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I am wondering if we can use the firm and industry fixed effects together in a regression (panel data).

In particular, in a regression, whether we should use such a code in STATA

reghdfe y x, a( firm industry)

where firm is the variable of firms and industry is the variable standing for industries.

I think we can use both fixed effects because it is like we control for the variables at firm and industry-levels that have not yet been in the equation. However, oppositely, I wonder if the industry fixed effect is redundant here because it may be overlapped by the firm fixed effects?

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To the extent that firms have only one unique industry, it is not necessary to include industry fixed effects in this regression. Running the regression with firm and industry fixed effects or firm fixed effects only will provide the same estimates for the coefficient of interest (on $x$). This is because, within firm, there is no variability in the variable industry.

It would be different if you worked with, say, worker-firm level data and wanted to include firm and occupation fixed effects. In this case, workers can have different occupations within firms and one can find workers of the same occupations in different firms. So including occupation fixed effects on top of firm fixed effects would matter.

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