where the vectors (ZMr 1 ZMr 2 ... ZMrJ )' and (Zr 1 Zr2 ... ZrJ )' are defined respectively in (8) and
(7). In the simulation experiment, J = 16, with r starting from r1 = .15 and then increasing by
.05 until r16 = .85. The critical values defined in (12) have been obtained with S = 1000.
Similarly, under the conditions stated in Corollary 1, we have that for m/n → 0,
Zn,m,1 -d→ MN 0,2 ∞ σ4 (a) LX (1,a)da
The empirical sizes (at 5% and 10% level) of the tests discussed above are reported in Table 1,
for κ = 0, η = 1, μ = —.8, ξn = n-10/13. The results for different values of the parameters needed to
generate (13) and the bandwidth ξn display a virtually identical pattern and therefore are omitted
for space reasons. Inspection of the Table reveals an overall good small sample behaviour of the
considered test statistics. The reported empirical sizes are everywhere very close to the nominal
ones, with a slight tendency to underreject for the test based on Zn,m . The zeros appearing in the
rows when n = m are not surprising; in fact, when using the statistic Zn , the critical values used in
the simulation exercise are just an upper bound of the true ones, and therefore one should expect
an undersized test.
Under the alternative hypothesis, the following model has been considered,
dXt
( κ + μXt )d t + η
√ exp ('7Z (√1 — ρ2dWι,t +
ρdW2,t
d σ2 = ( κ ι + μ ɪ σ2 )d t + η ɪ J σt2d W2 ,t.
(14)
A discretized version of (14) has been simulated using a Milstein scheme as above, with κɪ = 1,
ηɪ = 1, μɪ = —.2. Then, using the obtained values of σ2, the series for Xt has been generated,
with ρ = 0 and keeping the remaining parameters at the values used to generate Xt under (13).
The findings for the power of the tests based on Zn,m and Znm,ɪ are reported in Table 2. The
experiment reveals that the proposed tests has good power properties. The test based on Zn,m is
more powerful than the one based on Znm,ɪ; this is not surprising, given that Zn,m is specifically
constructed to highlight the differences between the local times of Xt and ft . In fact, in the case
of Zm,n the term driving the power is maxr ∣ J0r (LX(r, a) — Lf (r, a)) da∣, which is in general larger
than ∣∕01 (LX(1 ,a) — Lf(1 ,a))da∣, the term driving the power of Zn. Also, the power of the test
based on Zn,m is generally increasing in n and m, as one should expect. In some cases, however,
the power remains constant or even decreases when m approaches n (namely, the cases when
n = 144, 288, 576); this is due to the fact that, when n = m, we are not using the correct critical
values for the test, but just an upper bound, and this may decrease the resulting power of the test.
14