Page 134 - Contributed Paper Session (CPS) - Volume 2

CPS1461 Michal P. et. al
, = () , = () + (, ), < .
The limiting distribution depends on the unknown correlation structure of
the panel changepoint model, which has to be estimated for testing purposes.
The way of its estimation is shown in Maciak et al. (2018). Theorem 3.1 could
be extended for the bootstrap version of the test, where the correlation
structure need not to be known neither estimated. Thus, Theorem 3.1 can also
be viewed as a theoretical mid-step for justification of the bootstrap add-on.
Note, that in case of independent observations within the panel, the
correlation structure and, hence, the covariance matrix Λ, are both simplified
such that () = and (, ) = 0.
We proceed to the assumption that is needed for deriving the asymptotic
behaviour of the proposed test statistics under the alternative.

Assumption A 4
(∑ ) 2

lim =1 = ∞
→∞ ∑ 2
=1
Next, we show how the test statistics behave under the alternative.

Theorem 3.2 (Under alternative). If ≤ − 2, then under Assumptions A 1–
A 4, and alternative ,

p p
(4) R N () → ∞ ← S N ()
→∞ →∞

Assumption A 4 controls the trade-off between the size of breaks and the
variability of errors. It may be considered
as a detectability assumption, because it specifies the value of signal-to-noise
ratio. Assumption A 4 is satisfied, for instance, if 0 < ≤ , ∀ (a common

lower changepoint threshold) and ≤ , ∀ (a common upper variance

threshold). Another suitable example of ’s, for the condition in Assumption

A 4, can be 0 < = −1/2+ for some > 0 and > 0 together with

1 2 2
lim ∑ < ∞. Or, a sequence {∑ ⁄ } equibounded away from
→∞ =1 =1
infinity with = −1 √ may be used as well, where ≥ 0 and > 0. The

assumption ≤ − 2 means that there are at least two observations in the
panel after the changepoint.
Theorem 3.2 says that in presence of a structural change in the panel
means, the test statistics explode above all bounds. Hence, the procedures are
consistent and the asymptotic distributions from Theorem 3.1 can be used to
construct the tests.

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