1.3 Empirical processes
Following the above discussion the CUSUM and least squares tests are ap-
plied to the generic process Xt which represents the squared or absolute re-
turns, both of which are regarded as alternative measures of risk. A departure
from the limited number of applications found in the literature so far is to use
estimates of conditional volatility which are based on high frequency data.
The logic for considering such empirical processes is that squared returns
can be viewed as noisy realizations of the underlying conditional volatility
process (see Andersen and Bollerslev (1998) for a discussion). Hence, instead
of considering the daily return process and square it, we can take advantage
of high frequency intra-daily data to obtain daily estimates of volatility.8 Us-
ing the notation τ∖mγt to represent high frequency data on day t sampled with
frequency m we can study sums of squared returns r^m>j t for different values
of m, to produce the daily volatility measure: (i) σ^vι =
t = 1,...,T, where for the 5-minute sampling frequency the lag length is
m = 288 for financial markets open 24 hours per day (e.g. FX markets)
as in Andersen et al. (2001), Andreou and Ghysels (2002) and Barndorff-
Nielsen and Shephard (2000) or (ii) One-day Historical Quadratic Variation
(introduced in Andreou and Ghysels, 2002) defined as the sum of m rolling
QV estimates: σfz^yι = 1/m ΣjLι QUl(m),t+ι-j∙∕m, t = 1,...,T. The intra-
day volatilities are denoted as QVi, HQVi for window lengths i = 1,2,3.
Clearly, the regularity conditions for squared daily returns can be trans-
planted to these more efficient filtering schemes like QVi (as discussed in
section 1.1).
2 The Monte Carlo Design and Results
The aim of this section is to evaluate the performance of the Kokoszka and
Leipus (1998, 2000) in (1.4) as well as Inclan and Tiao (1994) tests in (1.5)
(also referred to as K&L and I&T tests, respectively) in detecting breaks in
the volatility dynamics of financial asset returns. The observed absolute or
squared returns transformations are the series monitored for single and mul-
tiple breaks. The simulation design examines the size and power properties
8We refrain here from a discussion of the diffusion details of this class of estimators as
well as definitions of quadratic variation. For details we refer the reader to Andersen et
al. (2001), Andreou and Ghysels (2002) and Barndorff-Nielsen and Shephard (2000).