nevertheless, by analogy with the test statistic
above, we consider testing based upon
n
1 1 Q(Xi,
ˆVn :=
K
n(n − 1)
ˆ θn) − Q(Xj, ˆ
θn)
ˆεiˆεj
hn
i=1 j=i
where ˆεi = Di − Q(Xi, ˆ θn). Unfortunately, the analysis of Zheng [18] does not apply directly
because in our case the estimated Q appears in the kernel function as well. Thus, we require
the following extension of Zheng’s analysis:
Theorem 4.1. Make the following assumptions:
(i) {Di, Xi} n i=1 is a random sample;
(ii) Q is Lipschitz continuous in (x, θ). Q(·, θ0) is continuously differentiable. Moreover, it
has density with respect to Lebesgue measure on [0,1] that is continuously differentiable;
(iii) K(·) is a nonnegative, bounded, symmetric and Lipschitz continuous function such that
K(u)du = 1;
(iv) The bandwidth sequence and the sequence of positive numbers {an} are such that:
9
hn