I got a large n (400) and large T (300) panel. I am interested in identifying the causal relation between X and Y. Both vary along n and T, so no treatment and control group, no event approach. Basically 400 time series. Any suggestions are appreciated. PVAR is what I am currently looking at, I also consider splitting T into smaller time periods, as the effect might not be time-invariant.
I am afraid that your question is not clear enough. What are X and Y? By PVAR you mean panel VAR, I suppose. I have seen several messages on RG about causality in panel data recently, also with GARCH effects. Did you have a look at them? I am unfortunately not an expert on panel data but well on time-dependent ARMA and VARMA models. But I am still struggling with 2 or 3 variables, so I fear it will not help you.
Guy Mélard thanks a lot for the answer and yes, I should have been clearer. X is just an independent variable (e.g. patents) and Y is the dependent variable (e.g. GDP). And yes, pVAR = panel vector autoregression. I will take a look at GARCH, but didn't find anything useful with respect to causality in panels (other than when a treatment and/or treatment group is available).
You can use a DH-test, the test and its implementation in Stata is described here:
https://www.unine.ch/files/live/sites/irene/files/shared/documents/Publications/Working%20papers/2017/WP17-03_V2.pdf
Other software implementations are available as well.
If you'd like to build a regression model, perhaps, a dynamic panel data model would be applicable here, where the Arellano–Bond estimator may be used to estimate its parameters. If you use non-negative data (patents, GDP), most likely, you'll need to transform variables. For count data try sqrt, for GDP - logs.
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