proved that À: is a consistent estimator of the unknown change-point k* with
F{∣fc* — k∖ > ε} ≤ C,∕(<5ε2λ∕n), where C is some positive constant and <5
depends on the ARCH parameters and ∖k* — k∖ = Op(l∕n) (Kokoszka and
Leipus, 1998, 2000). Under the null hypothesis of no break:
Uτ(k) →p[o,i] σ^B(k) (1.3)
where B(∕c) is a Brownian bridge and σ2 = ∑2 J*' ~ -x Cov(Xj, Xo)∙ Conse-
quently, using an estimator σ, one can establish that under the null:
sup{∣Uτ(fc)∣}∕σ →p[0,i] sup{B(fc) : ke[O,1]} (1.4)
which establishes a Kolmogorov-Smirnov type asymptotic distribution.5
The computation of the Kokoszka and Leipus (1998, 2000) test (hence-
forth K&L test) is relatively straightforward, with the exception of σ appear-
ing in (1.4). The authors suggest to use a Heteroskedasticity and Autocor-
relation Consistent (НАС) estimator applied to the Xj process. There are a
number of such estimators, depending on the kernel function one uses. Ex-
amples of kernels which have been used by econometricians include: Hansen
(1982) and White (1984) use the truncated kernel; the Newey and West
(1987) estimator uses the Bartlett kernel; and the estimator of Gallant (1987)
uses the Parzen kernel and that of Andrews (1991) uses the Quadratic Spec-
tral (QS) kernel. We have experimented with a number of estimators in
addition to the procedure of den Haan and Levin (1997) who propose a HAC
estimator without any kernel estimation, which is called the Vector Autore-
gression Heteroskedasticity and Autocorrelation Consistent (VARHAC) esti-
mator. This estimator has an advantage over any estimator which involves
kernel estimation in that the circular problem associated with estimating the
optimal bandwidth parameter can be avoided. This estimator involves fit-
ting a parametric autoregressive model and choosing the order of AR using
for instance the AIC. The Monte Carlo evidence reported in den Haan and
Levin (1997) indicates that the VARHAC estimator performs better than the
nonprewhitened and prewhitened kernel estimators in many cases. Although
we have not done a systematic study of various kernel HAC estimators versus
the VARHAC estimator, we found via simulations that the latter is reliable.
5Critical values can be found in most textbooks on nonparametric methods. The 90
%, 95 % and 99 % percentile (two-sided test) critical values are, respectively: 1.22, 1.36
and 1.63.