K-step ahead forecast for an AR(2) Given an AR(2) process $y_{t}=\phi_{1}y_{t-1}+\phi_2 y_{t-2}$, I understand that the two-step ahead forecast (that is, $\mathbb{E}[y_{t+2}|y_t]$ is given by $(\phi_{1}^2+\phi_{2})y_{t}+\phi_{1}\phi_{2}y_{t-1}$. However, I am unable to derive a general expression for the k-step ahead forecast (that is, for $\mathbb{E}[y_{t+k}|y_{t}]$. How might I go about doing this? Thank you.
Let's define $C_k$ such that:
$$
\mathbb{E}(y_{t+k}|y_t) = C_k y_t.
$$
Then:
$$
\begin{align*}
\mathbb{E}(y_{t+k+1}|y_t) &amp;= \phi_1 \mathbb{E}(y_{t+k}|y_t) + \phi_2 \mathbb{E}(y_{t+k-1}|y_t),\\
&amp;= \phi_1 C_k y_t + \phi_2 C_{k-1} y_{t},\\
&amp;= (\phi_1 C_k + \phi_2 C_{k-1}) y_t
\end{align*}
$$
This gives us the following recursive second order difference equation:
$$
C_{k+1} = \phi_1 C_k + \phi_2 C_{k-1}.
$$
Or, updated one period:
$$
C_{k+2}- \phi_1 C_{k+1} - \phi_2 C_k = 0.
$$
This is a second order homogeneous difference equation, which is of the form $C_k = \alpha a_1^k + \beta a_2^k$. Computing $a_1$ and $a_2$ can be done by solving the following characteristic function:
$$
(a^2- \phi_1 a - \phi_2) = 0
$$
This gives the following two roots:
$$
a_1 = \frac{\phi_1 + \sqrt{\phi_1^2 + 4 \phi_2}}{2} \text{ and } a_2 = \frac{\phi_1 - \sqrt{\phi_1^2 + 4 \phi_2}}{2}.
$$
So in general we have:
$$
C_k = \alpha a_1^k + \beta a_2^k
$$
To find $\alpha$ and $\beta$, we need two starting values, say $C_0 = 1$ and $C_1 = \phi_1$. In other words, we assume that $\mathbb{E}(y_t|y_t) = y_t$ and $\mathbb{E}(y_{t+1}|y_t) = \phi_1 y_t$.
This gives the following two conditions:
$$
\alpha + \beta = 1\\
\alpha \left(\frac{\phi_1 + \sqrt{\phi_1^2 + 4 \phi_2}}{2}\right) + \beta \left(\frac{\phi_1 - \sqrt{\phi_1^2 + 4 \phi_2}}{2}\right) = \phi_1.\\
$$
Solving this system in terms of $\alpha$ and $\beta$ gives:
$$
\beta = \frac{\sqrt{\phi_1^2 + 4 \phi_2} - \phi_1}{2 \sqrt{\phi_1^2 + 4 \phi_2}}\\
\alpha = \frac{\phi_1 + \sqrt{\phi_1^2 + 4 \phi_2} }{2 \sqrt{\phi_1^2 + 4 \phi_2}}
$$
This gives the following closed form expression condition:
$$
\begin{align*}
\mathbb{E}(y_{t+k}|y_t) &amp;= C_k y_t,\\
&amp;=\left[\begin{array}{ll} &amp;\frac{\phi_1 + \sqrt{\phi_1^2 + 4 \phi_2} }{2 \sqrt{\phi_1^2 + 4 \phi_2}}\left(\frac{\phi_1 + \sqrt{\phi_1^2 + 4 \phi_2}}{2}\right)^k\\
+ &amp;\frac{\sqrt{\phi_1^2 + 4 \phi_2} - \phi_1}{2 \sqrt{\phi_1^2 + 4 \phi_2}} \left(\frac{\phi_1 - \sqrt{\phi_1^2 + 4 \phi_2}}{2}\right)^k\end{array}\right] y_t
\end{align*}
$$