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CPS1461 Michal P. et. al
The sequence { } can be viewed as a part of a weakly stationary
, =1
process. Note that the dependent errors within each panel do not necessarily
need to be linear processes. For example, GARCH processes as error sequences
are allowed as well. The heteroscedastic random noise is modeled via the

nuisance variance parameters σi’s. For instance, they reflect the situation in
actuarial practice, where bigger insurance companies are expected to have
higher variability in the total claim amounts paid. The common factors ’s

introduce dependence among the panels. They can be though of outer drivers
influencing the stochastic panel behavior in the common way. E.g., the
common factors can represent impact of the economic/political/social
situation on the market. On one hand, there are no moment conditions on ’s

whatsoever. On the other hand, if the common factors have finite variance,

then the correlation between panel observations at the same time t, for i ≠ j,
becomes

( , )
( , ) = = .
,
,
2
2
2
2
2
2
2
2
√( + )( + ) √( / + )( / + )

Hence, the sign and the magnitude of the panel factor loadings ζi and ζj affect
the correlation between panels i ≠ j. If there is ζi =0 for some panel i, then the
panel is independent of the remaining ones due to Assumption A 1.

3. Result
Let us consider the model described in (1). For the practical utilization of
the model, we would like to construct a statistical test to decide whether there
is some common changepoint (with the corresponding jumps in the means
located at the changepoint time < ) across the given panels = 1, … , , or
not. The null hypothesis can be formulated as
(2) : =
0
Against a general alternative
(3) : = and ∃ ∈ {1, … , } such that ≠ 0

A graphical illustration of the change point model (1) in panel data under the
alternative, where the means change, can be seen in Figure 1.

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