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

To illustrate Theorem 2.1, consider as a special case the binary choice model d =
I(v + x ′ β0 + e ≥ 0) with data consisting of a sample of observations of di, vi, xi, zi. Let-
ting h(v, x, z, e, θ) = z[d − I(v ≥ 0)] gives, by equation (2.10),
E ∗
d − I(v ≥ 0)
z
f ∗
= E
(v|z)
∗ [z(x ′ β0 + e)] (2.12)
which, if E ∗ [ze] = 0, shows that β0 in the binary choice model can be estimated by linearly
regressing [di − I(vi ≥ 0)]/f ∗ (vi|zi) on xi using instruments zi. This is the binary choice
model identification result proposed in Lewbel (2000).
We will now apply Theorem 2.1 and its corollary to obtain identification results for
truncated regression models.
2.2 Exogenous Truncated Regression Model Identification
Our identification for the truncated regression model with exogenous regressors and possibly
heteroskedastic errors is based on the following assumptions:
ASSUMPTION A.1: Assume the truncated data are draws v, x, e, y conditional on yi ≥
0 as described by equations (1.1) and (1.2) with α0 = 0. The underlying, untruncated
conditional distribution of v given x is absolutely continuous with respect to a Lebesgue
measure with conditional density f ∗ (v|x).
ASSUMPTION A.2: Let Ω denote the underlying, untruncated support of the distribu-
tion of an observation of (v, x). Let F ∗ e (e|v, x) denote the underlying, untruncated condi-
tional distribution of an observation of e given an observation of (v, x), with support denoted
Ωe(v, x). Assume F ∗ e (e|v, x) = F ∗ e (e|x) and Ωe(v, x) = Ωe(x) for all (v, x) ∈ Ω.
ASSUMPTION A.3: The underlying, untruncated conditional distribution of v given x
has support [L, K] for some constants L and K, −∞ ≤ L < K ≤ ∞.
ASSUMPTION A.4: For all (x, e) on the underlying, untruncated support of (x, e),
[I(α0 > 0)L − I(α0 < 0)K]α0 + x ′ β0 + e < 0. Let k equal the largest number that satisfies
the inequality k ≤ [I(α0 > 0)K − I(α0 < 0)L]α0 + x ′ β0 + e for all (x, e) on the support of
(x, e). k > 0.
ASSUMPTION A.5: For some positive, bounded function w(x) chosen by the econome-
trician, E ∗ [exw(x)] = 0 and E ∗ [w(x)xx ′ ] exists and is nonsingular.
7