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.