ZHAPTER 12 TIME SERIES ANALYSIS LE. nc-7 C41 lieammxa AR TPEpr45a11 w-AFW_-1 R hPw__.o A-APz R HPZC1A-ZETA-HPZEC H al 5a, w ill 4r lpri4l pm. Alp pram, p51 TA- hElL e C( AR HPL R HPL2R…)et THFE wc>;R i=1 12. 4 Moving average processes C∑ T et N Ha(Hi )fodraura M, Va. crttrt Eb {e} Ce ruet-l- N MAIFE /. otcura do MA LEVa. crtt c er u R.R net C ARu1LR…RLF2oCHAPTER 12 TIME SERIES ANALYSIS 6 Example 7 Consider the AR (2) process yt − 1.2yt−1 + 0.2yt−2 = et . 1 − 1.2Z + 0.2Z 2 = (1 − Z) (1 − 0.2Z) = 0 gives Z = 1, 1 0.2 . Hence, yt is not stationary. As a matter of fact, yt − yt−1 = (1 − 0.2L) −1 et =  1 + 0.2L + 0.2L 2 + · · · et = ∞ i=0 (0.2)i et−i = ut and yt =  t i=1 ui + u0. 12.4 Moving average processes yt =  M j=−M θjet−j where et ∼ iid  0, σ2 (finite order MA processes). Example 8 yt = et + θ1et−1 + θ2et−2 ∼ MA (2) Consider an MA (q) process yt = et + θ1et−1 + · · · + θqet−q = (1 + θ1L + · · · + θqL q ) et