Non-stationary Time Series: Differencing and ARIMA Modelling

Abstract

Non-stationarity 3.1 The class of ARMA models developed in the previous chapter relies on the assumption that the underlying process is weakly stationary, thus implying that the mean, variance and autocovari-ances of the process are invariant under time shifts. As we have seen, this restricts the mean and variance to be constant and requires the autocovariances to depend only on the time lag. Many economic and financial time series, however, are certainly not stationary and, in particular, have a tendency to exhibit time-changing means and/ or variances. 3.2 In order to deal with such non-stationarity, we begin by assuming that a time series can be decomposed into a non-constant mean level plus a random error component: t t t x = +ε t t + t (3.1) A non-constant mean level μ t in (3.1) can be modelled in a variety of t ways. One potentially realistic possibility is that the mean evolves as a (non-stochastic) polynomial of order d in time. This will arise if d x t can be decomposed into a trend component, given by the polynomial , and a stochastic, stationary, but possibly autocorrelated, zero mean error component, which is always possible given Cramer's T. C. Mills, Time Series Econometrics