<p>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</p>

<p>$\tilde{\Delta y_1}$ : $c_1$</p>

<p>$\tilde{\Delta y_2}$: $c_2 + b_1c_1 = c_2 + b_1\tilde{\Delta y_1} $</p>

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

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

<p>Can you see the pattern? So the general rule is </p>

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

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

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

<p>$IRF_i = \sum^i_{j=0} \tilde{\Delta y_j}$</p>

<p>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.</p>

CC BY-SA 3.0
2016-11-14T10:35:48.373Z
Hi Andrew, thanks so much for getting back to me. I think I get the intuition now. I've re-run RR's regression and get exactly the same results as in the paper. However, when I plot the impulse response based on the formula you kindly provided, the results are a bit off after about 12 months. I have uploaded an example to EmpireBox (http://expirebox.com/download/10e1930362e430866e0fcd5b861308be.html). The simplest way in Excel seemed to be: $month_i=month_{i-1}+c_i+(b_{i-1}*month_{i-1})$. I think this is the same as the general rule you provided. Any thoughts would be greatly appreciated.
ts_highbury
CC BY-SA 3.0
2016-11-14T15:10:47.880Z
Hi I fixed my answer to account for the fact that the shocks accumulate both through the lagged shocks terms and through the lagged $\Delta$y's. Hopefully this should do the trick although it will be less easier to do in excel.
Andrew M
CC BY-SA 3.0
2016-11-13T21:59:23.733Z
2016-11-14T15:40:49.080Z
2016-11-14T15:40:49.080Z
Andrew M
<p>I am trying to replicate figure 2 from Romer and Romer's (2004) paper on monetary shocks (<a href="http://eml.berkeley.edu/~dromer/papers/AER_September04.pdf" rel="nofollow">http://eml.berkeley.edu/~dromer/papers/AER_September04.pdf</a>). Essentially, having generated a series for monetary shocks, they run the following ADL regression:</p>

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

<p>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."</p>

<p>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?</p>

<macroeconomics><econometrics><time-series><empirical-evidence><autoregressive>
Replicate Romer and Romer (2004) results
ts_highbury
# Replicate Romer and Romer (2004) results

(Asked by: username_0 on 2016-07-30)

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?

**Tags:** macroeconomics, econometrics, time-series, empirical-evidence, autoregressive


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## 1 answers

### Answer by username_1 on 2016-11-13. Score: 3

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 $i>36$ and $b\_r=0$ for $r>24$.

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.

> **Comments:**
> **username_0**: Hi Andrew, thanks so much for getting back to me. I think I get the intuition now. I've re-run RR's regression and get exactly the same results as in the paper. However, when I plot the impulse response based on the formula you kindly provided, the results are a bit off after about 12 months. I have uploaded an example to EmpireBox (http://expirebox.com/download/10e1930362e430866e0fcd5b861308be.html). The simplest way in Excel seemed to be: $month_i=month_{i-1}+c_i+(b_{i-1}*month_{i-1})$. I think this is the same as the general rule you provided. Any thoughts would be greatly appreciated.  
> **username_1**: Hi I fixed my answer to account for the fact that the shocks accumulate both through the lagged shocks terms and through the lagged $\Delta$y's. Hopefully this should do the trick although it will be less easier to do in excel.  

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