Role of Structural Coefficients in the Study of the Influence of an Additive Outlier on the Residuals in ARMA (1,1), AR(I), and MA(I) Models

Abstract

Motivation for this paper comes from our perception that the Simulation study on nature of Additive Outlier (AO) in ARMA(1,1) model by Zaharim et al. (2009) ought to be more extensive for a fuller understanding of the influence of the Outlier. Stated briefly, Zaharim et al. (2009) based on Simulations of ARMA(1,1) Time Series model with one AO, have concluded that the AO affects the observations only at its time of occurrence, and affects, noticeably, the residuals at the time instance of the AO and the one immediately next to it. In this paper, we have demonstrated, both analytically and through Simulation, that in the case of ARMA(1,1) with an AO one or more residuals can get affected depending on the values of the model parameters. The number of residuals that get noticeably affected depends on both the magnitude of AO and also the values of the parameters of the underlying model. In order to gain a deeper insight into this behavior of residuals, we narrow down the study to AR(1) and MA(1). This type of investigation reveals that in the case of AR(1), an AO affects the residuals at its time point of occurrence, and at the subsequent point. In contrast, in MA(1) model, all the residuals at and after the incidence of AO can get affected (as in the case of ARMA(1,1) model), emphasizing the crucial role of moving average parameter. These deeper implications have escaped the attention of the studies made by Zaharim et al. (2009).