| The autoregressive moving average (ARMA) model has been instrumental in the analysis of linear time series and has applications in many areas of scientific interest. In order to perform model estimation and produce reliable forecasts, practitioners require the model structure and specifications to remain constant over time. However in reality, assumptions of model stationarity are often violated for observed time series data. This motivates the use of statistical testing procedures developed for detection of structural breaks. Several new on-line procedures are proposed, which sequentially test for model stability as new data become available. The proposed methodology does not make the unrealistic assumption of known initial parameters which is commonly seen in the sequential literature. Rather, the approach incorporates an estimation step based on non-contaminated training data prior to the start of monitoring. Both asymptotic and finite sample properties are addressed through theory, simulations and applications. In particular, the methodology is demonstrated through a detailed analysis of pollution data. Additionally, the limit distributions of stopping times are derived for two residual-based cumulative sum (CUSUM) procedures in the open-ended monitoring setting. Special consideration is given to breaks in level and scale. Results indicate the precise form of the asymptotic depends both on the location of the break point and the size of the change implied by the drift. The theoretical results are accompanied by a simulation study and applications. |
