Introduction
On this twentieth anniversary of Rob Engle’s seminal paper on ARCH it is
worth reflecting on some of the outstanding questions in the literature. It has
long been conjectured that stock market volatility exhibit occasional breaks.
Diebold (1986), Hendry (1986) and Lamoureux and Lastrapes (1990) were
among the first to suggest that persistence in volatility may be overstated
with the presence of structural breaks. More recent and related evidence is
provided by Diebold and Inoue (2001), Granger and Hyung (1999), Mikosch
and Starica (1999), among others, which shows that the presence of breaks
may also explain the findings of long memory, particularly in volatility.
There is a substantial literature on testing for the presence of breaks in
linearly dependent stochastic processes (see for instance Bai (1994, 1997),
Bai and Perron (1998) interalia). There is a temptation to apply the tests
for ARMA-type processes in the context of ARCH or Stochastic Volatil-
ity (SV) models. For instance, one could view squared returns as an ARMA
process and proceed with the application of tests suggested for testing breaks
in the mean. Unfortunately, things are not so simple. The resemblance be-
tween ARMA and GARCH or discrete time SV models is deceiving (see e.g.
Francq and Zakoian (2000a,b)). It took many years of research after the orig-
inal work of Engle (1982) to clarify the asymptotics of GARCH(I5I) processes
(see, for instance, Lee and Hansen (1984) and Lumsdaine (1996)) and the
asymptotics of more general univariate and multivariate GARCH processes
(see Ling and McAleer (2002 a,b)). Recently, Carrasco and Chen (2001)
present a comprehensive study which shows that most univariate GARCH
processes are /!-mixing. This result precludes the application of many afore-
mentioned tests for structural breaks that require a much stronger mixing
condition.1
The purpose of this paper is to explore recent advances in the theory of
change-point estimation for strongly dependent processes including ARCH
and SV models. Some early attempts to test for a break in a GARCH are
found in Chu (1995) and Lundberg and Terasvirta (1998). A number of
recent papers have shown the consistency of CUSUM type change-point es-
timators for a single break and least squares tests for multiple breaks. The
tests are not model-specific and apply to a large class of (strongly) depen-
1 Most tests proposed for linear processes impose (/-mixing or strong mixing conditions
which are not satisfied by ARCH processes. For a general treatment of estimating the
weak GARCH models, see Francq and Zakoian (2000b).