Asymptotic properties of parameter estimation and change-point detection procedures for a generalized autoregressive process with conditional heteroscedasticity
Problem of change-point detection of the parameters of GARCH(p,q) process is considered. The autoregressive parameters of the process before and after the change point are supposed to be unknown. A sequential procedure for estimating the parameters based on the weighted least squares method is developed. The choice of the weights and the stopping rule allows one to construct an estimator with a preassigned mean square error depending on parameter H of the procedure. The asymptotic properties of the proposed estimator are studied. The asymptotic bound for the mean square error is determined. The procedure of change-point detection is based on comparison of the parameter estimators at different observation intervals. The upper bounds for probability characteristics of the proposed procedure: probabilities of the false alarm and the delay are found. The results of numerical simulation demonstrating the performance of the procedure are reported.