Test for parameter change in stochastic processes based on conditional least-squares estimator

In this paper, we consider the problem of testing for a parameter change in stochastic processes. In performing a test, we employ the cusum test considered in Lee et al. (Scand. J. Statist. 30 (2003) 651). The cusum test is based on the conditional least-squares estimator introduced by Klimko and Nelson (Ann. Statist. 6 (1978) 629). Special attention is paid to the nonlinear autoregressive processes including TAR and ARCH processes. It is shown that under regularity conditions, the test statistic behaves asymptotically the same as the sup of the squares of independent standard Brownian bridges. Simulation results as to ARCH(1) processes and an example of real data analysis are provided for illustration.