  This paper presents a bootstrapped p-value white noise test based on the
maximum correlation, for a time series that may be weakly dependent under the
null hypothesis. The time series may be prefiltered residuals. The test
statistic is a normalized weighted maximum sample correlation, where the
maximum lag increases at a rate slower than the sample size. We only require
uncorrelatedness under the null hypothesis, along with a moment contraction
dependence property that includes mixing and non-mixing sequences. We show
Shao's (2011) dependent wild bootstrap is valid for a much larger class of
processes than originally considered. It is also valid for residuals from a
general class of parametric models as long as the bootstrap is applied to a
first order expansion of the sample correlation. We prove the bootstrap
validity without exploiting extreme value theory (standard in the literature)
or recent Gaussian approximation theory. Finally, we extend Escanciano and
Lobato's (2009) automatic maximum lag selection to our setting with an
unbounded choice set, and find it works strikingly well in controlled
experiments. Our proposed test achieves accurate size under various white noise
null hypotheses and high power under various alternative hypotheses including
distant serial dependence.
