tutorial_5

# tutorial_5 - 1 3 Find the autocorrelation function of W t...

This preview shows pages 1–2. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online