We investigate our testing procedure in two simulation studies. In our first simulation we consider the time series model
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The first and the second parameter are the vectors of observations of and
. The third parameter model is the
name of a quantlet that implements the parametric model for the
null hypothesis. The optimal parameter kernel is the name of the
kernel
that is used to calculate the test statistic and h is
the bandwidth used to calculate
and
in (12.18). theta is directly forwarded to the
parametric model.
For the simulation study the sample sizes considered for each trajectory
are and
and
, the degree of difference between
and
,
takes value of 0,
and
.
As the simulation shows that
the two empirical likelihood tests
have very similar power performance, we will report the
results for the test based on the
distribution only.
To gauge the effect of the smoothing bandwidth
on the power,
ten levels of
are used for each simulated sample to
formulate the test statistic.
Figure 12.3 presents the power of the empirical
likelihood test based on 5000 simulation
with a nominal 5% level of significance.
We notice that when the simulated significance level of
the test is very close to the nominal
level for large range of
values which is especially
the case for the larger sample size
.
When
increases, for each fixed
the power increases as the distance between the null
and the alternative hypotheses becomes larger.
For each fixed
, there is a general trend of decreasing power
when
increases. This is due to the discretization
of
by binning as discussed at the end of the previous section.
We also notice that the power curves for
are a little erratic although they maintain the same trend as in
the case of
. This may be due to the fact that
when the difference between
and
is large, the
difference between the nonparametric and the parametric fits
becomes larger and the test procedure becomes more sensitive
to the bandwidths.
In our second simulation study we consider an Ornstein-Uhlenbeck process
fluctuating about 0 that satisfies the stochastic differential equation
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The number of observations is given by n+1/, a is the speed of
adjustment parameter , s is the diffusion coefficient
and delta is the time difference
between two observations.
The proposed simulation procedure and the Goodness-of-Fit test are illustrated
in
XFGelsim2.xpl
.