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I am confused whether the factors in a multi-factor model should be orthogonal or not. Google searches do not give a well documented answer and I couldn't find one in our library's limited catalog either. Intuition says they should. Moreover, the covariance of Mkt-RF with SMB and HML (on yearly data as obtained fron K. French's data library) is about 116 and 34 respectively, far from 0.

What am I missing?

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    $\begingroup$ That's a debate that has been going on for decades in Statistics. Orthogonal factors are easier to deal with, and from non-orthogonal factors you could construct orthogonal factors mathematically (Gram Schmidt). However the constructed factors are no-longer intuitive. FF factors are linked to specific accounting variables (Book to market etc.) and were discovered in this form, orthogonal factors might not make as much sense, some people say. The debate goes on... When FF create their models I am sure they try to choose factors that are at least somewhat orthogonal. $\endgroup$ – noob2 Jan 5 '16 at 16:07
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    $\begingroup$ To overcome this issue, you can apply principal component analysis to the 3 factors and then use the PC as factors. $\endgroup$ – Malick Jan 5 '16 at 16:18
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A factor model has the form $$r_{j,t}=\sum_n \beta_{j,n} f_{n,t}+\epsilon_{j,t}$$ Where $r_{j,t}$ is the return of stock $j$ at time $t$, $\beta_{j,n}$ is the sensitivity (factor loading) of stock $j$ to factor $n$, $f_{n,t}$ is the return of factor $n$ at time $t$, and $\epsilon_{j,t}$ is the idiosyncratic non-factor return. One factor can be the constant.

There are three ways to specify and/or estimate:

  1. The classical Capm/ Fama-French where you explicitly specify the factor series $f_{n,t}$ and use time series regressions, one per stock, to estimate the betas $\beta_{j,n}$. There is no reason for the factor time series to be orthogonal, although it is useful as a risk decomposition if they are close to orthogonal (as factor variances become additive).

  2. The Barra approach where you explicitly specify the loadings $\beta_{j,n}$ and use cross-sectional regressions, one per date, to estimate the corresponding factor moves $f_{n,t}$. There is no reason, again, for these estimated moves to be orthogonal, but being close to orthogonal is, again, desirable.

  3. The black box PCA approach, where both factor loadings and time series are estimated simultaneously. Then we have time series that are orthogonal by construction (because we have too many degrees of freedom we put orthogonality as a constraint). However, they do not map directly to an intutive set of macro factors, although they often resemble them. Also different time windows would give rise to different factors, which might not be desirable.

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  • $\begingroup$ Thank you, is there some sort of reference for (at least) the first case, please? $\endgroup$ – Konstantinos Jan 6 '16 at 11:18
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Using covariance to imply an inappropriate level of multicollinearity in a model can be very misleading, especially when the factors are measured in differing units or lack linear relationships. There will almost always be some level of collinearity in a multi-factor model (otherwise you run the risk of overfitting), especially one with a relatively small amount of explanatory variables like the FF equation. Remember, the FF model was really just an improvement on the CAPM to give a better ex-post fit to stock returns.

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