the RMSE of the two-stage estimator is less than half of the RMSE for the
FM-OLS or DOLS estimator.
Next, we compare the small sample properties of the tests of the cointe-
gration rank. The LR-bar test suggested by Lyhagen et al. (2001) is denoted
by “LLL” and the regression based test for the cointegration rank based on
(8) is labelled as “REG”. The asymptotic values for T, N → ∞ as reported
in Appendix B are used. From the simulation results displayed in Table
2(a) it turns out that for small values of T , the LLL test tends to be very
conservative, whereas the REG test performs much better in small samples.
To investigate the power of the test statistics we compute the rejection
rates under the local alternative
where Vnt = 10/(T√N). Note that if Vnt = 0, then rk(Π) = 2 and,
therefore, the system is stationary. Such sequence of local alternatives is
considered in order to make the power comparable for varying N and T .
Furthermore, such alternatives allow the study of the power against alterna-
tives that comes close to the null hypothesis of interest (“near-stationary”
alternatives), which seems to be a relevant situation in empirical practice.
Table 2(b) presents the size adjusted local power of the the LLL and the REG
test. It turns out that in small samples the REG test is much more pow-
erful against local alternatives even if the size bias of the tests is accounted
for. Furthermore it is interesting to note that both tests seem to converge to
roughly the same limiting power. For small T , however, the local power of
the REG test is much higher than the respective power of the LLL test.
Π=αβ0= -0.1
1
(1 — Vnt )
(16)
Finally, we study the performance of the robust estimator of the standard
errors of the parameters. To this end we compute the rejection frequencies of
a t-test for the hypothesis that β = [1, 1]0 in (16). Since the robust estimator
(15) is consistent under contemporaneous correlation and heteroskedastic er-
rors, the empirical size of a t-test based on the robust standard errors should
approach the nominal size for sufficient sample sizes. The respective esti-
mator is called “2S-HAC”. To generate contemporaneously correlated errors,
the matrix of contemporaneous errors Et = [εi1 , . . . , εiT]0 is multiplied by
the k × k matrix Q such that transformed errors εit result from the rows of
the matrix Et = QEt . In our simulations, the elements of the matrix Q are
generated by independent draws of U(0, 10) distributed random variable.
13