# T12 - STA4005B Time Series (2007-2008) Tutorial 12 (15/4,...

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Unformatted text preview: STA4005B Time Series (2007-2008) Tutorial 12 (15/4, 16/4) (Time & Venue: T9, BMS LT; W5, MMW 704) Correlation function between forecast errors The l step and l+j step ahead forecast errors are respectively et (l ) = a t +l + ψ 1 at +l −1 + ψ 2 at +l − 2 + L + ψ l −1 at +1 et (l + j ) = a t + l + j + ψ 1 a t + l + j −1 + ψ 2 a t + l + j − 2 + L + ψ j a t + l + ψ j +1 a t + l −1 + L + ψ l + j −1 a t +1 The correlation function between l step and l+j step ahead forecast error is l −1 corr[et (l ), et (l + j )] = ∑ψ ψ i =0 i i+ j ∑ψ ∑ψ m =0 where ψ 0 = 1 l + j −1 l −1 2 m n =0 2 n Example 1 Find the correlation between 2-step and 3-step ahead forecast error at time t for the MA(2) process such that Z t = at + θ 1 a t −1 + θ 2 at − 2 Example 2 Find the auto-covariance function of the process Z t = (1 − 0.6 B + 0.9 B 2 )(1 − 0.5 B 12 ) a t . Revision Exercise: Consider the process Z t = 3 + 0.7 Z t −1 + at 2 where σ a = 4 . Let Z 200 = 10 , Z 199 = 12 and Z 198 = 11 . ˆ ˆ a) Find Z 200 (1) and Z 200 (4) . b) Calculate the 95% prediction interval of Z 201 and the 95% prediction interval of Z 204 . Note that if X ~ N (0,1) , then P ( X ≤ 1.96) = 0.975 . ˆ c) Let e (k ) = Z − Z (k ) be the k-step ahead forecast error. Find Cov[e (k ), e (m)] . t t +k t t t ...
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## This note was uploaded on 11/02/2011 for the course STAT 4005 taught by Professor Wu,kaho during the Spring '08 term at CUHK.

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T12 - STA4005B Time Series (2007-2008) Tutorial 12 (15/4,...

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