19.5 AR(2)
An AR(2) process is built up from:
\[Y_t=\delta +\phi_1 Y_{t-1} + \phi_2 Y_{t-2}+\omega_t\] where \(\omega_t\) is a white noise with zero mean and variance \(\sigma^2_\omega, \,\, \rightarrow \,\, \omega_t \sim N(0,\sigma^2_\omega.\)
When a stationary process \(\left(Y_{t}\right)\) follows an \(\mathrm{AR}(2),\)
\[ Y_{t}=\mu+\phi_{1} Y_{t-1}+\phi_{2} Y_{t-2}+\omega_{t} \] with \(\phi_{2}+\phi_{1}<1, \quad \phi_{2}-\phi_{1}<1, \quad\left|\phi_{2}\right|<1 \quad\) (see 2.3.1) \(\mathrm{y} \quad\left(\omega_{t}\right) \sim \operatorname{IID}\left(0, \sigma_{\omega}^{2}\right), \quad\) it can be verified that:
Mean: \[ \begin{array}{c} \mu_{Y}=\frac{\mu}{1-\phi_{1}-\phi_{2}} \\ \rho_{k}=\phi_{1} \rho_{k-1}+\phi_{2} \rho_{k-2}(k \geq 1) \end{array} \]
ACF: \[ \rho_{k}=\phi_{1} \rho_{k-1}+\phi_{2} \rho_{k-2}(k \geq 1) \]
PACF:
\[ \phi_{k k}=\left\{\begin{array}{ll} \phi_{1} /\left(1-\phi_{2}\right) & \text { si } k=1 \\ \phi_{2} & \text { si } k=2 \\ 0 & \text { for all } k>2 . \end{array}\right. \]
- Variance:
\[ \begin{aligned} \sigma_{Y}^{2} &=\frac{\sigma_{A}^{2}}{1-\phi_{1} \rho_{1}-\phi_{2} \rho_{2}} \\&=\left[\frac{1-\phi_{2}}{1+\phi_{2}}\right]\left[\frac{\sigma_{A}^{2}}{\left(1-\phi_{1}-\phi_{2}\right)\left(1+\phi_{1}-\phi_{2}\right)}\right] \end{aligned} \]
More detailes
If the process is stationary in mean and variance then
- \(E[Y_t] = E[Y_{t-1}]\) and
- $Var(Y_t) = Var(Y_{t-1}), t , $ de forma que:
\[E[Y_t] = E[Y_{t-1}]=\mu=\delta +\phi_1\mu+\phi_2\mu \rightarrow \mu=\frac{\delta}{1-\phi_1-\phi_2}\] and
\[Var(Y_t) = Var(Y_{t-1})=\gamma_0=\phi_1^2\gamma_0 + \phi_2^2\gamma_0+\sigma_\omega^2 \rightarrow \gamma_0=\frac{\sigma_\omega^2}{1-\phi_1^2-\phi_2^2}\]
where \(\phi_1+\phi_2 \neq 1.\)
Moreover:
\[ \begin{array}{rl} cov(Y_t,Y_{t-1})&=cov(Y_{t-1},Y_t)=E[(Y_t-\mu)(Y_{t-1}-\mu)]=E[y_t y_{t-1}]=\gamma_1 \\ \gamma_1&=E(y_{t-1} y_t)=E[y_{t-1} (\phi_1y_{t-1}+\phi_2y_{t-2}+\omega_t)]=\phi_1\gamma_0+\phi_2\gamma_1\\ \gamma_2&=E(y_{t-2} y_t)=E[y_{t-2} (\phi_1y_{t-1}+\phi_2y_{t-2}+\omega_t)]=\phi_1\gamma_1+\phi_2\gamma_0\\ \ldots\\ \gamma_k&=E(y_{t-k} y_t)=E[y_{t-k} (\phi_1y_{t-1}+\phi_2y_{t-2}+\omega_t)]=\phi_1\gamma_{k-1}+\phi_2\gamma_{k-2} \end{array} \]
where \(y_t=(Y_t-\mu).\)
