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tutorial_8

# tutorial_8 - STA 4005 Tutorial 8 Method of Moment...

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STA 4005 Tutorial 8 Method of Moment Estimation (MME): For AR(p) process, the Yule-Walker equations are: ρ k = φ 1 ρ k - 1 + φ 2 ρ k - 2 + · · · + φ p ρ k - p for k 1 Replace ρ k by r k and solve ˆ φ 1 ˆ φ 2 . . . ˆ φ p = 1 r 1 r 2 . . . r p - 1 r 1 1 r 1 . . . r p - 2 . . . . . . . . . r p - 1 r p - 2 . . . . . . 1 - 1 r 1 r 2 . . . r p Example: From a series { Z t } of length 100, we have computed r 1 =0.7, r 2 =0.45, r 3 =0.3, ¯ Z = 2, and a sample variance 5. We consider an AR(2) process with a constant term θ 0 : Z t = θ o + φ 1 Z t - 1 + φ 2 Z t - 2 + a t where a t WN (0 , σ 2 ), estimate φ 1 , φ 2 , θ 0 and σ 2 by MME. Sol: Z t - μ = φ 1 ( Z t - 1 - μ ) + φ 2 ( Z t - 2 - μ ) + a t Z t = θ 0 + φ 1 Z t - 1 + φ 2 Z t - 2 + a t where θ 0 = μ (1 - φ 1 - φ 2 ). By MME, ˆ φ 1 ˆ φ 2 = 1 r 1 r 1 1 - 1 r 1 r 2 = r 1 (1 - r 2 ) 1 - r 2 1 r 2 - r 2 1 1 - r 2 1 = 0 . 75 - 0 . 08 ˆ σ 2 = S 2 (1 - ˆ φ 1 r 1 - ˆ φ 2 r 2 ) = 2 . 555 ˆ θ 0 = ¯ Z (1 - ˆ φ 1 - ˆ φ 2 ) = 0 . 66 Example: Consider the AR(2) process Z t = φZ t - 1 + φ 2 Z t - 2 + a t where a t WN (0 , σ 2 ) 1

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a . For what value of φ it is stationary? b . After observing 200 samples, the sample moments were computed:
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