I am currently trying to formulate an empirical design that works as follows. In a first step I regress a variable $y_t$ on another variable $x_{t-1}$ and controls $Z_t$: $$ y_t=\beta_1 x_{t-1}+Z_t\gamma+\varepsilon_t. $$ Then I test for Granger causality. I estimate these either in rolling window or with time-varying coefficients, so that for $t>T$ I get one coefficient per time period. I then create a vector $\xi_t$ of 1's and 0's, where a 1 means that there is Granger-causality at time $t$.
Then, in a second stage, I regress $\xi_t$ on a set of variables $W_t$: $$ \xi_t=W\delta+u_t. $$ The aim is to assess whether variation in $W_t$ can explain the Granger-causality patterns in $\xi_t$.
In addition, because I am working with firms, I can get a vector of Granger-causality dummies for each firm, $\xi_{i,t}$. Hence, I stack these vectors in a single vector and run the regression described above with time-fixed effects. The aim here is to see whether the variation in $W_t$ can explain cross-firm asymmetries in Granger-causalities.
Now, intuitively, this makes sense to me. However, I never saw this being done anywhere, which makes me suspicious about it. My questions:
- Does this sound plain wrong and why?
- Do you know of any similar methods?