Appendix E Solutions 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
        equation
        which is such that EX t  =  − 1 and ρ X (h) = 0.5h .
      3. (c) The stationary solution
        equation
        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
      equation

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

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

    which gives images , 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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