January 21.docx

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Department
Statistical Sciences
Course
STA457H1
Professor
Zhou Zhou
Semester
Winter

Description
January 21, 2014 {X}tis weakly stationary µ = E[X] γ (h) = Cov[X, X ] X t t+h correlation function: ρ (h) = CXrr[X, X ] = Cov[X,tX ]t+har(X)Var(X ) =tγ(ht+hγ(0)γ(0) = t t+h γ(h)/γ(0) Property: 1. |γ(h)| ≤ γ(0) for any h 2. γ(h) = γ(-h) for any h Proof: Cov[X, X ] t γ(ht+h γ(-h) = Cov[X, X ] t t-h Ex. White noise: WN(0, σ ) 2 {X}tis called a white noise if E[X] = 0 for any i i Cov[X, X]i= 0jfor any i≠j 2 Var[X] =iσ < ∞ White noise sequence is the building block for many more complicated time series models. Inputs -> filter -> outputs White noise -> linear time series model Ex. Moving average process of order 1: MA(1) model 2 Suppose Z , Z , Z0, …,1Z is 2 WN(0, σ n Let X = i + aZ i i-1 X is called a MA(1) process. E[X] =iE[Z i + aE[Z ] = i-1 2 Cov[X, X ]i= Ci+hZ+aZ , Z +aZ i i-1 i+h i+h-1 = Cov[Z, Z ] i aCi+hZ, Z i i+h-1] + aCov[Z , Z ]i-1a Coi+h i- 1, Z i+h-1] = A+B+C+D 1. If h=0, A=σ , B=0, C=0, D=σ => Cov[X, X] = σ + a σ = σ (a +1) 2 2 2 2 2 i i 2. If h=1, A=0, B=σ , C=0, D=0 => Cov[X, X ] = aσ i i+1 2 3. If h>1, A=0, B=0, C=0, D=0 => Cov[X, X ] = 0 i i+h To summarize, X is a statiInary process with covariance function γ(h) = {σ (a +1) if h=0 aσ if h=1 0 if h>1 Ex. Auto Regressive Process of order 1: AR(1) Let {Z} bi a WN(0, σ ), iϵZ 2 Let X = ij:0-∞ a Z j i-j X i AR(1) Need |a|<1 Observation: X = aX + Z i i-1 I Proof: RHS = a ∑j:0-∞ a Z j i-1-j Z =i∑j:1-∞ a Z + Z = ∑ji-j∞ a Zi= LHS j i-j HW: Check whether or not X = aX + Z is statioiary if i-1 0 I 0 {X} ts weakly stationary. Proof: E[X] =i∑j:0-∞ a E[Z ] = 0 i-j For all h≥0, j r j r Cov[X, X ]i= Ci+h∑ a Z , ∑ a Z i-j i+h-r] = ∑∑ a a Cov[Z , Z i-j i+h-r (Cov ≠ 0 iff i-j = i+h-r -> j = r-h) = ∑ a a j j+h σ = σ a ∑a = σ a /(1-a ) 2 h 2 γ Xh) = σ a /(1-a ) -> absolute value for h≥0 Suppose a>0 ACVF & A
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