Derive the condition of stationarity in AR(2) process.
The AR(2) process is defined as \[\begin{array}{c} \left(1-\phi_{1} B-\phi_{2} B^{2}\right) W_{t}=e_{t} \\ W_{t}-\phi_{1} W_{t-1}-\phi_{2} W_{t-2}=\epsilon_{t} \\ W_{t}-e_{t}=\phi_{1} W_{t-1}+\phi_{2} W_{t-2} \\ F_{t}=\phi_{1} W_{t-1}+\phi_{2} W_{t-2} \end{array}\]
where \(W_t\) is a stationary time series, \(e_t\) is the white noise error term, and \(F_t\) is the forecasting function.
The process defined in (1) can be written in the form \[\begin{array}{c} \left(1-\phi_{1} B-\phi_{2} B^{2}\right) W_{t}=e_{t} \\ W_{t}=\left(1+\psi_{1} B+\psi_{2} B^{2}+\psi_{3} B^{3}+\ldots\right) e_{t} \\ W_{t}=\Psi(B)_{t+} \end{array}\] and therefore, \[ \begin{array}{c} \left(1-\phi_{1} B-\phi_{2} B^{2}\right)^{\cdot 1}=\Psi(B)=\left(1+\psi_{1} B+\psi_{2} B^{2}+\psi_{3} B^{3}+\ldots\right.) \\ \left(1-\phi_{1} B-\phi_{2} B^{2}\right)\left(1+\psi_{1} B+\psi_{2} B^{2}+\psi_{3} B^{3}+\ldots\right)=1 \end{array} \]
Now, for (2) to be valid, it easily follows that
\(y_{1}-\phi_{1}=0 \Leftrightarrow \psi_{1}=\phi_{1}\) and that \(\psi_{2}-\phi_{1} \psi_{1}-\phi_{2}=0 \Leftrightarrow \psi_{2}=\phi_{1}^{2}+\phi_{2}\) and that \(\psi_{3}-\phi_{1} \psi_{2}-\phi_{2} \psi_{1}=0 \Leftrightarrow \psi_{3}=\phi_{1}^{3}+2 \phi_{1} \phi_{2}\) and finally that \(\text{for all ,} i>3 : \\ \psi_{i}=\phi_{1} \psi_{i .1}+\phi_{2} \psi_{i .2}\)
The model is stationasy if the \(y_t\) weight converges.This is the case when some conditions of \(f_1\) and \(f_2\) are imposed. These conditions can be found on using the solution of polynomial of the AR(2) process. The so called characteristics equation is used to find these solutions.
\[\left(1-\phi_{1} B-\phi_{2} B^{2}\right)=\left(1-\xi_{1} B\right)\left(1-\xi_{2} B\right)=0\]
The solution of \(X_1\) and \(X_2\) are
\[\xi_{1}, \xi_{2}=\frac{\phi_{1} \pm \sqrt{\phi_{1}^{2}+4 \phi_{2}}}{2}\]
which can be either real or complex. Notice that the roots are complex if \(\phi_{1}^{2}+4 \phi_{2}<0\).
When these conditions, in absoulte value, are smaller than 1, the AR(2) model is stationary