solutions5 - Matlab Solutions Time Series (2DD23) Exercise...

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Matlab Solutions Time Series (2DD23) Week 5 Exercise 4.4 Consider the AR(2) process X t = 1 3 X t - 1 + 2 9 X t - 2 + Z t In exercise 3.6 it is shown that the autocorrelationfunction is: ρ( k ) = 16 21 ± 2 3 ² | k | + 5 21 ± - 1 3 ² | k | Now we can compute the partial autocorrelation of the process. First we use the fact that the partial autocorrelation function at lag 1 always equals ρ( 1 ) (see page 62). π 1 = ρ( 1 ) = 16 21 ± 2 3 ² + 5 21 ± - 1 3 ² = 3 7 The partial autocorrelation function of an AR( p ) process at lag p is always α p . π 2 = α 2 = 2 9 The pacf of an AR( p ) model has a cut-off after lag p , so for all other lags the pacf is 0 . Exercise 5.1 Consider the MA ( 1 ) model X t = Z t + θ Z t - 1 We will compute the one-step-ahead forecast at time N : ˆ X N ( 1 ) = E ( X N + 1 | X N , Z N , X N - 1 , Z N - 1 ,...) = E ( Z N + 1 + θ Z N | X N , Z N , X N - 1 , Z N - 1 ,...) = E ( Z N + 1 | X N , Z N , X N - 1 , Z N - 1 ,...) + E Z N | X N , Z N , X N - 1 , Z N - 1 ,...) = E ( Z N + 1 ) + E Z N | Z N ) = 0 + θ Z N 1
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So ˆ x N ( 1 ) = θ
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solutions5 - Matlab Solutions Time Series (2DD23) Exercise...

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