12.4 Practice problems
Problem 12.4.1. A series of \(n = 200\) observations follows an AR(1) process with \(\alpha = 0.8\) and standard deviation \(\sigma = 5\). Find the approximate standard error of \(\bar x\) and the effective sample size.
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Solution. By Theorem 12.1.1, \(\Var (\bar X) \approx (25/200)(1.8/0.2) = 0.125 \times 9 = 1.125\), so \(\operatorname {se}(\bar x) = 1.06\), against \(5/\sqrt {200} = 0.35\) for independent data. The effective sample size is \(200/9 = 22\).
Problem 12.4.2. For a series of \(n = 150\) observations with \(c_0 = 2.5\), \(r_1 = 0.394\) and \(r_2 = -0.183\) (the series of Example 11.2.2), find the Yule–Walker estimates of an AR(2) model, including \(\hat \sigma _Z^2\).
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Solution. As in Example 11.2.2, \(\hat \alpha _1 = 0.552\) and \(\hat \alpha _2 = -0.400\). Then \(\hat \sigma _Z^2 = 2.5(1 - 0.552 \times 0.394 - (-0.400)(-0.183)) = 2.5(1 - 0.2175 - 0.0732) = 2.5(0.7093) = 1.77\).
Problem 12.4.3. An AR(1) model fitted to \(n = 400\) observations gave \(\hat \alpha = 0.9\). Find an approximate \(95\%\) confidence interval for \(\alpha \). Is there evidence against a random walk?
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Solution. \(\operatorname {se} \approx \sqrt {(1 - 0.81)/400} = 0.0218\), so the interval is \(0.9 \pm 0.043\), from \(0.857\) to \(0.943\), which excludes \(1\). (Near \(\alpha = 1\) the normal approximation is unreliable, and a unit-root test such as Dickey–Fuller is the proper tool; but an estimate this far below \(1\) with \(n = 400\) is strong evidence of stationarity.)
Problem 12.4.4. For the MA(1) data of Example 12.3.1, compute \(S(0.4)\) by the recursion.
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Solution. \(z_t = x_t - 0.4z_{t-1}\): \(0, 0, -2, -1 + 0.8 = -0.2, -2 + 0.08 = -1.92, 0 + 0.768 = 0.768, 2 - 0.3072 = 1.6928, 2 - 0.6771 = 1.3229\). The squares add to \(4 + 0.04 + 3.6864 + 0.5898 + 2.8656 + 1.7501 = 12.93\), as in the table.
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