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Suppose I have a factor model

$$ \rho_i = \sum_J \beta_{iJ} \rho_J + \epsilon_i $$

where $\rho_i$ is the excess return of asset i over the risk free rate and $\rho_J$ is the excess return of the factor J portfolio. My J's are fixed and i's are all the possible stocks.

How would I go about verifying the model? Concretely, it is a linear regression problem and I can find out the $\beta_{iJ}$ and their statistical significance. If it were just one stock I were regressing against the statistical significance would tell me if the model is good. However, for multiple stocks what is the criteria? I am sure this can be answered using stats but I want to know if this is a well known result or should I sit down and compute it.

The motivation for this is that CAPM and Fama-French were based on some intuition for the factors (and that is how it should be) but in the age of cheap computation and AI one might find a factor and it will be useful to verify it across a large universe of stocks.

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Just finding factors based on regression is a poor idea. A statistically significant factor in all honesty may mean nothing. Read the fama French original paper, they were not just trying to find factors which explained risk( like in CAPM, beta is a measure of systematic risk), but they were trying to find out factors that provided a "risk premia". There are a ton of factors in the BARRA risk models which don't have any premia attached to them ( no profits for taking those risks) but they just act as a simple factor to decompose returns. If you want to find factors on your own, you'll have to backtest them using a high-low bucket long-short strategy and see if it works, but again it may only work by chance in your specific case, so you'll have to check in markets all over the world for consistency. Finding factors with risk premia is a hard task, but you can read tons of papers on already existing factors and modify them according to your specific need.

And don't get swayed by the idea of using machine learning to fit factors, mostly those factors will be shit and unless you can explain the economic intuition behind them, it'll mean nothing

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