GARCH Models

Appendix C

Solutions to the Exercises

Chapter 1

1.1 1. (a) We have the stationary solution X_t = Σ_i≥0 0.5ⁱ(η_t−i+ 1), with mean EX_t = 2 and autocorrelations ρx(h) = 0.5^|h|.

(b) We have an ‘anticipative’ stationary solution

which is such that EX_t = − 1 and ρx(h) = 0.5^|h|.

(c) The stationary solution

is such that EX_t = 2 with ρx(1) = 2/19 and ρx(h) = 0.5^h−1 ρx(1) for h > 1.

2. The compatible models are respectively ARMA(1, 2), MA(3) and ARMA(1, 1).

3. The first noise Is strong, and the second is weak because

Note that, by Jensen’s Inequality, this correlation is positive.

1.2 Without loss of generality, assume X_t = for t < 1 or t > n. We have

which gives , and the result follows.

1.3 Consider the degenerate sequence (X_t)_t=0, 1, … defined, on a probability space (, , ), by X_t(ω) = (−1) ...