Lag selection and Granger causality: is the maximum lag the causal lag?

Question

When setting up the Granger causality (Wald) test, I determine the optimum lag using information criteria AIC, BIC, FPE to test for causality.

1. What does the lag selection really mean?
2. Does the selected lag describe the delay of the Granger cause effect? That is, if I select $\text{lag}=6$, does that mean that $X$ Granger-causes $Y$ with a delay of 6 time units (say, weeks)?

Answer 1

Lag selection is done before and independently of testing for Granger causality. Lag selection is about obtaining a "good" model, where "good" could have different meanings, e.g. efficient in forecasting (as due to AIC) or consistently selected (as due to BIC).

Given a selected model, you then test for Granger causality. That addresses the question, does knowing the history of $X$ help predict $Y$ beyond knowing the history of $Y$ itself. For that, any lag (or lag combination) of $X$ from 1 to the maximum lag could be important, not necessarily the maximum lag. The test involves assessing the significance of the contribution of all lags of $X$ in the equation for $Y$ jointly. Therefore, the following is not true:

...if I select $\text{lag}=6$, <...> $X$ Granger causes $Y$ with a delay of 6 time units (say, weeks)?