Appendix E Solutions to the Exercises
Chapter 1
 1.1

 (a) We have the stationary solution X _{ t } = ∑_{ i ≥ 0}0.5^{ i }(η _{ t − i } + 1), with mean EX _{ t } = 2 and autocorrelations ρ _{ X }(h) = 0.5^{∣h∣} .
 (b) We have an ‘anticipative’ stationary solution
 (c) The stationary solution
 The compatible models are, respectively, ARMA(1, 2), MA(3) and ARMA(1, 1).
 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
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 (Ω, 풜, ℙ), by X _{ t }(ω) = (−1)^{ t } for all
Get GARCH Models, 2nd Edition now with O’Reilly online learning.
O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.