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11_30_1 - STAT 420 Examples for(1 Fall 2010 AR p Y t = φ 1...

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Unformatted text preview: STAT 420 Examples for 11/30/2010 (1) Fall 2010 AR( p ): Y t = φ 1 Y t – 1 + φ 2 Y t – 2 + … + φ p Y t – p + e t ↓ Consider Cov ( … , Y t – k ), k = 1, 2, … , p . ⇒ γ 1 = φ 1 γ + φ 2 γ 1 + φ 3 γ 2 + … + φ p γ p – 1 γ 2 = φ 1 γ 1 + φ 2 γ + φ 3 γ 1 + … + φ p γ p – 2 … γ p = φ 1 γ p – 1 + φ 2 γ p – 2 + φ 3 γ p – 3 + … + φ p γ Divide by γ : ⇒ ρ 1 = φ 1 ρ + φ 2 ρ 1 + φ 3 ρ 2 + … + φ p ρ p – 1 ρ 2 = φ 1 ρ 1 + φ 2 ρ + φ 3 ρ 1 + … + φ p ρ p – 2 … ρ p = φ 1 ρ p – 1 + φ 2 ρ p – 2 + φ 3 ρ p – 3 + … + φ p ρ ( Yule-Walker equations ) The Yule-Walker equations for an AR(2) process: ρ 1 = φ 1 + φ 2 ρ 1 ρ 2 = φ 1 ρ 1 + φ 2 ρ 1 = φ 1 + φ 2 ρ 1 ⇒ ρ 1 = 2 1 1 φ φ- ⇒ ρ 2 = φ 1 ρ 1 + φ 2 = 2 2 2 1 1 φ φ φ +- k ≥ 2 ρ k = φ 1 ρ k – 1 + φ 2 ρ k – 2 MA(1) Y t = e t – θ e t – 1 E ( e t ) = 0, Var ( e t ) = 2 σ e for all t E ( e t e t' ) = 0, for t ≠ t' E (...
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11_30_1 - STAT 420 Examples for(1 Fall 2010 AR p Y t = φ 1...

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