# What is an I(k) random walk process?

Local linear trend - I(2) process: An extension of the random walk trend is obtained by including a stochastic drift component µt+1 = µt + βt + ηt, βt+1 = βt + ζt, ζt ∼ NID(0,σ2 ζ), (3) where the disturbance series ηt is as in (2). The initial values µ1 and β1 are treated as unknown coeﬃcients. Harvey (1989, §2.3.6) deﬁnes the local linear trend model as yt = µt + εt with µt given by (3). In case σ2 ζ = 0, the trend (3) reduces to an I(1) process given by µt+1 = µt+β1+ηt where the drift β1 is ﬁxed. This speciﬁcation is referred to as a random walk plus drift process. If in addition σ2 η = 0, the trend reduces to the deterministic linear trend µt+1 = µ1 +β1t. When σ2 η = 0 and σ2 ζ > 0, the trend µt in (3) remains an I(2) process and is known as the integrated random walk process which can be visualised as a smooth trend function.