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.

• Dear @Matthew Gunn, as per your earlier contribution, your answer is highly appreciated here. – M.Ba Feb 14 '19 at 22:14
• What is your concern about the coefficient of determination? Because you are not fitting to the central tendency one could easily expect $R^2$ to be negative. A portfolio that is far from the norm is not being priced by central measures. Could you explain the concern with the value of the coefficient of determination? – Dave Harris Feb 15 '19 at 15:29
• Dear @DaveHarris, as you can see in the two empirical examples mentioned at the end of Mitch Petersen's highly-praised 2009 paper that we are concerned about FM R2s, just as much as we are about that of the panel regression. Please correct me if I am missing anything here, but I have also read John Cochrane's Asset Pricing (2005) which elaborates on FM in its 12th chapter. – M.Ba Feb 16 '19 at 20:28
• is your question about the asymptotic properties of the FM coefficient of determination versus something like the small sample properties or are you concerned with the specifics of the coefficients in the partitions? I am trying to figure out how not to answer the wrong question. I don't necessarily think there is a contradiction. Frequentist decision rules, as opposed to Bayes actions, are algorithms conditioned on models. A slight change in the rule for construction can completely change the decision rule. This isn't intrinsically true for a Bayes action. What is the concern ? – Dave Harris Feb 18 '19 at 21:57