6 Small Sample Properties
In this section, the small sample properties of alternative estimators are
studied by means of Monte Carlo simulations. The data are generated by
the two-dimensional cointegrated VAR(1) model with error correction repre-
sentation
where εit ~ i.i.N.(0,I) and the individual effects are generated as μi ~
i.i.U (0, 1). If φ > 0 then yt is cointegrated with cointegration rank r = 0. To
save space, only the results for b = 1 and φ = 0.1 are presented. The results
for other values of the parameters are qualitatively similar.3
∆yit
μi-
[1
b] yi,t-1 + εit ,
The comparison includes the FM-OLS estimator with individual specific
constant and short-run dynamics proposed by Pedroni (1995, 2000) and
Phillips and Moon (1999), the dynamic OLS (DOLS) estimator suggested
by Kao and Chiang (2000), where the length of the lags and leads is two, the
(inefficient) OLS estimator and the two-step estimator suggested in Section
3. The FM-OLS and DOLS estimators are computed by using the GAUSS
library NPT 1.1 developed by Chiang and Kao (2000). The bias and root
mean-square errors (RMSE) for various sample sizes that are typical for em-
pirical work using country studies are reported in Table 1.
It is well known that the bias in the OLS estimator of the cointegration
parameters is O(T -1). As can be seen from the results reported in Table
1, the bias of the OLS estimator is severe if the number of time periods is
small. The nonparametric bias correction of the FM-OLS estimator seems to
be insufficient in short time series as it reduces the bias only marginally. For
T = 100 the bias reduction is more effective. The DOLS estimator removes
the bias by including future and past values of ∆yi(t2) . The results displayed in
Table 1 suggest that this approach performs slightly worse than the FM-OLS
estimator.
The bias of the two-stage estimator is much smaller in absolute value.
For T = 30 the two-step estimator is nearly unbiased, whereas the FM-OLS
and the DOLS estimator still possess a severe negative bias. For the root
mean square error (RMSE) the conclusions are similar. If T is small, then
3Additional simulation results and the GAUSS codes can be found on the homepage of
the authors (http://ise.wiwi.hu-berlin.de/~joerg/pancoint.html).
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