C, stoch ai or c ec hice s f s m ere HAL I PI2EIRSIM NMM A TEL TdSY Zt r tt ixp, p3 A T SE2b'SE TESnTaSIMSIAPAqullSAS/FL 1FMN, RIAFhS MYw 2g INISnr 2 m(2-dr N(KA-)p (SSS Fu 'SMC976 )) Remark 2 Estimating the coefficients of the M A processes is rather complicated, since 12.5 Autoregressi ve-moving average process tza1t1R… ROp tt p R1Rt1R…R1qtq~ABMA刘 gEL Ids INIYNTTFAINFES Squ1f2TnA r ez a-a1z-amz O PZH b亿EZAR1ZR.,ZqZH ES Tu/dS IRS un PENES.t 2A/F1 FTm1Ny Ind 2v SNF2b'S. WS RIvS vmxL-m团NTRE wESS RZi RUI RR r IPUt Z,txb er ztpCHAPTER 12 TIME SERIES ANALYSIS 8 Asymptotic theory for MA (1) model yt = et + θet−1, |θ| < 1 (invertible) The nonlinear least squares estimator of θ has the following property: 1. ˆθ p→ θ 2. √ T  ˆθ − θ  d→ N  0, 1 − θ 2 . (See Fuller (1976)) Remark 2 Estimating the coefficients of the MA processes is rather complicated, since the problem is nonlinear. 12.5 Autoregressive—moving average process yt = α1yt−1 + · · · + αpyt−p + et + θet−1 + · · · + θqet−q ∼ ARMA (p, q) model et ∼ iid  0, σ2 If all roots of the equations a (Z) = 1 − α1Z − α2Z 2 − · · · − αpZ p = 0 and b (Z) = 1 + θ1Z + · · · θqZ q = 0 lie outside the unit circle, {yt} is stationary and invertible. We have √ T (ηNLLS − η) → N (0, V ) where η =   α1 . . . αp θ1 . . . θq   , V = σ 2 ! EU1U ′ 1 EU1V ′ 1 EV1U ′ 1 EV1V ′ 1 "−1 , a (L)Ut = et , b (L) Vt = et