The Limits of Mathematics and NP Estimation in ... - Chichilnisky

Recent Developments in SeasonalRecent Volatility Models Developments in Seasonal Volatility Models377ɛ 2 t = ω +(1 − θB)(1 − ΘL)u tThavaneswaran et al. (2009) derive the moments for the RCA model with GARCH(p, q)errors.Here we propose the RCA model with seasonal GARCH innovations of the following form,y t =(β + b t )y t−1 + ɛ t (13)ɛ t = √ h t Z t (14)θ(B)Θ(L)h t = ω + α(B)ɛ 2 t (15)where Z t , θ(B), Θ(L), α(B) were defined in Section 2.The general expression for the kurtosis K (y) parallels the one in Thavaneswaran et al. (2009)for non-seasonal GARCH innovations and can be written as follows.Lemma 4.1. For the stationary RCA process y t with GARCH innovations as in (13)– (15) wehave the following relationships:(i) E(y 2 E(ɛ 2 tt )=1 − (β 2 + σb 2) , (16)(ii) E(y 4 t )= 6(β2 + σb 2)[E(ɛ2 t )]2 +[1 − (β 2 + σb 2)]E(ɛ4 t )1 − (3σb 4 + β4 + 6β 2 σb 2)[1 − (β2 + σb 2)] , (17)(iii) K (y) = 6(β2 + σb 2)[1 − (β2 + σb 2)][1 − (β 2 + σb 21 − (3σb 4 + β4 + 6β 2 σb 2) +1 − (3σb 4 + β4 + 6β 2 σb 2) K(ɛ) . (18)If Z t is normally distributed, then the above equations can be written as:(i) E(y 2 E(h t )t )=1 − (β 2 + σb 2) , (19)(ii) E(y 4 6(β 2 + σb 2t )=[1 − (β 2 + σb 2)](1 − 6β2 σb 2 − β4 − 3σb 4) [E(h t)] 2 3E(h 2 t+1 − 6β 2 σb 2 − β4 − 3σb4 , (20)(iii) K (y) = 6(β2 + σb 2)[1 − (β2 + σb 2)]3(1 − β 2 − σb 21 − 6β 2 σb 2 − β4 − 3σb4 +E(h 2 t )1 − 6β 2 σb 2 − β4 − 3σb4 [E(h t )] 2 . (21)Thavaneswaran et al. (2005a) show that:E(h 2 t )[E(h t )] 2 = 1E(Zt 4) − [E(Z4 t ) − 1] ,∑∞ j=0 Ψ2 j1which for a conditionally normally distributed ɛ t reduces to3 − 2 ∑ ∞ .j=0 Ψ2 jExample 4.1. RCA(1) with multiplicative seasonal GARCH (0,1)x(0,1) processy t =(β + b t )y t−1 + ɛ tɛ t = √ h t Z t