D . Wan Shin, B. Soo So Economics Letters 71 2001 181 –189
185
¯ ¯
for all uru 1, which produces a confidence interval of the form r 6z
se r .
iv a 2
iv
ˆ ˆ
ˆ In Theorem 3, we establish limiting normality of
t r 5 r 2 r ser .
iv iv
iv
Theorem 3. Consider model 3. Assume that all the characteristic roots of uB 5 1 2 u B
1 p 21
¯ 2 ? ? ? 2
u B
lie outside the unit circle. For any uru 1, the limiting distribution of t r is
p 21 iv
standard normal.
3. Numerical studies
We first compare empirical coverages and the average lengths of our confidence interval CI and
iv
the intervals CI of Andrews 1993 and CI of Stock 1991. The standard normal errors e are
a s
t
generated by RNNOA, a FORTRAN subroutine of IMSL 1989. In order to simulate stationarity for y , t 5 1, . . . , n in case of
uru , 1, the model is generated for t 5 2 19, . . . , n. Data y are generated
t t
by 2 with m 5 0; r 5 0, 0.3, 0.6, 0.9, 0.95, 0.99, 0.995, 1; n 5 50, 100; 10 000 replications. For
ˆ CI , we consider c 5 k
s, k 5 0, 1, 2, 3, 4. For CI , we only consider r 5 0.9, 0.95, 0.99, 0.995, 1
iv s
because Stock provided tables for CI only for r near unity. Nominal coverage is set to 90 because
s
Andrews prepared tables for only this coverage. All the estimators are adjusted for mean. We restrict r
to [21, 1] as Andrews 1993 did in constructing his confidence intervals. The upper limits of CI
iv
and CI are replaced by one if their right end-points are greater than one. The column under ‘NA’
s
represents cases in which Stock’s table Table 1 fails to provide a confidence interval. These are the cases where the Dickey–Fuller’s tau statistic is out of range [236.79, 1.67]. The coverage probability
and average length of CI are computed excluding these cases. In Table 1, empirical coverages
s
and the average lengths of the confidence intervals are reported. The coverage of CI is slightly
iv
smaller than the nominal coverage. However, the coverages are all close to the nominal coverage for all n and k considered here. When
r is close to one, the average length of CI is smaller than those of
iv
CI and CI . For example, when n 5 50 and r 0.99, the average length of CI is about 9 10 of that
a s
iv
of CI and 3 4 of that of CI . When r 5 0, 0.3, 0.6, length of CI
is almost as good as CI . The
a s
iv a
advantage of CI over CI and CI seems uniform for all k 5 1, 2, 3, 4. Hence, we may recommend
iv a
s
any of 1 k 4 for a good confidence interval because the performance of CI is similar for this
iv
wide range of k.
4. Concluding remarks
