Suppose is defined as in (12.6) and
let
be the conditional mean function,
be the density of the design points
, and
be the conditional variance function
of
given
,
a closed interval
.
Suppose that
is a parametric model
for the mean function
and that
is an estimator of
under this parametric model. The interest is to test the null hypothesis:
The problem of testing against a nonparametric alternative is not new for an independent and identically distributed setting, Härdle and Mammen (1993) and Hart (1997). In a time series context the testing procedure has only been considered by Kreiß et al. (1998) as far as we are aware. Also theoretical results on kernel estimators for time series appeared only very recently, Bosq (1998). This is surprising given the interests in time series for financial engineering.
We require a few assumptions to establish the results in this chapter. These assumptions are the following:
Assumptions (i) and (ii) are standard in nonparametric curve estimation
and are satisfied for example for bandwidths selected by
cross validation, whereas (iii) and (iv) are common in nonparametric
Goodness-of-Fit tests. Assumption (v) means the data
are weakly dependent. It is satisfied for a wide class of diffusion
processes.