Annual data VS Monthly data VS Quarterly data for a VAR model I have read blogs posts that say one should use monthly, quarterly or annual data depending on whether you want to predict monthly, quarterly or annual outcome respectively.
So I guess the same applies to studying the dynamics of two or more variables using a VAR model. For example, the response of $y_t$ to a shock in $x_t$ should be expected to be different at monthly, quarterly and annual frequencies, right? It appears so in my experiment in the attached figure.
So saying that &quot;the response of $y_t$ to a shock in $x_t$ is positive&quot; is incomplete, I guess. Perhaps, it is better to say, &quot;the response of $y_t$ to a shock in $x_t$ is positive at annual frequency, negative at quarterly frequency, and so on&quot;.
Or it is that the results should be similar irrespective of the frequency of the data?
For example, the response of $y_t$ to a shock in $x_t$ should be expected to be different at monthly, quarterly and annual frequencies, right?

Generally (and usually), yes.

Or it is that the results should be similar irrespective of the frequency of the data?

Generally (and usually), no. The answer to your question depends on the underlying data generating process (DGP). You can take an example process and analyze it at different levels of time aggregation analytically to figure out implications for what happens when we look at it using different frequencies. E.g. take a monthly AR(1) process
$$
y_t=\varphi_1 y_{t-1}+\varepsilon_t
$$
and see how it behaves when aggregated to quarterly or annually:
$$
\sum_{i=1}^{3}y_{t-i+1}=f\left(\sum_{i=1}^{3}y_{t-3-i+1}\right)+u_t
$$
and
$$
\sum_{i=1}^{12}y_{t-i+1}=g\left(\sum_{i=1}^{12}y_{t-12-i+1}\right)+v_t
$$
where you need to figure out $f(\cdot)$ and $g(\cdot)$ based on the original AR(1) model.