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

Recent Developments in SeasonalRecent Volatility Models Developments in Seasonal Volatility Models355Example 3.3. For a stationary autoregressive process of order one, AR(1), with multiplicativeseasonal GARCH (1, 0)x(1, 0) s errors of the form,y t = βy t−1 + ɛ tɛ t = √ h t Z t(1 − φB)(1 − ΦL)ɛ 2 t = ω + u twhere φ is the autoregressive parameter and Φ is the seasonal autoregressive parameter. TheΨ-weights given in Doshi et al. (2011) are as follows: Ψ 1 = φ, Ψ 2 = φ 2 ,...,Ψ s−1 = φ s−1 , Ψ s =φ 2 + Φ,...,Ψ j = φΨ j−1 + ΦΨ j−s − φΦΨ j−s .Itcanbeshownthat∑ ∞ j=0 Ψ2 j= 1 + 2φs Φ 2 + Φ 21 − φ 2 .Then, the kurtosis of y t is:K (y) = 6β2 (1 − β 2 )(1 − β 4 + (1 − β2 ) 2E(Zt 4)) (1 − β 4 ()1 + 2φE(Zt 4) − [E(Z4 t ) − 1] s Φ + Φ 2 ) ,1 − φ 2which for a conditionally normally distributed Z t reduces to:K (y) = 6β2 (1 − β 2 )(1 − β 4 + (1 − β2 ) 2 3(1 − φ 2 )) (1 − β 4 ) (1 − 3φ 2 − 4φ s Φ − 2Φ 2 ) .Forecast error varianceThavaneswaran et al. (2005a) derive the expression for the forecast error variance of variousclasses of zero mean GARCH(p, q) processes, in terms of the kurtosis and Ψ-weights.Thavaneswaran & Ghahramani (2008) extend the results for ARMA (p, q) processes withGARCH (P, Q) errors. In this section we extend the results to AR models with multiplicativeseasonal GARCH(p, q)x(P, Q) s errors.Theorem 3.1. Let y n (l) be the l-steps-ahead minimum mean square forecast of y n+l and lete (y)n (l) =y n+l − y n (l) be the corresponding forecast error. The variance of the l-steps-aheadforecast error of y n+l for the AR(1) model with seasonal GARCH errors as given in (2)- (4) is:Var[e (y)l−1ωn (l)] = ()pP∑ ψj=01 − ∑ φ i)(1 2 j . (10)− ∑ Φ ii=1i=1Proof. The theorem follows from the fact that for a stationary process with uncorrelated errornoise ɛ t the variance of the l-steps ahead forecast error is σɛ 2 ∑ l−1j=0 ψ2 j and from part (b) ofLemma 3.2.We now have expressions for the variance of the l-steps-ahead forecast error of y n+l for thepreviously discussed AR(1)-GARCH(p, q)x(P, Q) s models:AR(1)-GARCH(0, 1)x(0, 1) sVar[e (y)l−1n (l)] = ω ∑ β 2jj=0