Solution. With \(\Var (Z_t) = \sigma ^2\), \[ \gamma (0) = (1 + \beta ^2)\sigma ^2 , \qquad \gamma (1) = \beta \sigma ^2 , \qquad \gamma (\tau ) = 0 \ (\tau \ge 2) , \] the last because \(X_t\) and \(X_{t+\tau }\) then share no shock. Dividing, \[ \rho (1) = \frac {\beta }{1 + \beta ^2} , \qquad \rho (\tau ) = 0 \ (\tau \ge 2) . \] Differentiating, \(\rho (1)\) is extremised at \(\beta = \pm 1\), giving \[ |\rho (1)| \le \tfrac 12 . \] So a sample first autocorrelation beyond \(0.5\) in magnitude rules out an MA(1) outright, whatever \(\beta \) is chosen: a genuine identification constraint, not merely a guideline (Theorem 6.2.1).

Note also that \(\beta \) and \(1/\beta \) give the same \(\rho (1)\): the autocorrelations do not determine the model, which is why invertibility is imposed, choosing \(|\beta | < 1\), so that the root of \(\theta (B)\) lies outside the unit circle.

Solution. Stationarity requires the roots of \(1 - 0.5z - 0.3z^2 = 0\) to lie outside the unit circle. Solving, \[ z = \frac {0.5 \pm \sqrt {0.25 + 1.2}}{-0.6} = \frac {0.5 \pm 1.204}{-0.6} , \] giving \(z \approx -2.84\) and \(z \approx 1.17\). Both exceed one in modulus, so the process is stationary. (Equivalently, \(\alpha _1 + \alpha _2 = 0.8 < 1\), \(\alpha _2 - \alpha _1 = -0.2 < 1\) and \(\alpha _2 > -1\): the point is inside the triangle of Theorem 8.3.1.)

The Yule–Walker equations give \begin {align*} \rho (1) &= \frac {\alpha _1}{1 - \alpha _2} = \frac {0.5}{0.7} = 0.714 , \\ \rho (2) &= \alpha _1\rho (1) + \alpha _2 = 0.5(0.714) + 0.3 = 0.657 . \end {align*}

The slow decay is characteristic of an AR process: the ac.f. tails off rather than cutting off, and it is the partial ac.f. that would cut off after lag two.

Solution. A trend is removed by first differencing, \(\nabla X_t = X_t - X_{t-1}\). Yearly seasonality in monthly data is removed by seasonal differencing at lag twelve, \(\nabla _{12}X_t = X_t - X_{t-12}\). Applying both, \[ \nabla \nabla _{12}X_t = (X_t - X_{t-1}) - (X_{t-12} - X_{t-13}) . \] Fitting an ARMA model to the result gives a SARIMA \((p, 1, q) \times (P, 1, Q)_{12}\) model (Section 15).

Two cautions. Differencing more than necessary inflates the variance and introduces artificial negative autocorrelation (Theorem 10.2.1), so one difference of each kind is usually enough and the ac.f. should be inspected before applying a second. And a deterministic trend is better removed by regression than by differencing: the two are different models, and differencing a deterministic trend leaves a non-invertible moving-average term behind.

Solution. Uncorrelated residuals mean the model has captured the conditional mean correctly, so the ARMA orders need no revision. Correlated squared residuals mean the conditional variance is not constant (large errors follow large errors), which violates the constant-variance assumption of the ARMA model.

The remedy is to add a variance model: fit an ARCH or GARCH term to the residuals, so the mean follows the ARMA structure and the variance follows the volatility structure (Section 18). Ignoring it does not bias the point forecasts, but every interval built from the model will be wrong: too wide in quiet periods and far too narrow in turbulent ones, which is precisely when the interval matters.