Estimation in Financial Models - RiskLab

we assume that it follows an AR(1) E-ARCH process:
h i
ln [Y kh ] = ln Y (k,1)h + hh kh 2 + kh " kh ;
i h h i
= ln khi 2 + , ln
2
kh h + C12 " kh
ln h 2 (k+1)h
+ h j" kh j,(2h=) 1=2i ; (2.12)
where [(C 22 , C 2 12)=(1 , 2=)] 1=2 and " kh iid, N(0;h).
If ln( 2 0) is normally distributed, then the ln( 2 t ) process in (2.11) is Gaussian.
Otherwise, if > 0, a Gaussian stationary limit distribution for ln( 2 t ) exists.
Thus, the conditional variance in continuous time is lognormal, and as in
the GARCH(1,1) example above, from this information Nelson [54] infers
the distribution of the innovation process in the discrete time model (2.12).
He shows that in the discrete time model (with short time intervals) the
distribution of the innovations is approximately a normal-lognormal mixture.
Hence, we derived the approximate distributions of GARCH(1,1) and AR(1)
E-ARCH models for small sampling intervals by using the distributional results
available for the diusion limit.
15