18.2 Practice problems
Problem 18.2.1. In the local level model, find the steady-state gain when \(Q = R\), and when \(Q = 0.01R\). Interpret.
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Solution. Take \(R = 1\). With \(Q = 1\): \(P_\infty = (-1 + \sqrt 5)/2 = 0.618\) and \(K_\infty = 1.618/2.618 = 0.618\). With \(Q = 0.01\): \(P_\infty = (-0.01 + \sqrt {0.0401})/2 = 0.0951\) and \(K_\infty = 0.1051/1.1051 = 0.095\). When the level moves as much as the noise, the filter follows each new observation closely; when the level barely moves, it averages over many observations, like exponential smoothing with a small \(\alpha \).
Problem 18.2.2. A GARCH(1,1) model fitted to daily returns has \(\alpha _0 = 0.02\), \(\alpha _1 = 0.08\) and \(\beta _1 = 0.90\). Find the unconditional variance, and comment on the persistence of volatility.
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Solution. \(\Var (X_t) = 0.02/(1 - 0.98) = 1\). Since \(\alpha _1 + \beta _1 = 0.98\) is close to \(1\), a shock to volatility decays by only \(2\%\) a day: its half-life is \(\log 0.5/\log 0.98 \approx 34\) days. Turbulent periods last for weeks.
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