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CPS1461 Michal P. et. al
4. Discussion and Conclusion
The changepoint location problem for a very general panel data structure
is considered in this paper: the observations within each panel are dependent,
the given panels are allowed to be heteroscedastic, and even dependent
among each other. The mutual dependence is modeled using some common
random factors and some unknown panel specific loadings. The follow up
period is allowed to be extremely short and the changepoint magnitudes may
differ across the panels accounting also for a specific situation that some
magnitudes are equal to zero (thus, no jump is present in such case). Another
advantage of the proposed approach is that it does not require an apriori
estimation of the changepoint location. We considered two competitive self-
ormalized test statistics and their asymptotic properties are derived. Under the
null hypothesis of no change, the
test statistics weakly converge to a functional of the multivariate normal
random vector with the zero mean vector and the covariance structure
depending on the intra-panel covariances. Under the alternative hypothesis,
both test statistics are shown to converge to infinity with the increasing
number of panels and, thus, both procedures are proved to be consistent.
From the practical point of view, the general structure with heteroscedastic
and possibly dependent panels with extremely short follow up periods is a lot
more realistic scenario for practical utilization of the proposed changepoint
tests than a situation with independent or even homoscedastic panels.
A simulation study (which is not presented here) illustrates that even for
an extremely small panel size (10 observations only), both competitive test
statistics perform quite well: the empirical specificity of both tests is very close
to the theoretical value of one minus the significance level, but a slightly better
performance is observed for the test statistic R N () when considering some
heavy tailed common factors for the mutual panel dependence. On the other
hand, S N () slightly outperforms the previous one in terms of the power,
which is still comparable among various error dependence and panel
dependence structures. The power increases as the number of panels gets
higher. Furthermore, the sensitivity is also affected by the length of the follow
up period, the proportion of panels for which a non-zero jump magnitude is
observed, and the changepoint location. Longer follow up periods and higher
proportions of the panels with jumps in their means imply better powers for
both tests. When considering the changepoint location, then the highest
power is observed for the changepoint located close to the middle of the
follow up period.
The theory can be further extended to propose a consistent changepoint
estimate, which is otherwise not needed for the tests based on the self-
normalized statistics. Such estimate can be used to obtain a bootstrapped
counterpart for the asymptotic distribution of both test statistics. The self-

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