Appendix C

Solutions to the Exercises

Chapter 1

1.1 1. (a) We have the stationary solution Xt = Σi≥0 0.5iti+ 1), with mean EXt = 2 and autocorrelations ρx(h) = 0.5|h|.

(b) We have an ‘anticipative’ stationary solution


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

(c) The stationary solution


is such that EXt = 2 with ρx(1) = 2/19 and ρx(h) = 0.5h−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 Xt = X-bar_fmt for t < 1 or t > n. We have


which gives bapp03-ie365001_fmt, and the result follows.

1.3 Consider the degenerate sequence (Xt)t=0, 1, … defined, on a probability space (2126_fmt, , ), by Xt(ω) = (−1) ...

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