Are Fama-Macbeth R-Squared (R2) just assymptotically correct?

I have been doing a research on comparing Fama-MacBeth and panel regression procedures. Think of it as an emerging market case for Petersen (2009) link. My research consists of a route based on full-sample betas of Fama-French (for 90 months) and another based on 30 rolling 60-month windows. I have used this page for finding the R^2 in both routes.

I have noticed that the R^2 from the two procedures are very close in the full-sample case (up to 13 decimals!). But the story is very different for the rolling case (discussed below) and this has made me think of adding this to that page I referred above:

The problem with FM R^2 is that since it is not the original regression procedure, using each of formulae of R^2 (using residuals or fitted values) would not give you the same result. Of course, as the frequency of the data increase, they will converge significantly but algebraically they will never be the same, simply because you are not feeding the whole data set to a unique regression (i.e a panel regression) and are just running separate regressions and averaging their results. I have checked this numerically for different scenarios. Just try it for as many as 5*5 portfolios and 30 periods (with periodically changing FF betas). You might even see a negative R^2 from one and a more-than-1 R^2 from the other!

The problem with FM R^2 is that since it is not the original regression procedure, using each of formulae of R^2 (using residuals or fitted values) would not give you the same result. Of course, as the frequency of the data increase, they will converge significantly but algebraically they will never be the same, simply because you are not feeding the whole data set to a unique regression (i.e a panel regression) and are just running separate regressions and averaging their results.

I have checked this numerically for different scenarios. Just try it for as many as 5*5 portfolios and 30 periods (with periodically changing FF betas). You might even see a negative R^2 from one and a more-than-1 R^2 from the other!

Do you agree with my comment? I guess this would further enrich the fruitful discussion done there.

Also, the more important issue is that the answers on the page referred above and this page are contradictory. In my opinion, the former is correct. What do you think? Obviously, it can’t be both.