Partial sums of lagged cross-products of AR residuals and a test for white noise

Partial sums of lagged cross-products of AR residuals are defined. By studying the sample paths of these statistics, changes in residual dependence can be detected that might be missed by statistics using only the total sum of cross-products. Also, a test statistic for white noise is proposed. It is shown that the limiting distribution of the test statistic converges weakly to a vector Brownian motion with independent elements under the null hypothesis of no residual autocorrelation. An indication of the circumstances under which the asymptotic results apply in finite-sample situations is obtained through a simulation study. Some considerations are given to the empirical size and power of the test statistic vis-à-vis the Ljung-Box (Biometrika 65:297-303, 1978) portmanteau statistic, and a diagnostic test statistic proposed by Peña and Rodriguez (J. Am. Stat. Assoc. 97:601-610, 2002). An empirical example illustrates the importance of examining partial sums of time series residuals when inadequacies in model fit are anticipated due to a change in autocorrelation structure