d si aff n bnd ffib e b力 INIWMTPAINR Squ1f2ln }-612-HPbq2q{ T S TF/dS RSun4PENES, FES MA I NES/AEIn bSw MFISn 1 AIn 2finZES-IMSMAR I NESEA AlFRFRIP ∑的W班R刚<E WRSn Cet aeIn bs SxI/ASd 2n AIERI wly, wS EITYFES MA I NESAAArv SAF2b'S Example 9 t{eRet-1{小R The root of the equation FRI isz. Hence, this MA process is not invertible. In fact, RL {( v{}and∑|{ Example 10 gr et REfet-1 nf REflcet }R∈fz{∈ f L-FIP }R∈fL ∑吃 where <E RCHAPTER 12 TIME SERIES ANALYSIS 7 If all roots of the equation 1 + θ1Z + · · · + θqZ q = 0 lie outside the unit circle, the MA process can be written as an infinite—order AR process, such that et = ∞ j=0 ψjyt−j with  ψj   < ∞. When {et} can be expressed in such a way, we call the MA process invertible. Example 9 yt = et − et−1 = (1 − L) et . The root of the equation 1 − Z = 0 is Z = 1. Hence, this MA process is not invertible. In fact, et = 1 1 − L yt =  1 + L + L 2 + · · · yt = ∞ j=0 ψj yt−j , where ψj = 1 and  ψj   = ∞. Example 10 yt = et − 0.9et−1 = (1 − 0.9L) et . 1 − 0.9Z = 0 Z = 1 0.9 > 1. et = 1 1 − 0.9L yt =  1 + 0.9L + 0.9 2L 2 + · · · yt = ∞ j=0 ψjyt−j , where ψj = 0.9 j and ∞ 0  ψj   = 1 1 − 0.9 < ∞