Semiparametric Analysis to Estimate the Deal Effect Curve

Appendix 3: Model operationalizations for semiparametric model
We address three model operationalizations: the choice of the Kernel, the choice of
the bandwidth parameter, and the construction of confidence intervals.
Kernel choice
For a nonparametric model the Kernel function K(z) must be specified. It turns out
that for large samples it makes very little difference what the Kernel function is
(Silverman 1986). Often it can be chosen on the basis of tractability or convenience.
Like Rust (1988) and Abe (1995) we use the multivariate standard normal distribution
(with independent components):
K(z) = K(z1,...,zk)=(2π) −1 2k k
e −1 2z2 i . (16)
Bandwidth parameter choice
The bandwidth parameter h (see equation (8)) determines how fast the value of
K(.) falls as the absolute value of the argument increases (Abe 1995). It affects the
smoothness of the resulting response function, and its value involves the tradeoff
between bias and variance of the estimator. The value of h can be determined on the
basis of a criterion such as the minimization of the mean squared error. 12
i=1
We distinguish three different bandwidth selection techniques:
• Lee’s rule of thumb. An approximately mean integrated square error
minimizing h (see Lee 1996, p. 154) is given by h = N −1/(k+4) .
• Least squares cross validation (LSXV). Define a criterion variable y and a
predictor (k × 1) variable x, and the nonparametric regression function y =
m(x) + u. The “leave-one-out” Kernel estimator for m(xj) (j-th observation
of x) is computed for all j = 1,...,N by:
N
N
ˆmj (xj) ≡ K((xi − xj )/h)yi/ K((xi − xj )/h). (17)
i=j
Then h is chosen to minimize the cross-validation (CV) criterion (see Härdle
and Marron 1985):
N
(1/N) {yj −ˆmj(xj )} 2 . (18)
j=1
12 See Abe (1995, p. 306-7) for a more elaborate discussion about the choice of h.
34
i=j