8-16 GARCH Models(contd) Now substituting into(2)for o-2 Ot=00+ auf +aoB+aiBl-2 +B(ao auf3 +Bo-3) =00+a1l2-1+B+aB2 +ooB+aiBu, +Boil G2=00(1++)+a121(1+BL+B2L2)+B3o132 An infinite number of successive substitutions would yield 72=c(++f2+…)+a1121(1+BDL+2C2+.)+B002 σ4=y+c1ll-1(1++B22+…) y+y14-1+y2llt-2+ So the garch(l,)model can be written as an infinite order ARCh model
8-16 GARCH Models (cont’d) • Now substituting into (2) for t-2 2 • An infinite number of successive substitutions would yield • So the GARCH(1,1) model can be written as an infinite order ARCH model. t 2 =0 + 1 2 ut −1 +0 + 1 2 ut−2 + 2 (0 + 1 2 ut−3 +t-3 2 ) t 2 = 0 + 1 2 ut−1 +0 + 1 2 ut−2 +0 2 + 1 2 2 ut−3 + 3 t-3 2 t 2 = 0 (1++ 2 ) + 1 2 ut −1 (1+L+ 2 L 2 ) + 3 t-3 2 t 2 = 0 (1++ 2 +...) + 1 2 ut −1 (1+L+ 2 L 2 +...) + 0 2 = + + + = + + + + − − − 2 2 2 2 0 1 1 2 2 2 0 1 1 2 (1 ) t t t t u u u L L
8-17 GArCH Models(cont d) Why iS garch better than aRCH? more parsimonious-avolds overfitting less likely to breach fEknon-negativity constraints We can again extend the garch(l, 1)model to a garcH(p, 万=+u2 2 +6L +a1-q+11+02+…+AOp Mo +>a,u,+ ∑月σ i=1 j=1 But in general a garch(l, 1)model will be sufficient to capture the volatility clustering in the data
8-17 GARCH Models (cont’d) • Why is GARCH Better than ARCH? - more parsimonious - avoids overfitting - less likely to breach 违反non-negativity constraints • We can again extend the GARCH(1,1) model to a GARCH(p,q): • But in general a GARCH(1,1) model will be sufficient to capture the volatility clustering in the data. t 2 = = = + − + − q i p j i ut i j t j 1 1 2 2 0 t 2 = 0+1 2 ut−1 +2 2 ut−2 +...+q 2 ut−q +1t-1 2 +2t-2 2 +...+pt-p 2
8-18 TThe unconditional variance under the GARCH Specification The unconditional variance of u, is given by Var(uy 1+B<1 a t when ax1+B≥1 · is termed“non- stationarity” in variance a+B=l is termed intergrated GARCH For non-stationarity in variance, the conditional variance forecasts will not converge on their unconditional value as the horizon increases
8-18 The Unconditional Variance under the GARCH Specification • The unconditional variance of ut is given by when • is termed “non-stationarity” in variance • is termed intergrated GARCH • For non-stationarity in variance, the conditional variance forecasts will not converge on their unconditional value as the horizon increases. Var(ut ) = 1 ( ) 1 0 − + 1 + < 1 1 + 1 1 + = 1
8-19 5 Estimation of arch garch models Since the model is no longer of the usual linear form we cannot use OLs. Because rss depends only on the parameters in the conditional mean equation, not the conditional variance We use another technique known as maximum likelihood The method works by finding the most likely values of the parameters given the actual data. More specifically, a log-likelihood function is formed and the values of the parameters that maximise it are sought. ML can be employed to find parameter values for both linear and non-linear models
8-19 5 Estimation of ARCH / GARCH Models • Since the model is no longer of the usual linear form, we cannot use OLS. Because RSS depends only on the parameters in the conditional mean equation, not the conditional variance. • We use another technique known as maximum likelihood. • The method works by finding the most likely values of the parameters given the actual data. • More specifically, a log-likelihood function is formed and the values of the parameters that maximise it are sought. • ML can be employed to find parameter values for both linear and non-linear models
8-20 Estimation of arch garch models The steps involved in actually estimating an Arch or GARCH model are as follows 1. Specify the appropriate equations for the mean and the variance-eg an Ar(1)-GarCH(l, 1)model: =1+c21+t,t~N(0,O Bor 2 2. Specify the log-likelihood function to maximise: log(2z)-∑log(σ2) 2 2 R之V-4-1-1)2/0 3. The computer will maximise the function and give parameter values and their standard errors
8-20 Estimation of ARCH / GARCH Models • The steps involved in actually estimating an ARCH or GARCH model are as follows 1. Specify the appropriate equations for the mean and the variance - e.g. an AR(1)- GARCH(1,1) model: 2. Specify the log-likelihood function to maximise: 3. The computer will maximise the function and give parameter values and their standard errors yt = + yt-1 + ut , ut N(0,t 2 ) t 2 = 0 + 1 2 ut−1 +t-1 2 = − = = − − − − − T t t t t T t t y y T L 1 2 2 1 1 2 ( ) / 2 1 log( ) 2 1 log( 2 ) 2