EXERC ICE 2 Soit le modèle \( \operatorname{ARMA}(1,1) \) suivant: \( Y_{t}=0,5 Y_{t-1}+\varepsilon_{t}+0,4 \varepsilon_{t-1} \). avec \( \sigma \varepsilon^{2}=3 \) 1. Ce modèle est-il stationnaire? 2. Ce modèle est-il inversible? 3. Calculer la suite \( \gamma_{0}, \gamma_{1}, \gamma_{2}, \ldots, \gamma_{5} \) 4. Calculer la suite \( R_{1}, R_{2}, \ldots, R_{5} \). 5. Calculer les coefficients \( \psi_{1}, \psi_{2}, \ldots, \psi_{5} \) du modèle \( \mathrm{MA}(\infty) \) correspondants. 6. Calculer les coefficients \( \pi_{1}, \pi_{2}, \ldots, \pi_{5} \) du modèle \( A R(\infty) \) correspondants.

Step 1 ：1. Stationary: Check if \( |0.5| < 1 \), as the condition for \( Y_{t} = 0.5 Y_{t-1} + \varepsilon_{t} + 0.4\varepsilon_{t-1} \) to be stationary.

Step 2 ：2. Invertible: Check if \( |0.4| < 1 \), as the condition for \( Y_{t} = 0.5 Y_{t-1} + \varepsilon_{t} + 0.4\varepsilon_{t-1} \) to be invertible.

Step 3 ：3. Calculate \( \gamma_{k} \) for \( k=0 \) to \( k=5 \) using the formula \( \gamma_{k} = \sigma^2 \sum_{j=0}^{\infty} \psi_{j} \psi_{j+k} \), where \( \sigma^2 = 3 \) and \( \psi_{j} \) are the coefficients from the corresponding \( MA(\infty) \) model.

Step 4 ：4. Calculate \( R_{k} \) for \( k=1 \) to \( k=5 \) using the formula \( R_{k} = \frac{\gamma_{k}}{\gamma_{0}} \).

Step 5 ：5. Calculate \( \psi_{j} \) for \( j=1 \) to \( j=5 \) using the formula \( \psi_{j} = \frac{\pi_{j-1} + 0.5\pi_{j-2}}{1 - 0.5^2} \), where \( \pi_{j} \) are the coefficients from the corresponding \( AR(\infty) \) model.

Step 6 ：6. Calculate \( \pi_{j} \) for \( j=1 \) to \( j=5 \) using the formula \( \pi_{j} = \frac{\psi_{j-1} + 0.4\psi_{j-2}}{1 - 0.4^2} \), where \( \psi_{j} \) are the coefficients from the corresponding \( MA(\infty) \) model.