A Max-Correlation White Noise Test for Weakly Dependent Time Series

Abstract

This paper presents bootstrapped p-value white noise tests based on the max-correlation, for a time series that may be weakly dependent under the null hypothesis. The time series may be prefiltered residuals. Our test statistic is a scaled maximum sample correlation coefficient 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, and exploit two wild bootstrap methods for p-value computation. We operate either on a first order expansion of the sample correlation, or Delgado and Velasco's (2011) orthogonalized correlation for fixed lag length, both to control for the impact of residual estimation. A numerical study shows the first order expansion is superior, especially for large lag length. When the filter involves a GARCH model then the orthogonalization breaks down, while the first order expansion works quite well. We show Shao's (2011) dependent wild bootstrap is valid for a much larger class of processes than originally considered. Since only the most relevant sample serial correlation is exploited amongst a set of sample correlations that are consistent asymptotically, empirical size tends to be sharp and power is comparatively large for many time series processes. The test has non-trivial local power against local alternatives, and can detect very weak and distant serial dependence better than a variety of other tests. Finally, we prove that our bootstrapped p-value leads to a valid test without exploiting extreme value theoretic arguments, the standard in the literature.