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I was wondering about methods that could be used to compare different model specifications when dealing with panel data. In the case of cross-sectional data, a few applicable methods would be (in my opinion):

  • Comparing (adjusted) $R^2$
  • Comparing AIC or BIC
  • Comparing forecast accuracy

However, I am not sure if these measures are applicable in a panel data context, and I can only find measures that help you to decide between either a fixed effect (FE) or random effect (RE) specification (e.g. Hausman Test). I am concerned about the case in which we have different regressors such as the following:

Model 1: $wage_{it} = \alpha_i + educ_{it} + experience_{it} + city_{it} + \epsilon_{it}$

Model 2: $wage_{it} = \alpha_i +educ_{it} + experience_{it} + \epsilon_{it}$

Where $city_{it}$ indicates if individual $i$ lived in a city at time $t$, $\alpha_i$ denotes the individual specific effect and other variables are defined accordingly. I am not sure about the interpretation of the methods named above in this context, apart from comparing forecast accuracy. How would one go about comparing these models? (If need be, also assuming both models are FE/RE)

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