Appendix C
Solutions to the Exercises
Chapter 1
1.1 1. (a) We have the stationary solution Xt = Σi≥0 0.5i(ηt−i+ 1), with mean EXt = 2 and autocorrelations ρx(h) = 0.5|h|.
(b) We have an ‘anticipative’ stationary solution
![bapp03ue001_fmt](/api/v2/epubs/9780470683910/files/images/bapp03ue001_fmt.jpg)
which is such that EXt = − 1 and ρx(h) = 0.5|h|.
(c) The stationary solution
![bapp03ue002_fmt](/api/v2/epubs/9780470683910/files/images/bapp03ue002_fmt.jpg)
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](/api/v2/epubs/9780470683910/files/images/bapp03ue003_fmt.jpg)
Note that, by Jensen’s Inequality, this correlation is positive.
1.2 Without loss of generality, assume Xt = for t < 1 or t > n. We have
![bapp03ue004_fmt](/api/v2/epubs/9780470683910/files/images/bapp03ue004_fmt.jpg)
which gives , and the result follows.
1.3 Consider the degenerate sequence (Xt)t=0, 1, … defined, on a probability space (, , ), by Xt(ω) = (−1) ...