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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,...?

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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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