1 Test statistics for breaks in volatility dy-
namics
A classical statistical problem is to test the homogeneity of a process or the
parameter constancy of models. There is a substantial literature on this
question known as a change-point problem. The task is to test if a change or
structural break has occurred somewhere in a sample and, if so, to estimate
the time of its occurrence. The simplest form of departure from stationarity is
a change in mean at some (unknown) point in the sample. This problem has
received a great deal of attention, see for instance Csorgo and Horvath (1997)
for a literature review. Financial returns series typically have constant mean,
but exhibit noticeable and complex clustering patterns in volatility (see e.g.
Bollerslev et al. (1994) for a survey of stylized facts). Such processes pose
some non-trivial challenges as detecting a change in variance in an ARCH
model can be rather difficult.2 This section provides a brief discussion of
the Kokoszka and Leipus (2000) as well as the Lavielle and Moulines (2000)
tests for single and multiple breaks as well as the volatility series to which the
tests can be applied to in order to test for change points in the second-order
dynamics of a process.
1.1 CUSUM type tests
Let the asset returns process, rt, be a strongly dependent e.g. /!-mixing
process with finite fourth moment. A large class of ARCH and SV models
are /!-mixing (see, for instance, Carrasco and Chen, 2001) that satisfy these
assumptions.3 Define the process of interest Xt = ∣rt∣5 for <5 = 1,2 which
represents an observed measure of the variability of returns. Given that the
measurable functions of mixing processes are mixing and of the same size (see
White (1984, Theorem 3.49)) then Xt = G(rt,... , rt-τ), for finite τ, is also /!-
mixing. The choice of <5 is of course important. For <5 = 2 we look at squared
returns which is the parent process parametrically modelled in ARCH or
SV-type models. Alternatively, when <5 = 1, we examine absolute returns,
which is considered as another measure of risk, see for instance the Power-
2One could for instance think of extreme cases, where there is no change in the uncon-
ditional moments but only a perturbation in the conditional variance dynamics.
3Examples that form exceptions in this class are Integrated GARCH (IGARCH) or
Fractionally IGARCH (FIGARCH) models which are not covariance stationary.