CHAPTER 12 TIME SERIES ANALYSIS Chapter 12 Time Series Analysis 12.1 Stochastic processes A stochastic process is a family of random variables Xt, tETy Example 1 [S, t=0, 1, 2, . where St= 2i=o Xi and X; N iid (0, 0). S has a different distribution at each point t 12.2 Stationarity and strict stationarity If Xt, t eT is a stochastic process such that Var(X,)<o for each E T, the autocor- variance funct ion y of xt is defined by (r, s)=Cov(Xr, Xs )=E(Xr- EXr)(Xs-EXs Because Var(Xt)<oo for eachtE T 2(,)≤[E(x-Ex]1E(x,-EX)2]12 by the Cauchy-Schwarz inequality The autocorrelation function P(r, s)is defined by (,s)=-2a(,s) √x(r,n)n(8,s) Example 2 Let Xt=e+Bet-1, et N iid(0, 02) + 72(t+h, t)=Cou(Xt+h,X)=8o h=±1 1 h=0 Pr(t+h, t) The time series{X,t∈ z with index set Z={0,±1,±2,…} is said to be( weakly) stationary, if 1.E|X<∞ for all t∈z 2.EXt= m for all t∈z 3.Y(r, s)=%(r+t, s+t) for all T, s, tE ZCHAPTER 12 TIME SERIES ANALYSIS 1 Chapter 12 Time Series Analysis 12.1 Stochastic processes A stochastic process is a family of random variables {Xt ,t ∈ T} . Example 1 {St , t = 0, 1, 2, · · · } where St =
t i=0 Xi and Xi ∼ iid (0, σ2 ). St has a different distribution at each point t. 12.2 Stationarity and strict strationarity If {Xt , t ∈ T} is a stochastic process such that V ar (Xt) < ∞ for each t ∈ T, the autoco￾variance function γx (·, ·) of {Xt} is defined by γx (r, s) = Cov (Xr, Xs) = E (Xr − EXr) (Xs − EXs). Because V ar (Xt) < ∞ for each t ∈ T, γx (r, s) ≤ E (Xr − EXr) 2 1/2 E (Xs − EXs) 2 1/2 < ∞ by the Cauchy—Schwarz inequality. The autocorrelation function ρx (r, s) is defined by ρx (r, s) = γx (r, s)  γx (r, r) γx (s, s) Example 2 Let Xt = et + θet−1, et ∼ iid (0, σ2 ). γx (t + h, t) = Cov (Xt+h, Xt)    =  1 + θ 2 σ 2 , h = 0 = θσ2 , h = ±1 = 0, |h| > 1 ρx (t + h, t)    = 1, h = 0 = θ (1+θ 2 ) , h = ±1 = 0, |h| > 1 The time series {Xt , t ∈ Z} with index set Z = {0, ±1, ±2, · · · } is said to be (weakly) stationary, if 1. E |X2 t | < ∞ for all t ∈ Z 2. EXt = m for all t ∈ Z 3. γx (r, s) = γx (r + t, s + t) for all r, s, t ∈ Z