J. Carlos Escanciano and Ignacio N. Lobato 979Qpstatistic by introducing a consistent estimator of the asymptotic null covariance
matrix of the sample autocorrelations,̂T, so that the modifiedQpstatistic retains
theχp^2 asymptotic null distribution. Lobato, Nankervis and Savin (2002) define this
statistic as ̃Qp=
(√
n̂ρ)′̂
T−^1(√
n̂ρ). The main drawback of this approach is that,
in order to construct̂T, a bandwidth parameter has to be introduced (seeibid.,
for details). This approach works for general dependence structures that allow for
the asymptotic covariance matrix of the firstpautocorrelations to take any form.
The second modification has been studied by Horowitzet al.(2006), who employ
a bootstrap procedure to estimate consistently the asymptotic null distribution of
theQptest for the general case. They compare two bootstrap approaches, a sin-
gle and a double blocks-of-blocks bootstrap, and their final recommendation is
to employ a double blocks-of-blocks bootstrap after prewhitening the time series.
This solution presents a similar problem, though, namely that the researcher has
to choose arbitrarily a block length number. The previous papers considered raw
data, but Francq, Roy and Zakoïan (2005) have addressed the use of theQpstatistic
with residuals. They propose to estimate the asymptotic null distribution of the
Qptest statistic for the general weak dependent case. However, their approach still
requires the selection ofp, and of several additional arbitrary numbers necessary
to estimate consistently the needed asymptotic critical values.
These references represent an effort to address the problem of testing for
m.d.s. using standard linear measures (autocorrelations) but allowing for nonlin-
ear dependence. Lobato (2001) represents an alternative approach with a similar
spirit. The target is to avoid the problem of introducing a user-chosen number and
the idea is to construct an asymptotically distribution free statistic. Although this
approach delivers tests that handle nonlinear dependence and control properly the
Type I error in finite samples, its main theoretical drawback is its inefficiency in
terms of local power.
A related statistic, which has been commonly employed in the empirical finance
literature (see Cochrane, 1988; Lo and MacKinlay, 1989), is the variance ratio,
which takes the form:
VRp= 1 + 2p∑− 1j= 1( 1 −j
p
)̂ρj.Under independence,
√
np(VRp− 1 )is asymptotically distributed asN(0, 2(p− 1 )).
Although this test can also be robustified and can be powerful on some occasions,
it presents the serious theoretical limitation of being inconsistent. For instance,
González and Lobato (2003) considered a moving average of order 2 (MA(2)) pro-
cessyt=et−0.4597et− 1 +0.10124et− 2. For this processVR 3 =0, in spite of the
first two autocorrelations being non-zero. The problem with variance ratio statistics
resides in the possible existence of compensations between autocorrelations with
different signs, and this may affect power severely. Related to variance ratio (VR)
tests, Nankervis and Savin (2007) have proposed a robustified version of Andrews
and Ploberger’s (1996) test that appears to have very good finite sample power with
common empirical finance models. In related work, Delgado and Velasco (2007)