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My question is simple: what is the best practice in moving known variables to the LHS of Fundamental Factor Model regression?

I am seeing different approaches. $R_{it}=\alpha_i + \beta_{i,1} f_{1,t}+ \beta_{i,2}f_{2,t}+ \dots + \beta_{i,k}f_{k,t} + \epsilon_{i,t} \quad \forall i = 1, \dots, N$

Where $R_{it}$ are the GROSS stock returns and $\beta_{i,1}$ are the factors premia. the factors loadings are normalized.

Now, the constant $\alpha_i$ in this regression should capture the risk free rate and the market premium (or the market returns) and with that in mind many practitioners (most) deduct it so that this what they run as a regression to determine crosssectionally the size of the premia:

$ER_{it}=\alpha_i + \beta_{i,1} f_{1,t}+ \beta_{i,2}f_{2,t}+ \dots + \beta_{i,k}f_{k,t} + \epsilon_{i,t} \quad \forall i = 1, \dots, N$

where: $ER_{it}= R_{it} - Rf_{it}$

That makes sense to me, now why stop there? why not also deduct the average excess return of the market? (that i see many less practioners doing)

And in big picture, what are the implications of leaving known variables to be captured as 'constants'. This particularly worries me in a context with a lot of dummies to neutralize for sector, country, etc.

Thanks

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At a minimum, because there's more ambiguity in 'market return'. What constitutes the market? In the US, LC or SC? For LC, S&P 500 or Russell 1000? And even though they're both meant to represent broad-based LC US equities, S&P skews larger than the R1, so the R1 will show a tilt to small cap, slightly impacting your size factor exposures if you include one.

Perhaps obviously, this will impact your coefficient estimates as well of goodness-of-fit stats. By leaving the market returns as a stand alone factor, you can at least see the relative weighting of various risk factors and assess quality of your model with all available information.

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