GARCH Models

Appendix D

Problems

Problem 1

The exercises are independent. Let (η_t) be a sequence of iid random variables satisfying E(η_t) = 0 and Var(η_t) = 1.

Exercise 1: Consider, for all t , the model

where the constants satisfy ω > 0, α_i ≥ 0,i = 1,…,q and β_j ≥ 0, j = 1,…, p. We also assume that η_t is independent of the past values of _t. Let μ = E|η_t|.

1. Give a necessary condition for the existence of E|_i|, and give the value of m = E|_t|.

2. In this question, assume that p = q = 1.

(a) Establish a sufficient condition for strict stationarity using the representation

and give a strictly stationary solution of the model. It will be assumed that this condition is also necessary.

(b) Establish a necessary and sufficient condition for the existence of a second-order stationary solution. Compute the variance ...