High Moment Partial Sum Processes of Residuals in ARMA Models and their Applications

In this article, we study high moment partial sum processes based on residuals of a stationary autoregressive moving average (ARMA) model with known or unknown mean parameter. We show that they can be approximated in probability by the analogous processes which are obtained from the i.i.d. errors of the ARMA model. However, if a unknown mean parameter is used, there will be an additional term that depends on model parameters and a mean estimator. When properly normalized, this additional term will vanish. Thus the processes converge weakly to the same Gaussian processes as if the residuals were i.i.d. Applications to change-point problems and goodness-of-fit are considered, in particular, cumulative sum statistics for testing ARMA model structure changes and the Jarque-Bera omnibus statistic for testing normality of the unobservable error distribution of an ARMA model. Copyright 2007 The Author Journal compilation 2007 Blackwell Publishing Ltd.