tutorial_5

tutorial_5 - 1 3. Find the autocorrelation function of { W...

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STA 4005 Tutorial 5 Def : A series { Z t } is said to follow an integrated autoregressive moving average model if the d -th difference W t = d Z t is a stationary ARMA process where =1-B, d = ( d - 1 ) If W t is ARMA(p,q) model, then Z t is ARIMA(p,d,q) model. In general, the ARIMA(p,d,q) model can be expressed as φ ( B )(1 - B ) d Z t = θ ( B ) a t Example 1: Given a process Z t = 1 . 25 Z t - 1 - 0 . 25 Z t - 2 + a t 1. Identify the model as a specified ARIMA model 2. Find the value of the ψ k , k=1 , 2 , 3 , ··· , if the process is written as the form Z t = (1 + k =1 ψ k B k ) a t Example 2: Consider the following infinite MA process { Z t } , Z t = a t + c ( a t - 1 + a t - 2 + ··· ) where c is a fixed constant and a t WN (0 2 ). 1. Show that { Z t } is nonstationary. 2. Let W t = Z t - Z t - 1 , show that { W t } is a stationary MA(1) model.
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Unformatted text preview: 1 3. Find the autocorrelation function of { W t } . Example 3: Let Z t = 0 . 4 Z t-1 + 0 . 45 Z t-2 + a t + a t-1 + 0 . 25 a t-2 , a t ∼ WN (0 ,σ 2 ). 1. Write the model in terms of B and identify the model as a specific ARIMA model. 2. Simplify the model and identify it as a specific ARIMA model. 3. Determine if the model is stationary and/or invertible. 4. If the model is stationary(invertible), find the general form of the coefficients ψ j ’s( π j ’s), s.t., Z t = ∑ ∞ j =0 ψ j a t-j ( a t = ∑ ∞ j =0 π j Z t-j ) 2...
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This note was uploaded on 01/20/2012 for the course STA 4005 taught by Professor ? during the Spring '08 term at CUHK.

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tutorial_5 - 1 3. Find the autocorrelation function of { W...

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