![]() |
We now apply the empirical likelihood test procedure on the
S&P 500 data presented in Figure 12.1 to test the parametric
mean function
given in the Cox-Ingersoll-Ross diffusion
model (12.33).
The process
is restored from the observed residuals by the
approach introduced in Härdle et al. (2001).
The parametric
estimate for
is
by using
methods based on the marginal distribution and
the autocorrelation structure of
. For details about the procedure
see Härdle et al. (2001).
The cross validation is used to find the bandwidth
.
However, the score function is
monotonic decreasing for
and then become
a flat line for
.
This may be caused by the different intensity
level of the design points.
Further investigation shows that a
-value larger (smaller) than 0.06 (0.02) produces an
oversmoothed (undersmoothed) curve
estimate.
Therefore, the test is carried out for a set of
values ranging from 0.02 to 0.06.
The P-values of the test as a function of
are plotted in Figure
12.2.
The P-values indicate that
there is insufficient evidence to reject the diffusion model.