17.9

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

c) = 0, that var(ut  ut - 1, ut - 2,

c) follows the ARCH(1) model s2t

= a0 + a1u2t

- 1, and that the process for ut is stationary. Show that var(ut) = a0>(1 - a1). (Hint: Use the law of iterated expectations, E(u2t) = E3E(u2t

 ut - 1)4.)

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.