Semiparametric Analysis to Estimate the Deal Effect Curve

Appendix 2: Estimation procedure for semiparametric model
The estimation procedure we use is described in Lee (1996). The semiparametric
model (1) can be simplified as follows:
y = m(z) + x ′ β. (9)
Taking E(·|z) on (9), we obtain:
E(y|z) = m(z) + E(x|z) ′ β. (10)
Subtracting (10) from (9) gives:
y−E(y|z) ={x−E(x|z)} ′ β+u ⇔ y = E(y|z)+{x−E(x|z)} ′ β+u,(11)
which excludes m(z). Here the deterministic component of y is decomposed into two
parts: one is the effect of z on y, and the other is the effect of x on y net of z. The
three-step estimation procedure first eliminates the influence of x on y and determines
next the influence of z on y:
Step 1 Estimate E(y|z) and E(x|z), respectively, as:
N
N
EN(y|zi) ≡ K((zj − zi)/hy,1) · yj/ K((zj − zi)/hy,1), (12)
and
EN(x|zi) ≡
j=i
j=i
j=i
N
N
K((zj − zi)/hx,1) · xj/ K((zj − zi)/hx,1). (13)
Step 2 Substitute (12) and (13) into (11) to define a new criterion variable y −
EN(y|z) and predictors x − EN(x|z). Apply Least Squares Estimation (LSE)
to the new model y − EN(y|z) ∼ = {x − EN(x|z)} ′ β + u to get:
bN = [
{xi − EN(x|zi)}·{xi−EN(x|zi)} i
′ ] −1 ·
[
{xi − EN(x|zi)}·{yi−EN(y|zi)}] (14)
32
i
The least squares estimate bN has the following asymptotic distribution
(which is used to compute standard errors):
√ N(bN −β) = d N(0,A −1 BA −1 ),
A = p AN ≡ 1
N
B = p BN ≡ 1
N
j=i
{xi − EN(x|zi)}·{xi−EN(x|zi)} ′ ,
i
{xi − EN(x|zi)}·{xi−EN(x|zi)} ′ · v 2 i ,
i
vi ≡{yi−EN(y|zi)}−{xi−eN(x|zi)} ′ bN.