How Market-Return Choices Affect Fundamental Factor Regressions
Summary
The document discusses what belongs on the left-hand side of a fundamental factor model regression. It contrasts gross stock returns with returns in excess of the risk-free rate, then asks whether the average excess market return should also be removed before estimating factor premia. The key modeling concern is that constants may absorb known effects, especially when the regression includes many sector or country indicator variables.
The response recommends keeping market returns as a separate factor, in part because the definition of the market is ambiguous. Different broad equity indexes can have different size tilts, which may change estimated size-factor exposures and other coefficients. Including the market factor also lets the researcher inspect its relative weight and assess model fit using the available information. These are practical considerations rather than a universal specification rule: the document does not compare models empirically or prescribe one benchmark for all markets, universes, or research goals.
Key ideas
- A fundamental factor regression can use gross stock returns or returns in excess of the risk-free rate.
- Removing additional market returns changes what the intercept and factor coefficients represent.
- The choice of market benchmark can affect estimated exposures because broad indexes may have different tilts.
- Keeping market returns as a standalone factor helps reveal its weight and supports assessment of model fit.
- The discussion presents modeling considerations rather than an empirically tested universal rule.
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Full text
# Fundamental factor models: what to move to the LHS
# 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
## Answer by Chris (score 0)
https://quant.stackexchange.com/a/49592
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.