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I have a question for estimating factor return. I’ve found that there seems to be 2 methods for estimating factor return.

First, with return of an asset i(r_i) and factor loadings such as PER, EPS, Momentum etc(B_i), factor return can be estimated by doing regression - cross sectional regression

Second, sorting assets into deciles according to a factor value, and by taking long on the 1st decile and shorting on 10th decile, we can also get factor return.

I think ultimate goals of both methods are same - estimating the return I can expect when I expose myself into a certain factor, the factor return. But what is the difference between them? Are they catching different aspects of factor return? If I can get a factor return just by building Long-Short Portfolio, what is the need of doing a cross sectional regression?

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You are confusing two different things. Let's say you have a factor that you identified, call it: $\lambda_t$. There are also other factors out there that are widely know. Let me call them: $F_t$ (potentially a vector of factors).

Now the first thing you mention is:

  1. Run a time-series regressions on the factors $F_t$ and $\lambda_t$. Then run a cross-section regression of the loadings on those factors. This will get you the factor risk-premium for $\lambda_t$.
  2. The second thing you mention which is a portfolio sort, does not give you a factor risk-premium. But will give you an $\alpha$.
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  • $\begingroup$ Thank you for your answer, but I'm not sure if I understood the answers correctly. 1. Run time-series regression on factor F and lambda and get 'B' matrix for all assets and for all timesteps. -> Run cross-sectional regression on the loadings of those factors and get 'Factor Return' matrix for all timesteps. -> With 2 estimated matrix - 'B' and 'Factor Return', calculate risk premium of those factors F and lambda by simply solving r=Bf+s. $\endgroup$
    – geonhwa
    Commented Sep 6, 2022 at 23:16
  • $\begingroup$ My question is why do we have to do time-series regression first? Can't we just get factor return matrix by making long-short portfolio and then directly go into cross-section regression? And may I know the reason why you used the widely know factor F and new factor lambda together? Can't we just use the lambda factor alone to get risk premium (r=Bf+s) of lambda? $\endgroup$
    – geonhwa
    Commented Sep 6, 2022 at 23:16
  • $\begingroup$ @geonhwa - The underlying theory is that cross-sectional average returns are driven by each asset's $\beta$ (aka covariance/correlation with) the factors in question. As such - your cross sectional regression should be factor $\beta$ on the right-hand side, and average return on the left-hand side. $\endgroup$ Commented Sep 9, 2022 at 14:14
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    $\begingroup$ We don't know the values of these $\beta$s (which are so called "population parameters"), so we need to estimate them - the most common method being Time-Series regression of asset return onto the factor. $\endgroup$ Commented Sep 9, 2022 at 14:16

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