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366 Chapter 10 Further Topics
10.2. Verify the calculations of Example 10.2.1 to fit an intervention model to the
series SB.TSM.
10.3. If {Xt} is the linear process (10.3.1) with {Zt} ∼IID 0,σ 2 and η EZ 3 t ,how
that the third-order cumulant function of {Xt} is given by
C3(r, s) η
∞
i−∞
ψiψi+rψi+s.
Use this result to establish equation (10.3.4). Conclude that if {Xt} is a Gaussian
linear process, then C3(r, s) ≡ 0 and f3(ω1,ω2) ≡ 0.
10.4. Evaluate EZ4 t for the ARCH(1) process (10.3.10) with 0 p, βj 0 for j>q, and Ut Z2 t −ht
is white noise if EZ4 t < ∞.