I want to test whether cross-sectional dependence in one specific variable (y) in panel data format follows a known structure (W) (e.g. network, spatial dependence), after controlling for individual fixed effects and time fixed effects. The aim is simply to show that this structure is describing cross-sectional dependence in a statistically significant way.
My first idea was to look into the spatial econometrics literature. Here, W would be the spatial weight matrix. For purely cross-sectional data, Moran's I would probably be the relevant test but I am unsure how and whether this measure can be applied to panel data.
What would be the correct way of testing a known cross-sectional dependence structure on a panel of one variable, after controlling for individual fixed effects and time fixed effects? Any guidance and help would be very much appreciated! Thank you very much in advance.