Description

2. Rewrite the following time series models using the backward shift operator B and the di↵erence operator and in terms of the time series xt and at (that is, no xt 1, yt or at 1 should show up in the final expression).
(1) xt = 1xt 1 + at + ✓1at 1
(2) yt = 1yt 1 + at + ✓1at 1 and yt = xt xt 1.
(3) xt = 1xt 4 + at
(4) yt = 1yt 1 + at + ✓1at 1 and yt = xt xt 4.
3. Define the operator 4 as 4xt := xt xt 4. Rewrite the following time series models in the original forms WITHOUT the operator B, the di↵erence operator and 4.
(1) (1 1B)xt = (1 + ✓B)at
(2) (1 1B)(1 1B4)xt = at
(3) 4xt = (1 + ✓1B)at
(4) (1 1B) 4xt = (1 + ✓1B)at
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