Chapter 3.docx

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University of Toronto St. George
Statistical Sciences
Zhou Zhou

Chapter 3 – ARMA Models 3.1 –  ARMA(p,q)  Process Definition A weakly stationary TS  { }t  is called an RMA(p,q)  process  (p≥0,q≥0)  if X tϕ X1 t−1−…−ϕ X p t−p=Z +t Z1 t−1+…+θ Zq t−q(¿) Z WN 0,σ 2) { t ARMA (p,q)  can be written as  Φ(B)X =t(B)Z t 2 p Φ B =1−ϕ B−ϕ1B −…−ϕ2B p Θ B =1+θ B+1 B +…2θ B q q Theorem (¿) A stationary solution of   exists and is unique iff  ϕ z =1−ϕ z−…1ϕ z ≠0∀ zpZ , z =1 ∣ ∣ Example AR(1) (1−aB)X =Z t t ϕ(z)=1−az=0 1 z= ⇒ ∣a∣≠1 a Definition An  ARMA (p,q)  process is causal if there exist constants   such that  ∞ ∑ ∣ j∞ j=0 ∞ X j ∑ ψiZ j−i i=0 Theorem Causality is equivalent to  ϕ(z)≠0∀ z∣≤1 AR(1)causal⇔ ∣a∣<1 Find  ψ j ’s! Write Ψ B =1+ψ B+ψ1B +… 2 2 We know X =Ψ (B)Z t t Meanwhile X = Θ(B) Z t Φ(B) t Θ(B) ⇒Ψ (B)= Φ(B) Θ(B)=Ψ (B)Φ(B) 1+θ 1+…+θ B =q1+ψ B(ψ B 1… 1−ϕ2B−…−ϕ B)( 1 p p) 1  order coefficients: LHS=θ 1 RHS=ψ −ϕ 1 1 ⇒θ 1ψ −ϕ1 1 ψ =θ +ϕ 1 1 1 nd 2  order coefficients: LHS=θ 2 RHS=ψ −ϕ −ψ ϕ 2 2 1 1 ⇒θ 2ψ −ϕ2−ψ ϕ2 1 1 ψ 2θ +2 +ψ2ϕ =θ1+1 + θ2+ϕ 2 ( 1 1) 1 ψ Recursively solve for all  j ’s Definition ARMA (p,q) { } { } An   model  t  is invertible if there exist constants   such that  ∞ ∑ ∣ ∣j∞ j=0 ∞ Zt= ∑ πjX t−j j=0 Theorem θ z =1+θ z+…+θ z q Let  1 q Invertibility is equivalent to θ(z)≠0∀ ∣z∣≤1 Example AR(2) Xt=0.7X t−10.1X t−2+Z t 2 ZtWN 0,σ ) I
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