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notes10d - NOTES 10 Seasonal Models : Let s denote the...

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Unformatted text preview: NOTES 10 Seasonal Models : Let s denote the known seasonal period (eg. for monthly series, s = 12). Definition : A Seasonal MA( Q ) s model is defined by Z t = a t- Θ 1 a t- s- Θ 2 a t- 2 s- ...- Θ Q a t- Qs Seasonal MA(1) 12 Z t = a t- Θ a t- 12 , | Θ | < 1 Cov ( Z t ,Z t- 1 ) = Cov ( a t- Θ a t- 12 ,a t- 1- Θ a t- 13 ) = 0 . Cov ( Z t ,Z t- 12 ) = Cov ( a t- Θ a t- 12 ,a t- 12- Θ a t- 24 ) =- Θ σ 2 a . ρ k = 1 , k = 0 ,- Θ 1+Θ 2 , | k | = 12 , , | k | 6 = 0 , 12 Note that the Seasonal MA( Q ) s model can be rewritten as Z t = Θ( B ) a t where Θ( x ) = 1- Θ 1 x s- Θ 2 x 2 s- ...- Θ Q x Qs is the seasonal MA characteristic polynomial. Also we have ρ = 1 ρ ks =-- Θ k + Θ 1 Θ k +1 + Θ 2 Θ k +2 + ... + Θ Q- k Θ Q 1 + Θ 2 1 + Θ 2 2 + ... + Θ Q , k = 1 , 2 ,...,Q. and ρ h = 0 for h is not a multiple of s or h > Qs . Note that seasonal MA( Q ) s is always stationary and invertible if and only if all roots of Θ( x ) = 0 all exceed one in absolute value. Also, the model can also be viewed as an MA( Qs ) model with all θ ’s equals to zero except at the seasonal lags s, 2 s,...,Qs ....
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notes10d - NOTES 10 Seasonal Models : Let s denote the...

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