# Appendix ESolutions to the Exercises

## Chapter 1

1. 1.1
1. (a) We have the stationary solution X t  = ∑ i ≥ 00.5 i (η t − i  + 1), with mean EX t  = 2 and autocorrelations ρ X (h) = 0.5h .
2. (b) We have an ‘anticipative’ stationary solution
which is such that EX t  =  − 1 and ρ X (h) = 0.5h .
3. (c) The stationary solution
is such that EX t  = 2 with ρ X (1) = 2/19 and ρ X (h) = 0.5 h − 1 ρ X (1) for h > 1.
1. The compatible models are, respectively, ARMA(1, 2), MA(3) and ARMA(1, 1).
2. The first noise is strong, and the second is weak because

Note that, by Jensen's inequality, this correlation is positive.

2. 1.2 Without loss of generality, assume for t < 1 or t > n . We have

which gives , and the result follows.

3. 1.3 Consider the degenerate sequence (X t ) t = 0, 1, … defined, on a probability space (Ω, , ), by X t (ω) = (−1) t for all

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