∞ i=0 i = 0, and it is called I (d), d = 1, 2, . . . , if d y t is I (0). To simplify matters, all variables are assumed to be either I (0) or I (1) in the following, if not explicitly stated otherwise.Notice that a K-dimensional vector of time-series variables y t = (y 1t , . . . , y Kt ) is I (d), in short, y t ∼ I (d) if at least one of its components is I (d). In that case, d−1 y t will still have a stochastic trend while d y t does not. This definition does not exclude the possibility that some components of y t may be I (0) individually if y t ∼ I (1). Moreover, it is often convenient to define an I (0) process y t for t ∈ Z rather than t ∈ N in the same way as before, and I will repeatedly make use of this possibility in the following. In fact, I will assume that I (0) processes are defined for t ∈ Z in a stationary context, if not otherwise stated. On the other hand, if I (d) processes y t with d > 0 are involved, it is often easier to define them for t ∈ N, and this is therefore done here.More general definitions of integrated processes can be given. In particular, an I (0) process does not have to be a linear process, and I (d) processes for non-integer d can Kabaila P 1993 On bootstrap predictive inference for autoregressive processes. Journal of Time Series Analysis 14, 473-484. Kemp GCR 1999 The behavior of forecast errors from a nearly integrated AR(1) model as both sample size and forecast horizon become large. Econometric Theory 15, 238-256. Kilian L 1998a Confidence intervals for impulse responses under departures from normality. Econometric Reviews 17, 1-29. Kilian L 1998b Small-sample confidence intervals for impulse response functions. Review of Economics and Statistics 80, 218-230. Kilian L and Chang PL 2000 How accurate are confidence intervals for impulse responses in large VAR models?. Economics Letters 69, 299-307. Kim JH 1999 Asymptotic and bootstrap prediction regions for vector autoregression. International Journal of Forecasting 15, 393-403. King RG, Plosser CI, Stock JH and Watson MW 1991 Stochastic trends and economic fluctuations.