Page 133 - Contributed Paper Session (CPS) - Volume 2

CPS1461 Michal P. et. al
An alternative way for testing the change in panel means could be a usage
of CUSUM type statistics. For example, a maximum or minimum of a sum (not
a ratio) of properly standardized or modified sums from our test statistics R N
() or S N (). The theory, which follows, can be appropriately rewritten for
such cases.
Prior to deriving asymptotic properties of the test statistics, we provide
assumptions on the relationship between the heterogeneous volatility and the
mutual dependence of the panels.
Assumption A 2. For some > 0,
(∑ 2+ ) 2
lim =1 = 0
→∞ 2 2+
(∑ =1 )

and | | 2+ < ∞, for ∈ {1, … , }.
1,

Assumption A 3.

(∑ ) 2

lim =1 = 0
2
→∞ (∑ )
=1

If there exist constants , > 0, not depending on , such that

≤ ≤ , = 1 … ;

then the first part of Assumption A 2 is satisfied. Additionally, suppose that,
e.g.,| | ≤ −1/2− for all ’s and some, > 0, then Assumption A 3 holds

as well.
Now, we derive the behavior of the test statistics under the null hypothesis.
Theorem 3.1 (Under null). Under Assumptions A 1 – A 3 and hypothesis H0,

D | − |
R N () → max −
→∞ =1,…,−1 max | − |+ max | − |
=1,…, =,…−1 −

and

D
S N () →
→∞
−1 2
( − )

∑ 2 2

=1 ∑ ( − ) + ∑ −1 ( − − )
=1 = −

where : − and [ , … , ] is a multivariate normal random vector with
⊺

1
zero mean and covariance matrix Λ = { } , such that
, ,=1
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