15.4 Practice problems
Problem 15.4.1. Write the airline model \(\nabla \nabla _{12}X_t = (1 + \theta B)(1 + \Theta B^{12})Z_t\) as an equation for \(X_t\).
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Solution. \(\nabla \nabla _{12}X_t = X_t - X_{t-1} - X_{t-12} + X_{t-13}\), so \(X_t = X_{t-1} + X_{t-12} - X_{t-13} + Z_t + \theta Z_{t-1} + \Theta Z_{t-12} + \theta \Theta Z_{t-13}\).
Problem 15.4.2. For the airline model with \(\theta = -0.4\) and \(\Theta = -0.6\), compute the non-zero autocorrelations of \(W_t\).
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Solution. \(\rho (1) = -0.4/1.16 = -0.345\), \(\rho (12) = -0.6/1.36 = -0.441\), and \(\rho (11) = \rho (13) = (-0.345)(-0.441) = 0.152\) (Figure 13, panel 1).
Problem 15.4.3. The sample ac.f. of a seasonally differenced monthly series (\(n = 144\)) has \(r_1 = 0.45\), \(r_2 = 0.21\), \(r_3 = 0.08\), \(r_{12} = -0.38\), and every other value up to lag \(36\) is within \(\pm 0.17\). Suggest a model.
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Solution. At the low lags the ac.f. decays (\(0.45\), \(0.21\), \(0.08\)), suggesting a non-seasonal AR(1) (check the partial ac.f.). At the seasonal lags there is a single spike at \(12\) and nothing at \(24\) or \(36\): a seasonal MA(1). So try SARIMA \((1, 0, 0) \times (0, 1, 1)_{12}\), the model of Example 15.2.3, and check its residuals.
Problem 15.4.4. For the model of Example 15.2.3 with \(\alpha = 0.5\) and \(\theta = -0.6\), the data give \(x_n = 120\), \(x_{n-11} = 105\), \(x_{n-12} = 110\) and the residual \(z_{n-11} = 3\). Find \(\hat x(n, 1)\).
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Solution. \(\hat x(n, 1) = x_{n-11} + \alpha (x_n - x_{n-12}) + \theta z_{n-11} = 105 + 0.5(10) - 0.6(3) = 105 + 5 - 1.8 = 108.2\). The forecast starts from the value a year earlier and adjusts it by half of the latest year-on-year change, less part of last year’s surprise.
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