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 "the response of $$y_t$$ to a shock in $$x_t$$ is positive" is incomplete, I guess. Perhaps, it is better to say, "the response of $$y_t$$ to a shock in $$x_t$$ is positive at annual frequency, negative at quarterly frequency, and so on".

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

• how are the different models specified? The inclusion of different numbers of lags can really alter the results Jan 10 at 23:37
• Thanks!. Same model, same everything...only data changes. They are all 2-variable VAR(1). Jan 11 at 3:03
• Do you include seasonal dummies? Because you probably should for both the quarterly and monthly data. Also it is important to note that 100 steps ahead is not the same as 100 steps ahead for monthly data or quarterly data i.e. 100 years $\neq$ 100 quarters $\neq$ 100 months Jan 11 at 22:15
• Also 1 lag in annual terms would be similar to the 4th lag in quarterly terms of the 12th lag in monthly terms. So doesn't really make much sense to compare identical models. Either way none of the effects are significant either which suggests all your models are pretty poor. Therefore I wouldn't trust the signs on the impacts that you see in the figures. Jan 11 at 22:50
• But my question is should I expect similar results using quarterly, monthly, or annual data...after doing all the stuff you have suggested? I mean, is there some statistical proof or evidence that data frequency doesn't matter much for a VAR model if indeed you do all the necessary stuff...seasonality, lag length, etc, appropriately for each model. In others words, which frequency should I choose for my VAR model? Based on what? Based on which frequency I want to forcast? But I am not interested in forcast here, just capturing dynamic relationships. Does data frequency matter? Jan 12 at 23:26

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), 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.