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

Recent Developments in SeasonalVolatility Models 9Recent Developments in Seasonal Volatility Models39∑ ∞ j=0 βj a t−j ,wherea t is an uncorrelated sequence with variance σa 2 . By equating the varianceof y ∗ t to the variance of y t we have σɛ 2/(1 − β2 − σb 2)=σ2 a /(1 − β2 ),andσa 2 = σɛ 2 (1 − β 2 )/(1 −β 2 − σb 2).Note: When σb 2 = 0, var[e(y) n (l)] in Theorem 4.1 reduces to var[e (y)n (l)] in Theorem 3.1 for theAR model with seasonal GARCH errors.We now have expressions for the variance of the l-steps-ahead forecast error of y n+l for thepreviously discussed RCA(1)-GARCH(p, q)x(P, Q) s models:RCA(1)-GARCH(0, 1)x(0, 1) sRCA(1)-GARCH(0, 1)x(1, 0) sRCA(1)-GARCH(1, 0)x(1, 0) sVar[e (y)n (l)] = ω(1 − β2 l−1)(1 − β 2 − σb 2) ∑Var[e (y)n (l)] =Var[e (y)n (l)] =β 2jj=0ω(1 − β 2 l−1)(1 − Φ)(1 − β 2 − σb 2) ∑ β 2j ,j=0ω(1 − β 2 l−1)(1 − φ)(1 − Φ)(1 − β 2 − σb 2) ∑ β 2j .j=0Theorem 4.2. Let Y t =[y t − ( ˆβ + b t )y t−1 ] 2 . Also, let Y n (l) be the l-steps-ahead minimummean square forecast of Y n+l and let e (Y)n (l) = Y n+l − Y n (l) be the corresponding forecasterror. The variance of the l-steps-ahead forecast error of Y n+l for the RCA(1) model withseasonal GARCH errors as given in (13)- (15) is:⎡ ⎤Var[e (Y)n (l)] = σu2 l−1∑ Ψ 2 ω 2j = ⎡ ⎤ [ ]j=0⎣∑∞ p 2 ]Ψ 2 P 2[K⎦j 1 − ∑ φ i[1 (ɛ) − 1] ⎣ l−1∑ Ψ 2 ⎦j (23)j=0− ∑ Φ ij=0i=1i=1where, from (18), K (ɛ) = 1 − (3σ4 b + β4 + 6β 2 σb 2)[1 − (β 2 + σb 2 K (y) − 6(β2 + σb 2))]2 1 − (β 2 + σb 2) .Proof. The proof follows from part (b) of Lemma 3.2.Note: When σb 2 = 0, K(ɛ) in Theorem 4.2 reduces to K (ɛ) in Theorem 3.2 for the AR model withseasonal GARCH errors.We now have expressions for the variance of the l-steps-ahead forecast error for the previouslydiscussed RCA(1)-GARCH(p, q)x(P, Q) s models:RCA(1)-GARCH(0, 1)x(0, 1) sRCA(1)-GARCH(0, 1)x(1, 0) sRCA(1)-GARCH(1, 0)x(1, 0) sVar[e (Y)n (l)] = (K(ɛ) − 1)μ 2 l−1(1 + θ 2 )(1 + Θ 2 )∑ Ψ 2 jj=0Var[e (Y)n (l)] = (K(ɛ) − 1)μ 2 (1 − Φ 2 l−1)1 + θ∑2 Ψ 2 jj=0Var[e (Y)n (l)] = (K(ɛ) − 1)μ 2 (1 − φ 2 l−1)1 + 2φ s Φ +∑Φ 2 Ψ 2 jj=0which are similar to the expressions given in Doshi et al. (2011). Here, K (ɛ) is given in Theorem4.2. and expressions for K (y) for a conditionally normally distributed ɛ t are given in Examples4.1, 4.2, and 4.3.