J. Carlos Escanciano and Ignacio N. Lobato 983An alternative solution recently explored by Escanciano and Lobato (2007) con-
sists of modifying the Box–Pierce statistic using an adaptive Neyman test that takes
the form:
AQn=Q ̃p∗,
where:
̃p=min{m:1≤m≤pn;Lm≥Lh,h=1, 2,...,pn}, (20.4)
and where:
Lp=Qp∗−π(p,n,q).
pnis an upper bound that grows slowly to infinity withn, and:
π(p,n,q)=⎧
⎪⎪
⎪⎪⎪
⎪⎨⎪⎪⎪
⎪⎪⎪
⎩plogn, if max 1 ≤j≤pn∣∣
∣∣
∣∣̂ρj^2
τj∣∣
∣∣
∣∣≤√
qlogn2 p, if max 1 ≤j≤pn∣∣
∣∣
∣∣̂ρ^2 j
τj∣∣
∣∣
∣∣>√
qlogn,whereqis some fixed positive number. We denote this automatic portmanteau test
AQn. Our choice ofqis 2.4 and is motivated by an extensive simulation study in
Inglot and Ledwina (2006) and from simulations in Escanciano and Lobato (2007).
Small values ofqresult in the use of Akaike’s criterion, while largeq′slead to choos-
ing Schwarz’s criterion. Moderate values, such as 2.4, provide a “switching effect”
which combines the advantages of the two selection rules: when the alternative is
of high frequency (that is, when the only significant autocorrelations are at large
lagsj), Akaike is used whereas, if the alternative is of low-frequency (i.e., if the first
autocorrelations are different from zero), Schwarz is chosen. The adaptive test is
an improvement with respect to the traditional Box–Pierce and Hong approaches
because theAQntest is more powerful and less sensitive to the selection of the
bandwidth numberpnthan these approaches and, more importantly, it avoids the
estimation of the complicated variance-covariance matrixTsince its asymptotic
distribution isχ 12 for general m.d.s. processes.
Summarizing, testing the MDH using linear measures of dependence presents
two challenging features. The first aspect is that the null hypothesis implies that
an infinite number of autocorrelations are zero. This feature has been addressed
successfully in the frequency domain under severe restrictions on the dependence
structure of the process. The second feature is that the null hypothesis allows the
time series to present some form of dependence beyond the second moments.
This dependence entails that the asymptotic null covariance matrix of the sample
autocorrelations is not diagonal, so that it hasn^2 non-zero terms (contrary to
Durlauf, 1991, and Deo, 2000, who consider a diagonal matrix, which has onlyn
non-zero elements). This aspect has been handled by introducing some arbitrary
user-chosen numbers whose selection complicates statistical inference.
However, all these tests are suitable for testing for lack of serial correlation but
not necessarily for the MDH and, in fact, they are not consistent against non-
martingale difference sequences with zero autocorrelations. This happens when