Solution 10

(a) \(y_t = \phi y_{t-2} + u_t - \theta u_{t-1}\). Stationarity \(|\phi|<1\), invertibility \(|\theta|<1\).

(b) \(\gamma_0 = \frac{1+\theta^2}{1-\phi^2}\sigma_u^2,\; \gamma_1 = -\frac{\theta}{1-\phi}\sigma_u^2,\; \gamma_2 = \phi\gamma_0\).

(c) For \(h>q\): \(\hat{Y}_{t+h} = \phi \hat{Y}_{t+h-2}\).

Second part (10 continued): \(Y_t = X_t - X_{t-1}\) with \(X_t = \phi X_{t-1}+Z_t\).

\(\gamma_Y(0)=\frac{2\sigma^2}{1+\phi},\; \gamma_Y(1)=\phi\gamma_Y(0)-\sigma^2,\; \gamma_Y(h)=\phi\gamma_Y(h-1)\) for \(h\ge2\).

Forecasts: \(\hat{Y}_{t+1}=\phi Y_t - Z_t\), and for \(h\ge2\), \(\hat{Y}_{t+h}=\phi^{h-1}\hat{Y}_{t+1}\).

Solution 11

1. \((1-L)(1-0.9L)Y_t = (1-L)(1-0.9L)(10+0.02t) + (1-L)\varepsilon_t\) ⇒ ARMA(1,1).

2. At \(t=80\): trend \(T_{80}=11.6\), observed \(y=11.2\) ⇒ \(x_{80}=-0.4\).
\(\hat{y}_{81}=11.26\), variance \(0.062\) ⇒ CI \(11.26\pm0.12\).

3. For stationary process, \(\lim_{h\to\infty} \mathbb{V}\text{ar}_t(x_{t+h}-\hat{x}_{t+h}) = \mathbb{V}\text{ar}(x_t)\).

4. Long horizon forecast: \(\hat{y}_{160}\approx T_{160}=13.2\); unconditional variance \(0.019\) ⇒ CI \(13.2\pm0.3\).

Solution 12

1) \(\hat{Y}_{t+1} = (1+\phi)Y_t - \phi Y_{t-1} + \theta\varepsilon_t\).
Recurrence for \(k\ge2\): \(\hat{Y}_{T+k} = (1+\phi)\hat{Y}_{T+k-1} - \phi Y_{T+k-2}\).

2) Characteristic roots \(1\) and \(\phi\).

3) \(\hat{Y}_{T+k} = A + B\phi^k\) determined by initial values.

4) Limit as \(k\to\infty\): \(A = Y_T + \frac{\phi}{1-\phi}(Y_T - Y_{T-1}) + \frac{\theta}{1-\phi}\varepsilon_T\).

Solution 13

(a) Trend \(T_t=33+1.90t\); at \(t=240\), \(T=489\), observed GDP 506 ⇒ above trend (favorable).

(b) ARIMA(0,1,1).

(c) Deterministic trend.

(d) Forecast: \(\mathbb{E}_{240}[PIB_{243}] = T_{243} - 0.61 u_{240} = 484.33\).
Forecast error variance \(\approx 338.4\) ⇒ CI \([448,521]\).

Solution 14

1. \(Y\) is not stationary because its mean \(\beta t + S_t\) depends on time.

2. \((1-B^4)Y_t = 4\beta + (1-B^4)U_t\). The autocovariance of \((1-B^4)U\) is a filter of \(U\).

3. SARIMA\((0,0,1)(0,1,1)_4\).

4. \((1-L^p)\).

5. By induction.