Assume I have data at the country ($c$)-sector ($s$)- firm ($i$) level. I can introduce separated country and sector fixed effects, or I can introduce country-sector fixed effects, as shown in the two equations below.

$$Y_{csi}=X_{csi}\beta+\alpha_{c}+\lambda_{s} + \epsilon_{csi}$$

$$Y_{csi}=X_{csi}\beta+\theta_{cs} + \epsilon_{csi}$$

I would welcome the chance to understand what assumptions are made in each case, and hence understand why they are different. The second model seems more restricticve, but I cannot clearly explain why.




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