Integrated Time Series

Abstract

Earlier in Chapter 3 we touched on random trends and described two aggregation schemes to decompose series into trends and other, weakly stationary, components. Series with random walk components are called integrated (of order one) because they are the sums (integrals) of weakly-stationary components. When some linear combinations of components of an integrated vector-valued series become weakly stationary rather than being integrated, we say that these components are cointegrated. This notion is proposed by Granger—see Granger [1981], Granger and Weiss [1983], and Engle and Granger [1987], for example. In state space modeling of cointegrated series, some components of time series at different time points may enter as state variables. We therefore interpret the notion of cointegration broadly and allow for the possibility of linear combination of not-necessarily-contemporaneous components of time series being weakly stationary, as in Aoki [1990], for example. We devote this chapter to modeling of integrated time series and related series such as cointegrated or nearly integrated series.