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sample_finalsol

sample_finalsol - AMS 316 Final Solution 1 x t = a bt ct 2...

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Unformatted text preview: AMS 316 Final Solution December 15, 2010 1. x t = a + bt + ct 2 + s t + y t (a) trend: a + bt + ct 2 , seasonal: s t (b) s t ≡ , (1- B ) 2 x t = (1- 2 B + B 2 ) x t = a + bt + ct 2 + y t- 2 a- 2 b ( t- 1)- 2 c ( t- 1) 2 + a + b ( t- 2) + c ( t- 2) 2 = 2 c + y t- 2 y t- 1 + y t- 2 . Since y t is stationary, (1- B ) 2 x t is stationary. 2. x t = z t + θ 1 z t- 1 + θ 2 z t- 2 (a) ACVF: γ x ( k ) = cov ( z t + θ 1 z t- 1 + θ 2 z t- 2 ,z t + k + θ 1 z t + k- 1 + θ 2 z t + k- 2 ) = k > 2 σ 2 · θ 2 k = 2 σ 2 · θ 1 (1 + θ 2 ) k = 1 σ 2 · (1 + θ 2 1 + θ 2 2 ) k = 0 γ x (- k ) k < 1 ACF: ρ x ( k ) = γ x ( k ) /γ x (0) = k > 2 θ 2 / (1 + θ 2 1 + θ 2 2 ) k = 2 θ 1 (1 + θ 2 ) / (1 + θ 2 1 + θ 2 2 ) k = 1 1 k = 0 ρ x (- k ) k <...
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sample_finalsol - AMS 316 Final Solution 1 x t = a bt ct 2...

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