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I created an impulse response function from a VAR model. I used data transformed by taking the first difference of logarithms. I am now in trouble with giving a substantive interpretation of the scale of the impulse response function. How can I properly interpret it?

The IRF is an Orthogonal impulse response function, and an example is below:

enter image description here

As said here:

An IRF indicates what is the impact of an upward unanticipated one-unit change in the "impulse" variable on the "response" variable over the next several periods (typically 10).

In this case, data are differentiated. Thus, I would say that the values at each time point on the x-axis represent the difference between that time point's value and the previous one. For instance, the value at time $t3$ is the difference between the values at time $t3$ and $t2$ ($t3 - t2$). This difference is then logged.

Therefore, I would interpret the above chart, for example, as follows, starting from the first time period on the x-axis:

A one-log-unit increase from time $t-1$ to time $t0$ in the impulse variable $s1$, leads to a change of about 0.002 log-units in the response variable $s2$ from the time period $t-1$ to $t$.

The maximum effect can be seen at time $t3$, where one-log-unit increase from the time $t-1$ to $t0$ in the impulse variable $s1$, leads to about 0.005 log-units change in the response variable $s2$ from period $t2$ to $t3$.

I could then transform back the logged units by taking the exponential of the values. In this case, I would say, for instance:

The increase in the response $s2$ at time $t1$ is about 1 ($exp(0.002)$) for each 2.718282 units increase ($exp(1)$) in the impulse variable $s1$.

Is that correct?

There are similar questions on CrossValidated, like this one and this one, but they are still without answers.

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Your impulse response function graph shows time evolution of how s2 responds to a shock that originated in s1. A one-unit (one-standard-deviation of s1) change (shock) in your variable s1 at time t results in a change in your dependent variable s2 whose magnitude is shown on the y axis of your graph.

For example, your log differences s2 would go up by 0.002 at t=1 in response to a shock that originates in s1 whose magnitude is one standard deviation of s1.

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