weighted and two stage least squares estimation of ... - Boston College

and let δi denote the mean 0 vector
Mκψi + i
f ∗ i
where
ϕb(·, ·) = E
ϕa(·) = E
− Ξ0 − cpϕb(v ∗ i , z ∗ i ) + cpϕa(z ∗ i ) (A.8)
i
rvzi
f ∗ i
i
rzi
f ∗ i
then we assume that
vi
= ·, zi = ·
zi
= ·
C4.1 ˜ E [δni 2 ] < ∞ uniformly in n ∈ N.
C4.2 1
n
n
i=1 δi − δni = op(n −1/2 )
(A.9)
(A.10)
C5 We let Z = Zc × Zd denote the support of zi, which we assume to be the same for the
truncated and untruncated populations. We assume the support set Zc is an open,
convex subset of R Zc and assume the support of vi, denoted by V is an open interval
in R. Let f ∗ (v, z (c) |z (d) ) denote the (population untruncated) conditional (Lebesgue)
density of vi, z (c)
i
given z (d)
i , and let f ∗ (z (d) ) denote the probability mass function of
z (d)
i . Furthermore, let f ∗ (v, z) denote f ∗ (v, z (c) |z (d) ) · f ∗ (z (d) ). Then we assume:
C5.1 f ∗ (v, z (c) |z (d) ), considered as a function of v, z (c) , is p times continuously differ-
entiable, with bounded p th derivatives on V × Z.
C5.2 There exists a constant µ0 > 0 such that for all zi ∈ Z, f ∗ (z (d)
i ) ∈ (0, µ0).
C6 The kernel functions K1(·), K2(·) satisfy the following properties:
C6.1 They are each the product of a common univariate function which integrates to
1, has support [−1, 1], and is assumed to be p times continuously differentiable.
C6.2 For two vectors of the same dimension, u, l, we let u l denote the product of
each of the components of u raised to the corresponding component of l. Also,
for a vector l which has all integer components, we let s(l) denote the sum of its
components. The kernel functions are assumed to have the following property:
Kj(u)u l du = 0 j = 1, 2 l ∈ N, 1 ≤ s(l) < p (A.11)
27