Abstract :
This paper characterizes the bandwidth value (h) that is optimal for estimating parameters
of the form η = Eω/f V|U ( V|U), where the conditional density of a
scalar continuous random variable V, given a random vector U, f V|U, is replaced
by its kernel estimator. That is, the parameter η is the expectation of ω inversely
weighted by f V|U, and it is the building block of various semiparametric estimators
already proposed in the literature such as Lewbel (1998), Lewbel (2000b), Honor´e
and Lewbel (2002), Khan and Lewbel (2007), and Lewbel (2007). The optimal bandwidth
is derived by minimizing the leading terms of a second-order mean squared
error expansion of an in-probability approximation of the resulting estimator with
respect to h. The expansion also demonstrates that the bandwidth can be chosen on
the basis of bias alone, and that a simple “plug-in” estimator for the optimal bandwidth
can be constructed. Finally, the small sample performance of our proposed
estimator of the optimal bandwidth is assessed by a Monte Carlo experiment.
