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

Proof: Our proof strategy will be to show the probability limit of the term ˆ Hn is the
matrix H0 and use the results from the previous section to derive a linear representation for
H −1
0 ˆ Sn. Finally, with the asymptotic distribution theory for π0 ≡ (a0, b ′ 0) ′ we can apply the
delta method to attain the asymptotic distribution theory of our estimators of θ0 = (α0, β ′ 0) ′ .
As a first step note that the conditions on the bandwidth sequence and the kernel
functions (i.e. Assumption WLS3) we can replace kernel density estimates with true values
and the remainder term is uniformly (in the support of xi, vi) op(1). (See, e.g. Newey
and McFadden(1994), Lemma 8.10 for the bounded support and trimming in C5’,C8’ or
Sherman(1994), Corollary 5A, for the support and trimming in C5,C8). Thus we are left
with the term:
1
n
n
i=1
τniwkixix ′ i
(A.57)
Thus by the law of large numbers and Assumptions WLS8,WLS9 the above term converges
to H0 ≡ E[wkixix ′ i]. Note that using (2.14) we can alternatively express H0 as:
H0 = k
E[w(xi)]
|α0|
k2
3 E[w(xi)x ′ i] k
2
E[w(xi)xi] k
2 E[w(xi)xix ′ i]
By assumption WLS8 it follows that ˆ H −1
n
p
→ H −1
0 .
Next we apply results from the previous section to derive a linear representation for
H −1 1
0
n
(A.58)
n
τni ˆwkixi(vi − x ′ iπ0) (A.59)
i=1
First we note that (by (2.14)) and the conditions on the trimming function behavior (specif-
ically WLS6.2), E[τniwkixi(vi − x ′ iπ0)] = o(n −1/2 ). Conditions C1-C8 follow immediately
from Assumptions WLS1-WLS9. To state the form of the limiting variance matrix of the
estimator of π0 we note i/f ∗ i is mean 0 by (2.14), and so δi in WLS9 is mean 0.
Finally, to get the asymptotic variance of the parameters of interest θ0 = (α0, β0) we
can simply apply the delta method and pre and post multiply the matrix H −1
0 ΩH −1
0 by the
Jacobian of transformation. This (J + 1) × (J + 1) matrix, referred to here as Q, is of the
form:
Q =
α 2 0 01×J
α0β0 α0IJ×J
(A.60)
where 01×J refers to a (1 × J) vector of 0’s and IJ×J refers to a (J × J) identity matrix.
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