Location and scale-based CUSUM test with application to autoregressive models

This study aims to refine the residual cumulative sum (CUSUM) test for location-scale time series and to develop the location and scale-based CUSUM (LSCUSUM) test. This test clarifies the role of residuals in the CUSUM test well. In addition, it provides a much more convenient method for analysing time series models with more complicated structure and copious parameters when compared with the conventional estimate- and score vector-based CUSUM tests. The test comprises the two CUSUM test statistics featuring the location and scale parts, which are asymptotically independent and converge to the supremum of a Brownian bridge. The LSCUSUM test is applied to the vector AR, scalar ARMA, and infinite order AR models to demonstrate the merit of the method. In particular, special attention is paid to the vector AR model owing to its importance in applications. We show its validity through Monte Carlo simulations and conduct real data analysis for illustration.