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

bapp03ue001_fmt

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

(c) The stationary solution

bapp03ue002_fmt

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

bapp03ue003_fmt

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

bapp03ue004_fmt

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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