Page 319 - Contributed Paper Session (CPS) - Volume 7

CPS2110 Johann Sebastian B. C. et al.
The parametric p-values will be computed from the null distribution of (4),
while the nonparametric p-values will be computed based on the bootstrap
distribution of (4). For the bootstrap p-value, the following formula is used
(Davison & Hinkley, 1997):

From the estimated p-values, the size of the test will be computed as the
proportion of ejections under a true null hypothesis, i.e. when there is truly no
overdispersion. Similarly, the power of the test will be computed as the
proportion of rejections under a false null hypothesis, i.e. when overdispersion
is truly present.
It must also be noted that, by the original method of Cameron and Trivedi
(1990), no restrictions on is made, so that there is a non-zero probability that
a negative coefficient may be estimated, so that as a consequence, a negative
variance might be attained. For the final part of the simulations, the proportion
of negative coefficients will be recorded and compared with the computed
size and power as an added measure of reliability of the test.

3. Result
The first table presents the parametric and nonparametric size of the score
test for overdispersion, i.e. the proportion of rejections when the null
hypothesis of no overdispersion is true:
Constant Linear Quadratic Polynomial

Bootstrap Bootstrap Bootstrap Bootstrap
10% 28% 0% 27% 13% 26% 29% 27% 24%
5% 26% 0% 26% 9% 25% 16% 25% 18%
1% 21% 0% 24% 0% 23% 10% 23% 8%
Table 1: Size of the test
From the table, it can be seen that the parametric -test for overdispersion
tends to reject 0 more than the reference level – thus, the test is not properly
sized. However, if the bootstrap distribution were to be used instead, the size
tends to be smaller than the former, and increases as the assumed model of
the variance becomes more complicated. The next table presents the
parametric and nonparametric power of the test, i.e. the proportion of
rejections when the null hypothesis of no overdispersion is false, under the
weak form:

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