ARIMA Time Series Models

Abstract

The autoregressive moving average (ARMA) model defined as $X_t = \nu + \alpha _1 X_{t - 1} + \ldots + \alpha _p X_{t - p} + \beta _1 \varepsilon _{t - 1} + \ldots + \beta _q \varepsilon _{t - q} + \varepsilon _t$ deals with linear time series. That means, the time series should be covariance stationary processes, see Franke et al. (2008). The model consists of two parts, an autoregressive (AR) part of order p and a moving average (MA) part of order q. When an ARMA model is not stationary, the methods of analyzing stationary time series cannot be used directly. In order to handle those processes within the framework of the classical time series analysis, we must first form the differences to get a stationary process. The autoregressive integrated moving average (ARIMA) models are an extention of ARMA processes by the integrated (I) part. Sometimes ARIMA models are refered to as ARIMA(p; d; q) whereas p and q denote the order of an autoregressive (AR) respective a moving average (MA) part and d describes the integrated (I) part.