Chapter2_LN

2 2 = [−θ1 + (−θ1 )(−θ2 )]σa = (−θ1 + θ1 θ2 )σa ,
– The auto-covariance function at lag 2 is γ2 = cov(Zt , Zt−2 ) = cov(at − θ1 at−1 − θ2 at−2 , at−2 − θ1 at−3 − θ2 at−4 )
– ρ1 (θ) = ρ1 (1/θ). • Example 3: Consider the MA(2) process, Zt = θ0 + at − θ1 at−1 − θ2 at−2 . – Clearly, µ = E(Zt ) = θ0 . – The variance is then γ0 = var(Zt ) = var(at − θ1 at−1 − θ2 at−2 )
∞ ∑ j =0 2 σa ∞ ∑ j =0 2 ψj < ∞,
γk = cov(Zt , Zt−k ) = and cov(Zt , Zt−k ) ρk = = var(Zt ) Exercise 2. Let
2 σaБайду номын сангаас
ψj ψj +k ,
k ≥ 0,
∑∞ j =0 ψj ψj +k ∑ , ∞ 2 j =0 ψj
• Definition: A linear, causal and stationary time series is also called a general linear process. – A time series is also called a stochastic process or simply a process since all time series form a special class of stochastic processes. – The general linear process has the form of Zt =
– Note that the MA process is a special general linear process. (why?) Hence, it is linear, causal and stationary. • Example 2: Consider the MA(1) process, Zt = θ0 + at − θat−1 .
k > 0.
Zt = at + ϕat−1 + ϕ2 at−2 + ϕ3 at−3 + · · · ,
2 where |ϕ| < 1, and {at } ∼ W N (0, σa ).
– Is it a general linear process? – How about its probabilistic properties, i.e. (1) mean, (2) variance and (3) autocorrelation?
+∞ ∑ j =−∞ 2 ψj var(at−j ) ∞ ∑ j =−∞
=
2 σa
2 ψj < ∞.
3. For the auto-covariance function,
2 cov(Zt , Zt−k ) = σa +∞ ∑ j =−∞
ψj ψj +k ,
k ≥ 0.
. Hence, {Zt } is stationary since the mean function and the auto-covariance function are both independent of t. Exercise 1. Show that time series {Zt } is also strictly stationary. • Definition: A time series {Zt } is linear if the value of Zt is a linear function of a white noise sequence. (Is the time series in Example 1 linear?) • Definition: A time series {Zt } is causal if the value of Zt is affected only by the information up to now, i.e. Zt is independent of the future information as with s > t. – Time series models are usually driven by the white noises, say {at }, i.e. they are the functions of {at } such as {Zt } in Example 1. The random variable at can be interpreted as the information at time t, and the white noise is also called ”innovation sequence” or ”information sequence”. 1
+∞ ∑ j =0
ψj at−j = ψ0 at + ψ1 at−1 + ψ2 at−2 + · · · ,
∑∞ 2 2 2 where {at } ∼ W N (0, σa ) and σa j =0 ψj < ∞. Without loss of generality, we can set ψ0 = 1. (why?) – The mean, auto-covariance and autocorrelation of a general linear process: µ = E(Zt ) = 0, γ0 = var(Zt ) =
THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT3301/4601/6002 Time Series Analysis
Chapter 2 Stationary linear time series models
2 ). where q ≥ 0 is an integer and {at } ∼ W N (0, σa
– A moving average process of order 0 is actually a white noise sequence if θ0 = 0. – The MA process is also called the MA model. – The MA model was first proposed by E. Slutsky in 1927 to explain some phenomena in economic data such as period behaviors etc. 2
1
General linear processes
∑+ ∞ • Example 1: Suppose that Zt = j =−∞ ψj at−j for all t ∈ Z, where {at } ∼ ∑+ ∞ 2 2 2 W N (0, σa ) and σa j =−∞ ψj < ∞. (Is {Zt } stationary?) ∑ ∞ 1. For the mean function, E(Zt ) = + j =−∞ ψj E(at−j ) = 0. 2. For the variance function, var(Zt ) =
−0.4
−0.2
0.0
ρ1
0.2
0.4
−4
−2
0 θ
2
4
Figure 1: The plot of ρ1 against the value of θ. Exercise 3. For a MA(1) process, Zt = at − θat−1 , show that – maxθ ρ1 (θ) = 0.5 and minθ ρ1 (θ) = −0.5; 3
2 = cov(−θ2 at−2 , at−2 ) = −θ2 σa .
– Moreover, as before, γk = 0, ∀ k ≥ 3.
– Thus, the auto-correlation function for the MA(2) process is −θ1 + θ1 θ2 ρ1 = , 2 2 1 + θ1 + θ2 −θ2 ρ2 = , 2 2 1 + θ1 + θ2 ρ = 0, ∀ k ≥ 3. k • Consider a general MA(q ) process, Zt = θ0 + at − θ1 at−1 − θ2 at−2 − · · · − θq at−q . – The mean is µ = θ0 and the variance is
2 , = cov(−θat−1 , at−1 ) = −θσa
and then the auto-correlation function at lag 1 is ρ1 = −θ . 1 + θ2
– The auto-covariance function at lag 2 is γ2 = cov(Zt , Zt−2 ) = cov(at − θat−1 , at−2 − θat−3 ) = 0, since there are no a’s with subscripts in common between Zt and Zt−2 . Then, ρ2 = 0. – Similarly, γk = cov(Zt , Zt−k ) = 0, and ρk = 0, whenever k ≥ 2; that is, the process has no correlation beyond lag 1. This fact will be important later when we need to choose suitable models for real data (model specification). – We can summarize these properties as follows, E (Zt ) = θ0 , 2 γ0 = var(Zt ) = σa (1 + θ2 ), −θ ρ1 = , 1 + θ2 ρ =0 , ∀ k ≥ 2. k