<p>The impulse response function simply tells you the path that <span class="math-container">$Y_t$</span> follows if kicked by a single unit shock <span class="math-container">$\epsilon_t$</span>. I think that your notation is misleading. Consider the following:</p>
<p>If a shock (<span class="math-container">$\epsilon_t$</span>) occurs at time <span class="math-container">$t$</span> (say today), then <span class="math-container">$\frac{\partial y_t}{\partial \epsilon_t}$</span> denotes the impact multiplier, that is, how <span class="math-container">$y_t$</span> responds to the impact (immediate response) of that shock.
On the other hand, <span class="math-container">$\frac{\partial y_{t+j}}{\partial \epsilon_t}$</span>, with <span class="math-container">$j&gt;0$</span> and denoted as dynamic multiplier, tells you which is the effect of a shock occurred at time <span class="math-container">$t$</span> on the variable at time <span class="math-container">$y_{t+j}$</span>.</p>
<p>For instance, if you have quarterly data, and you simulate a shock <span class="math-container">$\epsilon_t$</span> today, and you want to see how <span class="math-container">$y_t$</span> responds to that shock over time, <span class="math-container">$\frac{\partial y_{t}}{\partial \epsilon_t}$</span> simply tells you today's response of <span class="math-container">$y_t$</span> to the simulated shock <span class="math-container">$\epsilon_t$</span>. Then, <span class="math-container">$\frac{\partial y_{t+1}}{\partial \epsilon_t}$</span> tells you what the persistence of today's shock is after 3 months, that is, how that shock still affects <span class="math-container">$y_t$</span> after 3 months.</p>

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2023-02-01T22:20:21.413Z
2023-02-05T21:48:07.050Z
2023-02-05T21:48:07.050Z
Tony
<p>When you output IRFs to a monetary shock using a VAR, are IRFs to one standard deviation monetary shock just differences between path of y with a one-time shock and and path of y without the shock?</p>
<p>For example, if the shock occurs at say <span class="math-container">$t=1$</span>, what does <span class="math-container">$\frac{\partial y_{t+1}}{\partial \epsilon_{t}}$</span> really mean?</p>
<p>So before the monetary shock, is there a path of <span class="math-container">$y_{t}$</span> t=1,2,3,...?</p>

<econometrics>
The Interpretation of the VAR Impulse Response Functions
EHMJ
# The Interpretation of the VAR Impulse Response Functions

(Asked by: username_0 on 2023-02-01)

When you output IRFs to a monetary shock using a VAR, are IRFs to one standard deviation monetary shock just differences between path of y with a one-time shock and and path of y without the shock?

For example, if the shock occurs at say $t=1$, what does $\frac{\partial y\_{t+1}}{\partial \epsilon\_{t}}$ really mean?

So before the monetary shock, is there a path of $y\_{t}$ t=1,2,3,...?

**Tags:** econometrics

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## 1 answers

### Answer by username_1 on 2023-02-01. Score: 1

The impulse response function simply tells you the path that $Y\_t$ follows if kicked by a single unit shock $\epsilon\_t$. I think that your notation is misleading. Consider the following:

If a shock ($\epsilon\_t$) occurs at time $t$ (say today), then $\frac{\partial y\_t}{\partial \epsilon\_t}$ denotes the impact multiplier, that is, how $y\_t$ responds to the impact (immediate response) of that shock.
On the other hand, $\frac{\partial y\_{t+j}}{\partial \epsilon\_t}$, with $j>0$ and denoted as dynamic multiplier, tells you which is the effect of a shock occurred at time $t$ on the variable at time $y\_{t+j}$.

For instance, if you have quarterly data, and you simulate a shock $\epsilon\_t$ today, and you want to see how $y\_t$ responds to that shock over time, $\frac{\partial y\_{t}}{\partial \epsilon\_t}$ simply tells you today's response of $y\_t$ to the simulated shock $\epsilon\_t$. Then, $\frac{\partial y\_{t+1}}{\partial \epsilon\_t}$ tells you what the persistence of today's shock is after 3 months, that is, how that shock still affects $y\_t$ after 3 months.


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