N5

Let rt denotes the return of a financial asset and t denotes the standard deviation of returns at time t. Suppose rt follows rt = + et with et = ztt where zt N(0, 1). (a) Write down an ARCH(q) model with q=3 for 2t . (b) Write down an GARCH(q,p) model with q=1 and p=2 for 2t . (c) Derive the unconditional variances of the ARCH model in (a) (show all necessary steps). (d) Derive the unconditional variances of the GARCH model in (b) (show all necessary steps). (e) Discuss and compare the two ARCH-type models in (a) and (b).