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

To complete the proof we apply theorem A.1 to derive a linear representation for
1
n
n
zi(ˆy ∗ i − x ′ iβ0) (A.79)
i=1
In this context, i = µ −1
0 zi(yi − viα0)I[0 < yi < k] − zix ′ iβ0. The preliminary estimators are
ˆµ and ˆα. We note that:
i
E ∇µ = − µ −2 k
E[zix ′
i]β0
and
E
f ∗ i
i
∇α
f ∗ i
0
α0
1
= −
2α2 (k
0
2 E[zi] − kE[zix ′
i]β0 · µ −1
0
(A.80)
(A.81)
Hence the limiting distribution follows from this linear representation, the convergence of ˆ ∆
to ∆, and Slutsky’s theorem.
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