Ch. 19 Models of Nonstationary Time Series In time series analysis we do not confine ourselves to the analysis of stationary time series. In fact, most of the time series we encounter are nonstationary. How to deal with the nonstationary data and use what we have learned from stationary model are the main subjects of this chapter 1 Integrated Process Consider the following two process oXt-1+t,|o<1: where ut and ut are mutually uncorrelated white noise process with variance o and of, respectively. Both Xt and Yt are Ar(1) process. The difference between two models is that Yt is a special case of a Xt process when =1 and is called a random walk process. It is also refereed to as a ar(1) model with a unit root since the root of the AR(1)process is 1. When we consider the statistical behavior of the two processes by investigating the mean(the first moment), and the variance and autocovariance(the second moment ), they are completely different. Although the two process belong to the same AR(1) class, Xt is a stationary process, while yt is a nonstationary process Assume that tE T*, T*=10, 1, 2,1,I the two stochastic pr rocesses can be expressed ad Similarly. in the unit root case 0 Suppose that the initial observation is zero, Xo=0 and Yo=0. The means of the two (Xt) I This assumption is required to derive the convergence of integrated process to standard Brownian Motion. A standard Brown Motion is defined on t E0, 1.Ch. 19 Models of Nonstationary Time Series In time series analysis we do not confine ourselves to the analysis of stationary time series. In fact, most of the time series we encounter are nonstationary. How to deal with the nonstationary data and use what we have learned from stationary model are the main subjects of this chapter. 1 Integrated Process Consider the following two process Xt = φXt−1 + ut , |φ| < 1; Yt = Yt−1 + vt , where ut and vt are mutually uncorrelated white noise process with variance σ 2 u and σ 2 v , respectively. Both Xt and Yt are AR(1) process. The difference between two models is that Yt is a special case of a Xt process when φ = 1 and is called a random walk process. It is also refereed to as a AR(1) model with a unit root since the root of the AR(1) process is 1. When we consider the statistical behavior of the two processes by investigating the mean (the first moment), and the variance and autocovariance (the second moment), they are completely different. Although the two process belong to the same AR(1) class, Xt is a stationary process, while Yt is a nonstationary process. Assume that t ∈ T ∗ , T ∗ = {0, 1, 2, ...}, 1 the two stochastic processes can be expressed ad Xt = φ tX0 + X t−1 i=0 φ i ut−i . Similarly, in the unit root case Yt = Y0 + X t−1 i=0 vt−i . Suppose that the initial observation is zero, X0 = 0 and Y0 = 0. The means of the two process are E(Xt) = 0 and E(Yt) = 0, 1This assumption is required to derive the convergence of integrated process to standard Brownian Motion. A standard Brown Motion is defined on t ∈ [0, 1]. 1