I am reading the paper and get the following question. I think here is how the regression is constructed:

  1. First step:
    $R_t^i = \alpha^i + \beta^i \cdot MarketBeta_t + \gamma_i\cdot Size_t + \nu \cdot Value_t$
    $t = 1....T, i = 1....N$

The first step, for each portfolio, regress the portfolio return $R_t$ on three factors, and get $\beta, \gamma, \nu$.

However, this portfolio is formed based on Size, Value. So, you would expect the loading to factors are pretty high.

  1. Second step: $E[R^i] = \beta^i \cdot RiskPremium1 + \gamma^i \cdot RiskPremium2 + \nu^i \cdot RiskPremium3$

with the second regression, you get the risk premium for each factor loading.

Here is what confuse me:
Using Value as an example, you form the portfolios based on Value, and find higher value has higher expected return. Then you regress portfolio returns on Value, of course the higher return has higher factor beta, and the higher factor beta has higher expected return.
Isn't the regression redundant? Isn't comparing the portfolio returns made the point?

  • $\begingroup$ @noob2 Please see Table IV in the 1992 paper $\endgroup$
    – JOHN
    Aug 23, 2018 at 19:59
  • $\begingroup$ OK, I'll delete my remark until I can check that Table. $\endgroup$
    – nbbo2
    Aug 23, 2018 at 20:05

1 Answer 1


No, looking at the returns of the value factor does not make the point. The second stage regression of returns on $\beta^i$, $\gamma^i$ and $\nu^i$ allows you to separate out whether the returns of the Value portfolio are due to a Value risk premium, or due to another confounding factor.

For example, it is quite possible that building the portfolios based on Value could be giving you a market exposure. Given that, we cannot be sure whether the higher return of the Value portfolio is due to the market or the Value factor. The second stage regression separates those effects.

  • $\begingroup$ I guess this would imply that we always need to have the market factor while performing the cross-sectional regression? Since we want to see if value exposure is obtained after conditioning on the market exposure. $\endgroup$ Sep 26, 2018 at 7:59

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