Detecting Multiple Breaks in Financial Market Volatility Dynamics

ARCH models (Ding et al., 1993). It is worth mentioning that long memory
features have also been established in the absolute returns (e.g. Ding et al.,
1993, Granger and Ding, 1996). Although the tests analyzed here apply to
some long memory volatility models they are beyond the scope of this paper.⁴
Without an explicit functional form for the second conditional moment, the
tests discussed in this section will examine whether there is evidence of struc-
tural breaks in the dynamics of stock returns volatility. If we find a break,
one must conclude that when fitting ARCH or SV-type processes, there will
be instability in their parametric structure. We can take this reasoning a
step further and think of sampling returns intra-daily, denoted r^y_t for some
intra-day frequency i = 1,. . . , m, and form data-driven estimates of daily
volatility by taking sums of squared intra-day returns. This is an exam-
ple of X_t = G(r(₁)_ιt,. . . , Γ(_m)_jt). The high frequency process is /!-mixing, and
so is the daily sampled sum of intra-day squared returns, or various other
empirical measures of quadratic variation. Using the notation of Andreou
and Ghysels (2002) X_t = (QVi)_t which are locally smoothed filters of the
quadratic variation using i days of high-frequency data. The case of QVl
corresponds to the filters studied by Andersen et al. (2001) and Barndorff-
Nielsen and Shephard (2000). The details of the various specifications for
the X_t process will be discussed in the last subsection.

In order to test for breaks in an ARCH(∞) Kokoszka and Leipus (1998,
2000) consider the following process:

(k                 τ   ∖

l∕√r∑V-⅛∕(Γ√Γ)∑V        (1.1)

J=I                J=I   /

where 0 < к < T, X_t = r_t. The returns process {r_t} follows an ARCH(∞)
process, r_t = u_ty∕K_t, h_t = a + ∑2y)ι ⅛^rt-ρ ^a — θ> ðj ≥ θ>J = ɪ, 2, with finite
fourth moment and errors u_t that can be non-Gaussian. The CUSUM type
estimator к of a change point к* is defined as:

к = min{∕c : ∣U_τ(k)∣ = max ∖U_τ(j)∣}                (1∙2)

l≤j≤T

The estimate к is the point at which there is maximal sample evidence for a
break in the squared returns process. In the presence of a single break it is

⁴Results on change-point tests for volatility models with long memory can be found in
Andreou (2002).

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