How to calculate impulse responses for a given autoregressive process? Is there a possibility to use a recurrent equation to calculate the impulse responses for an AR(p) process?
$$Y_t = \rho_1Y_{t-1} + \ldots +  \rho_pY_{t-p} + e_t $$
It is quite a theoretical question, but I do not know how to start with the AR(1) process.
 A: Elaborating on Martin's answer, you will want to compare coefficients in the general AR(p) case.
First, write the AR(p) process in compact lag operator notation, using $\rho(L)=1-\rho_1L-\ldots-\rho_pL^p$. 
We have from $\rho(L)Y_t=\epsilon_t$ and the lag operator statement of an $MA(\infty)$ process, $Y_t=\psi(L)\epsilon_{t}$, that
$$\rho(L)Y_t=\rho(L)\psi(L)\epsilon_t=\epsilon_t$$
Hence, $\rho(L)\psi(L)=1$. Now the two polynomials $\rho(L)\psi(L)$ and 1 (the latter of order 0) are identical if and only if the coefficients of each power are identical.
Example for an $AR(2)$:
We obtain
$$
(1-\rho_1L-\rho_2L^2)(\psi_0+\psi_1L+\psi_2L^2+\psi_3L^3+\ldots)=1
$$
Matching powers of $L$ yields
\begin{align*}
\psi_0&=1\\
\psi_1-\rho_1\psi_0&=0\Rightarrow\psi_1=\rho_1\\
-\rho_2\psi_0-\rho_1\psi_1+\psi_2&=0\Rightarrow\psi_2=\rho_2+\rho_1^2\\
\ldots&
\end{align*}
A: Not sure if your equation is correct. I guess you meant
$$
y_t = \rho_1 y_{t-1} + \dots + \rho_p y_{t-p} + \epsilon_t
$$
In case of a AR(1) process you have to cast it into its MA($\infty$) (or 'covariance stationary') representation by reinserting the past observations $y_{t-j}$ where $j=1,\dots,\infty$:
$$
y_t = c + \rho_1 y_{t-1} + \epsilon_t .
$$
With $y_{t-1} = c + \rho_1 y_{t-2} + \epsilon_{t-1}$ it follows that
$$
y_t = c + \rho_1 (c + \rho_1 y_{t-2} + \epsilon_{t-1}) + \epsilon_t.
$$
If you do that infinitely you end up with 
$$
y_t = \frac{c}{1-\rho} + \sum_{j=0}^{\infty} \psi_j \epsilon_{t-j} \quad \text{with} \quad \psi_j = \rho^j
$$
where we used that $\sum_{j=0}^{\infty}\rho^j c = \frac{c}{1-\rho}$ and the condition that $\rho < 1$.
The impsule-responses can now be calculated by 
$$
\frac{\partial y_{t+j}}{\partial \epsilon_t} = \psi_j.
$$
You basically have to do the same for higher order AR(p) processes. There you can compute the $\psi_j$'s by 'comparing coefficients'. An important condition for an $AR(p)$ process to have a covariance stationary representation is that its roots all lie outside the unit circle (thats why we assumed that $\rho < 1$).
I hope this rudimentary info helps you...
