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

where B c n denotes the complement of the event Bn. We note that
P (B c n) ≤ P (|vmx ˆ − vmx| ≥ n −(1/2+δ) ) + P (|vmn ˆ − vmn| ≥ n −(1/2+δ) )
+
Zc
j=1
P (|z [j]
mx ˆ − z[j]
mx| ≥ n −(1/2+δ) ) + P (|z [j]
mn ˆ
− z[j]
mn| ≥ n −(1/2+δ) ) (A.22)
and the right hand side goes to 0 by the well known n-rate of convergence of the extreme
estimators under the compact support conditions. Also, we note that
where
P (n 1/2 |An| > ɛ, Bn) ≤ P (Cn > ɛ) (A.23)
Cn = 1
√ n
+
Zc
j=1
+ I[z [j]
i
n
I[vi ∈ [vmx − hn − n −(1/2+δ) , vmx − hn]] + I[vi ∈ [vmn + hn − n −(1/2+δ) , vmn + hn]]
i=1
I[z [j]
i
∈ [z[j]
mx − hn − n −(1/2+δ) , z [j]
mx − hn]]
∈ [z[j] mn + hn − n −(1/2+δ) , z [j]
mn + hn]] > 0]
We note that by the assumption that vi, z (c)
i
has positive density everywhere on the rectangle,
E[Cn] = o(1) and Var(Cn) = o(1), so P (Cn > ɛ) → 0, establishing the desired result.
We now prove the theorem for the density weighted closed form estimator. The proof
applies to either of the two trimming assumptions, and their corresponding support assump-
tions. For clarity of exposition, we focus on the first set of assumptions, and simply note
that Assumptions C5’,C7’, and C8’ could be used whenever C5,C7, and C8 are referred to
in the proof.
Proof: We work with the relationship:
ˆΞ − Ξ0 = 1
n
ˆi
τni
n ˆf i=1
∗ −
i
i
f ∗
i
+ 1
n
i
τni
n f
i=1
∗
− E
i
i
+ E
τni
f ∗ i
− Ξ0
31
i
τni
f ∗ i
(A.25)
(A.26)
(A.27)
(A.24)