The Misspecification of Arma Models

The object of this paper is to assess the effects of fitting a model of the wrong order to a time series which is generated by an autoregressive moving–average process. The method is to examine the spectral density functions which are indicated by the probability limits of the least–squares estimators of the misspecified models. The least–squares estimates are asymptotically equivalent to the maximum–likelihood estimates. The experiments reported in this paper suggest that, if the spectral density function of the data–generating process displays prominent modes, then there is a danger of being seriously misled about the nature of the process whenever one fits a model with too few parameters. It also transpires that, in such cases, the criterion function is liable to have several local minima. Autoregressive moving–average models are being used increasingly in object detection applications where the spectrum of a signal serves to identify the nature of its source. Our results suggest that radical misidentifications can result from the use of incorrectly parametrised models.