# Fundamental factor models: what to move to the LHS

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