Replicate Romer and Romer (2004) results I am trying to replicate figure 2 from Romer and Romer's (2004) paper on monetary shocks (http://eml.berkeley.edu/~dromer/papers/AER_September04.pdf). Essentially, having generated a series for monetary shocks, they run the following ADL regression:

$\Delta y_t=a_0+\sum_{k=1}^{11}a_kD_{kt}+\sum_{i=1}^{24}b_i \Delta y_{t-1}+\sum_{j=1}^{36}c_jS_{t-j}+e_t$

where $y$ is the log of industrial production, $S$ are monetary shocks and the $D_k$'s are monthly dummies. The say that the "estimated response of log output after one month is $c_1$, the coefficient on the first lag of $S$; the estimated response of log output after two months is $c_1+(c_2+b_1c_1)$; and so on."

My question is: how do I iterate this forward to produce figure 2, i.e., what is the formula for the response after three months etc?
Sorry for any confusion in my previous answer but there are 2 steps to this process 1) tracing out the impact on the $\Delta$y's and the lags of the shocks and then 2) accumulating the shocks to get the cumulative response (Note that my original answer tried to do this simultaneously). So in the first step (ignoring the intercept, the dummies and the residual terms) we get

$\tilde{\Delta y_1}$ : $c_1$

$\tilde{\Delta y_2}$: $c_2 + b_1c_1 = c_2 + b_1\tilde{\Delta y_1} $

$\tilde{\Delta y_3}$ : $c_3+ b_1(c_2 + b_1c_1) + b_2c1 = c_3 + b_1\tilde{\Delta y_2} + b_2 \tilde{\Delta y_1}$

$\tilde{\Delta y_4}$ : $c_4 + b_1\tilde{\Delta y_3} + b_2 \tilde{\Delta y_2} + b_3 \tilde{\Delta y_1}$

Can you see the pattern? So the general rule is 

$\tilde{\Delta y_i}$= $c_{i}+\sum_{r=0}^{i-1}b_{r}\tilde{\Delta y_{i-r}}$

When $\tilde{\Delta y_0}=c_0=b_0=0$ and where $c_i=0$ for $i&gt;36$ and $b_r=0$ for $r&gt;24$.

Now the next step to is to accumulate the shocks so that the IRF value for a given month is

$IRF_i = \sum^i_{j=0} \tilde{\Delta y_j}$

This should give you the results you are looking for. Technically if you were to trace out the exact values of the regression that you specified then you would also need to include the dummy variables but from the sounds of it they only care about tracing out the effects of the monetary shocks.