The Effect of Serial Correlation on the Performance of CUSUM Tests

Abstract

We employ the theory of weak convergence of cumulative sums to the Wiener Process to obtain large sample theory for cusum tests. These results provide at theoretical basis for studying the effects of serial correlation on the performance of the one-sided cusum test proposed by Page (1955). Particular attention is placed on the first, order auto-regressive and first order moving average models. In order to treat the sequential version of the test, we employ the same Wiener process approximation. This enables us to study the effect of correlation not only on the average run length but, more importantly, on the run length distribution itself. These theoretical distributions are shown to compare quite favorably with the true distribution on the basis of a Monte Carlo study using normal observations. The results on the changes in the shape of the run length distributions show that more than average run length should be considered. Our primary conclusion is that the cusum test is not robust with respect, to departures from independence. The use of cusum tests is now widespread and the presence of serial correlation common so that attention should be drawn to the seriousness of this lack of robustness.