16.9

a. Suppose that E(ut  ut - 1, ut - 2,

c) = 0, that var1ut  ut - 1, ut - 2, c2 follows the ARCH(1) model s2t

= a0 + a1u2t- 1, and that the process for ut is stationary. Show that var1ut2 = a0> 11 - a12. (Hint: Use the law of iterated expectations E(u2t

) = E[E(u2t

 ut - 1)].)

b. Extend the result in

(a) to the ARCH(p) model.

c. Show that gpi

= 1 ai 6 1 for a stationary ARCH(p) model.

d. Extend the result in

(a) to the GARCH(1,1) model.

e. Show that a1 + f1 6 1 for a stationary GARCH(1,1) model.