Small sample distributions of chose

Small sample distributions of chosen unit root tests
Next, we follow Rudebusch (1993) to estimate the
finite sample distributions of the chosen unit root
tests under the null and alternative hypotheses. This
approach helps us discern which distribution is more
likely to generate the test value. We describe the steps
as follows.
First, artificial ridt series were generated for each
country according to the estimated models in
Equations 17 and 18. The sample length of each
generated series is consistent with the original length.
To prevent our results from the effect of initial values,
100 more observations were generated and then
dropped from the beginning of the generated series.
Second, for each country, the chosen test statistics
were calculated for each of the 5000 generated series,
to establish the simulated distributions for the two
hypotheses. Based on the two distributions, we
calculated the finite-sample sizes and size-adjusted
powers for each test, where the sizes were calculated
according to the 5% asymptotic critical values, while
the powers were based on the 5% small sample
critical values in order to control for size distortions.
Finally, the p-values of the test statistics for each of
the estimated processes were also calculated as
follows:
p-value of test statistics under the null hypothesis
¼ P ^  ^samplej fDSð^Þ ð19Þ
p-value of test statistics under the alternative
hypothesis ¼ P ^  ^samplej fLSð^Þ ð20Þ
where ^ and ^sample are the test statistics calculated
from the artificial and actual data; fLSð^Þ and fDSð^Þ
are the simulated distributions of ^, conditional on
the AR(p) and the AR(p 1) models.
Note that instead of using the estimated lags from
the best-fitting models like Kuo and Mikkola (1999)
did, we chose the lag p for the unit root tests with the
MAIC in each of the calculations to avoid possible
size distortions.
IV.