Predictions by non-invertible ARMA models

Abstract

In the time series analysis one of the most used predicting models are of so called auto regressive moving average (ARMA) type. These models are well studied in numerous monographies and research papers. One of the basic assumptions used in the derivation process of the prediction equations is the invertibility of the underlying process. Usually invertibility is assumed as a prerequisite and very little attention is paid to the forecasting of non-invertible processes. Recent papers [1, pp. 227-229.] shows that nowadays more and more researchers consider and examine the case when the underlying process does not satisfy invertibility condition. Non-invertible processes have been studied also quite a long time ago, but they have become the object of interest due to new applications in sciences (signal detection, nancial analysis) also the rapid development of computer sciences and computation possibilities take part in the growing interest of such processes. In basic time series course non-invertible processes usually are discussed very briefly, but globally the interest in such processes is increasing, therefore the aims of this thesis are: 1) to investigate theoretically the questions related to predicting further values of non-invertible ARMA processes; 2) to do the computer simulations and compare di erent methods. To cover these aims both theoretical and simulation studies are provided. Therefore in the beginning we give a very short introduction and necessary background of stationary ARMA processes needed to give the de nition of the non-invertibility of ARMA process. We proceed with another natural assumption used in the derivation process of the prediction equations. The assumption of Gaussian distributed random variables (innovations) gives some prerogatives and simplifies the derivation of the prediction equations also in case of non-invertible process. We briefly discuss the gains which are represented as a useful collection of consecutive theorems that leads to the minimum mean square error predictor in case of non-invertible process with Gaussian distributed data. To extend our studies of non-invertible processes we continue with studies of non-Gaussian, non-invertible process. This situation requires more specific analysis which is provided by a case study of a non-invertible moving average MA(1) process with uniformly distributed innovations (error process). The thesis consists of 3 main sections with suitable subsections. In the first section the basic concept of a non-invertibility is given. The second section is dedicated to the forecasting of an ARMA process. In this section the derivation of prediction equations in case of invertible process is given, then the derivation of the forecast of a non-invertible process with Gaussian distributed data is described and the sections concludes with the derivation of the minimum mean square error predictor in case of the non-invertible process with uniformly distributed innovation series. Results are illustrated with computer simulations and corresponding graphs. In the last section a real world application is considered and corresponding results are given. Since all sections require some computational work and appropriate programming, the collection of suitable codes written in the R language [2] and scripts of the open-source mathematical software system Sage [3], which provides the symbolic calculations needed for this thesis, can be found in the appendix.