A Novel Correction for the Multivariate Ljung-Box Test

Abstract

We propose a new analytical improvement to the multivariate Ljung-Box test that addresses the inherent pronounced deviations of the original test from nominal type I error rates under almost all scenarios. Prior attempts to mitigate this issue have been directed at modification of the test statistics or correction of the test distribution to achieve precise results in finite samples. In previous studies, focused on designing corrections to the univariate Ljung-Box, a method that specifically adjusts the test rejection region has been the most successful of attaining the best type I error rates. We adopt the same approach for the more complex, multidimensional time series scenarios. We use large sample simulation data (multivariate normal) for a range of sample sizes (n = 40 to 300), lags (m = 2 to n − 1), and multivariate time series dimension (k = 2 to 10) to obtain an empirical estimation of the correct rejection regions for the particular combination of these variables. Furthermore, we use regression modeling with interactions and covariate power combinations to parametrically extend these precise rejection regions to all combinations of sample sizes, lags, and number of time series. Thus, we design and implement the correction based on regression models fit to empirically estimate rejection thresholds. Our results are validated to independent validation simulations and show that we attain almost perfect type I error rates with mean absolute deviation of 0.0011 across all scenarios compare to 0.0276 for the classical Ljung-Box test, which represents a 96% improvement. These findings will enhance the goodness-of-fit diagnostics for multivariate time series.