Chapter 19_Models of Nonstationary Time Series.pdf

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
X = φX 1 + u , φ< 1;
t t− t | |
Yt = Yt−1 + vt,
where ut and vt are mutually uncorrelated white noise process with variance
2 2
σu and σv , 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 ∗, ∗= 0, 1, 2, ... , 1 the two stochastic processes can be
∈ T T { }
expressed ad
t−1
t i
Xt = φ X0 + φ ut−i.
i=0
X
Similarly, in the unit root case
t−1
Yt = Y0 + vt−i.
i=0
X
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
and variances are
1
t− 1
V ar(X ) = φ2iV ar(u ) σ2
t t−i −→ 1 φ2 u
i=0
X −
and
t−1
V ar(Y ) = V ar(v ) = t σ2.
t t−i · v
i=0
X
The autocovariance of the two series are
t−1 t−τ−1
X i i
γτ= E(XtXt−τ) = E φ ut−i φ ut−τ−i
" i=0 ! i=0 !#
1 X τ Xt−1 1 t−τ−1
= E[(ut + φ ut−1 + ... + φ ut−τ+ ... + φ u1)(ut−τ+ φ ut−τ−1 + ... +