# Understanding Fama Macbeth Regressions of Returns

I'm trying to understand what the Fama-Macbeth regressions of returns actually mean. The source of confusion is a 2013 Novy-Marx paper, in which he states the following:

"The first specification of Panel A shows that gross profitability has roughly the same power as book-to-market predicting the cross section of returns ... The second and third specifications replace gross profitability with earnings-to-book equity and free cash flow-to-book equity, respectively. These variables have much less power than gross profitability"

The corresponding table is here, with info highlighted. Up to this point, my understanding is that the highlighted numbers are the monthly (annual?) Fama-Macbeth regressions of returns.

He then goes on to state the following:

Appendix A.2 performs similar regressions employing alternative earnings variables. In particular, it considers earnings before interest, taxes, depreciation, and amortization (EBITDA)... These regressions show that EBITDA-to-assets [has] significant power predicting the cross section of returns, but that gross profits-to-assets subsumes [its] predictive powers

However, Table A2 (pic 2) shows a different story, with the value of EBITDA/assets greater than GP/assets.

Can someone explain where I'm having a gap in understanding? It seems that EBITDA/assets in his study results in stronger returns, yet his wording and explanation makes it seem the opposite. I am also not sure what the highlighted numbers even mean. Do they signify abnormal returns from certain variables? Or something totally different?

"The other side of value: The gross profitability premium" is the study in question.

First, you are right that monthly return data is used. Table 1 concludes, that a one unit increase of gross profitability accounts for an additional 0.75% stock return per month after controlling for book-to-market ratio, size, stock reversal ($$r_{1,0}$$) and momentum ($$r_{12,2}$$).

Table A2 is based on the following statement on p.17:

Table A2 shows results of Fama and MacBeth regressions employing gross profits-to-assets into EBITDA-to-assets and XSGA-to-assets. [...] The table shows that both variables have power explaining the cross section of average returns, either individually or jointly. [...] The table also shows that while XSGA-to-assets has no power to predict returns in regressions that include gross profits-to-assets, EBITDA-to-assets retains incremental power after controlling for gross profitability.

So in fact, Earnings before interest, taxes, depreciation, and amortization (EBITDA) is gross profits minus operating expenses, which largely consist of selling, general, and administrative expenses. Table A2 shows this decomposition, that EBITDA-to-assets drives profitability more than XSGA-to-assets does, but both are economically and statistically very significant drivers of the profitability anomaly.

The first specification (1) in Table A2 is the same is shown in Table 1. Specification (2) and (3) show the decomposition of gross profitability from specification (1) (which is EBITDA minus operating expenses. The latter mainly consists of selling, general, and administrative expenses (XSGA); all scaled by assets). The 0.75% increase in monthly stock returns can therefore be subsumed in an underlying increase of 1.42% (accounting for EBITDA-to-assets) and an underlying decrease of 0.65% (accounting for XSGA-to-assets). Both account for the shown gross-profitability effect from specification (1).

• Thank you for the clarification, hoping you can further clarify some things; I'm not a econometrics/stats expert so I really appreciate it! (1) Can you explain how you know the returns are monthly? I was not able to find a specification in the paper. (2) Would it be correct to say that the Fama-Macbeth returns are +1.42% for EBITDA and -0.65% for XSGA, thus resulting in +0.75% for GP? The 1.42% is the number I'm most interested in; (3) is this claiming that the Fama-Macbeth returns for EBITDA are almost double than those of GP? – user44007 Jan 14 at 4:53
• (1) It is the most common case in empirical asset pricing, that monthly return data is used if not otherwise mentioned. You may find a hint on p.3, where $r_1,0$ is defined as the "one month performance measure". (2) Yes, that is right. More precise (instead of "Fama/MacBeth returns) would be to say that a one-unit increase in EBITDA-to-assets results in a 1.42% increase of monthly stock returns, after controlling for book-to-market ratio, size and momentum. (3) In general yes, but EBITDA-to-assets is part and therefore subsumed by gross profitability (and the effect is lowered by XSGA) – skoestlmeier Jan 14 at 7:35
• Furthermore, welcome on Quantitative Finance SE! If you find the answer above helpful, please upvote/accept it. Otherwise comment/edit for further clarification. – skoestlmeier Jan 14 at 7:37
• Thanks again for the continued clarification. Some follow on questions: (1) What does "one-unit" increase mean? An increase by 1 standard deviation or an increase of literally 1 (for instance, GP/TA = 0.25 vs GP/TA = 1.25)? (2) The way the data is presented leads me to believe that the risk-adjusted returns to an EBITDA/TA sorted portfolio would significantly outpace that of GP/TA. Would you agree with this? – user44007 Jan 15 at 4:31
• (1) Yes, it means an increase of literally one. (2) Well, the regression results support your statement, which should not be very surprising because higher EBITDA/TA just ignores the costs of XSGA. Again, it is much more the "econ" part in econometrics, than the math part. The papers aims to find a more robust variable of "profitability". You are fine to ignore the cost and just look at EBITDA/TA. If the relationship between EBITDA/TA and monthly stock returns is assumed to be linear, yes, you would find higher excess-returns in corresponding portfolio sorts. – skoestlmeier Jan 15 at 9:07