# The Interpretation of the VAR Impulse Response Functions

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,...?

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.