Interpretation of within R2 when there are multiple FE

[Chengmou Lei]

Hi all,

I have a question related to within R2.

With the help of this post , together with other source, it is helpful to get me started with models with one dimension FE. For instance, if I add firm FE (firm is my panel variable), then within R2 tells me how much variation within a firm is explained by my model.
However, I wonder what is the interpretation when I have multiple FE? Say I have a panel where my panel var is city-establishment, my variation is at establishment-time level, if I add city-establishment FE and city-time FE, is the interpretation of within R2:
1) how much variation within a city-establishment cell is explained by my model?
2) how much variation within a city-time cell is explained by my model?
3) how much variation within a city-establishment-time cell is explained by my model?

BTW, is there any statistical explanation about the interaction FE that I can refer to?

Thank you in advance

[Andrew Musau]

The within-R2 is the R2 statistic from the regression of the demeaned variables. See the documentation of xtreg for the definition. So no matter how many dimensions you use to demean, the definition does not change. The act of demeaning can be viewed as purging the fixed effects. Therefore, how much variation in the outcome is explained by variation in the RHS variables after purging the fixed effects.