EXERC ICE 2
Soit le modèle $\mathrm{ARMA}(1,1)$ suivant: $Y_t=0,5Y_{t-1}+{\epsilon }_t+0,4{\epsilon }_{t-1}$. avec $\sigma {\epsilon }^2=3$
1. Ce modèle est-il stationnaire?
2. Ce modèle est-il inversible?
3. Calculer la suite ${\gamma }_0,{\gamma }_1,{\gamma }_2,\dots ,{\gamma }_5$
4. Calculer la suite $R_1,R_2,\dots ,R_5$.
5. Calculer les coefficients ${\psi }_1,{\psi }_2,\dots ,{\psi }_5$ du modèle $\mathrm{MA}(\mathrm{\infty })$ correspondants.
6. Calculer les coefficients ${\pi }_1,{\pi }_2,\dots ,{\pi }_5$ du modèle $AR(\mathrm{\infty })$ correspondants.

Step 1 ：1. Stationary: Check if $|0.5|<1$, as the condition for $Y_t=0.5Y_{t-1}+{\epsilon }_t+0.4{\epsilon }_{t-1}$ to be stationary.

Step 2 ：2. Invertible: Check if $|0.4|<1$, as the condition for $Y_t=0.5Y_{t-1}+{\epsilon }_t+0.4{\epsilon }_{t-1}$ to be invertible.

Step 3 ：3. Calculate ${\gamma }_k$ for $k=0$ to $k=5$ using the formula ${\gamma }_k={\sigma }^2\sum _{j=0}^{\mathrm{\infty }}{\psi }_j{\psi }_{j+k}$, where ${\sigma }^2=3$ and ${\psi }_j$ are the coefficients from the corresponding $MA(\mathrm{\infty })$ 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 $AR(\mathrm{\infty })$ 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 $MA(\mathrm{\infty })$ model.