The multiple testing problem for Box-Pierce statistics

Abstract

We derive the exact joint asymptotic distribution for multiple Box-Pierce statistics, and use these results to determine appropriate critical values in joint testing problems of time series goodness-of-fit. By sequentially testing at various lags, we can identify specific problems with a model, and identify superior models. A novel $\alpha$-rationing scheme, motivated by the sequence of conditional probabilities for the statistical tests, is developed and implemented. This method can be used to produce critical values and p-values both for each step of the sequential testing procedure, and for the procedure as a whole. Efficient computational algorithms are discussed. Simulation studies assess the impact of finite samples on the real Type I error. It is also demonstrated empirically that the conventional $\chi^{2}$ critical values for the Box-Pierce statistics are too small, with a Type I error rate greater than the nominal; the new method does not suffer from this defect, and allows for more rigorous model-building. We illustrate on several time series how model defects can be identified and ameliorated.