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I am using panel data consisting of macroeconomic variables on country level and for the time period between 2009 and 2019. I have a balanced panel data set. I performed a Chow test, a Hausman test and a Lagrange Multiplier test to select the best panel data approach and it turns out to be fixed effects. Now, I am wondering how to formally determine if I should use only individual-fixed effects or only time-fixed effects or both. Is there a formal test to solve this issue?

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In this case, you should test for heteroskedasticity and autocorrelation primarily. One key reason to use any fixed effect model is to solve for heterogeneity, which is not exactly as heteroskedasticity, but close. Don't follow R2 or F-statistic to much, as two-fixed effect could win either way.

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