8-10 ARCH Models(contd) One example of a full model would be D,=B1+B2x2t+.+Bkk +u, u,N(O, o) where ot We can easily extend this to the general case where the error variance depends on g lags of squared errors: o=a+a,u +au2+... +a This is an arCH(@ model. Instead of calling the variance ot, in the literature it is usually called h. so the model is D,=B1+ Bix2r+.+ BkXkt+ut, u N(o,h) where h,=0+a1 u-1+a2 u 22+. +aa u-q8-10 ARCH Models (cont’d) • One example of a full model would be yt = 1 + 2x2t + ... + kxkt + ut , ut  N(0, ) where = 0 + 1 • We can easily extend this to the general case where the error variance depends on q lags of squared errors: = 0 + 1 +2 +...+q • This is an ARCH(q) model. • Instead of calling the variance , in the literature it is usually called ht , so the model is yt = 1 + 2x2t + ... + kxkt + ut , ut  N(0,ht ) where ht = 0 + 1 +2 +...+q t 2 t 2 t 2 ut−1 2 ut −q 2 ut −q 2 t 2 2 t −1 u 2 ut −2 2 t −1 u 2 t −2 u