Model misspecification in the time series analysis - Davies, Neville (1977)
The Box and Jenkins (1970) methodology of time series model building using an iterative cycle of identification, estimation and diagnostic checking to produce a forecasting mechanism is, by now, well known and widely applied. This thesis is mainly concerned with aspects of the diagnostic checking and forecasting part of their methodology. For diagnostic checking a study is made of the overall or 'portmanteau' statistics suggested by Box and Pierce (1970) and Ljung and Box (1976) with regard to their ability for detecting misspecified models; analytic results are complemented by simulation power studies when the fitted model is known to be misspecified. For forecasting, a general approach is proposed for determining the asymptotic forecasting loss when using any fitted model in the class of structures proposed by Box and Jenkins, when the true process follows any other in that same class. specialisation is made by conducting a thorough study of the asymptotic loss incurred when pure autoregressive models are fitted and used to forecast any other process. In finite samples the Box-Pierce statistic has its mean well below that predicted by asymptotic theory (so that true significance levels will be below that assumed) whilst the Box-Ljung statistic has its mean approximately correct. However, both statistics are shown to be rather weak at detecting misspecified models, with only a few exceptions. Asymptotic forecasting loss is likely to be high when using even high order autoregressive models to predict certain simple processes. This is especially the case when allowance is made for estimation error in the fitted models. Finally, some outstanding problems are outlined. One of these, namely the problem of misspecified error structures in time series regression analysis, is examined in detail.