The following example is from the book Active Portfolio Management by Grinold and Kahn. Suppose we have the factor returns and want to estimate the exposures/factor loadings. Say the factor returns are returns on the value-weighted NYSE, gold, a government bond index, a basket of foreign currencies, and a basket of traded commodities. The authors mention that to estimate the exposures, we can set the exposure of each stock to the NYSE equal to 1, and then run a regression to estimate the exposures to other factors. I don't understand why we should set the exposure to NYSE to 1. Why can't we just run a regression of the stock return on all the factor returns and take the coefficients as the exposures?
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The proposal in the book reduces the degrees of freedom of the problem by one versus your proposal. Seeing as you are only interested in the the relative exposure allocation, it makes sense to reduce the problem's degrees of freedom by 1.
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$\begingroup$ Why are we only interested in the relative exposures? $\endgroup$ Jul 8, 2021 at 12:59
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$\begingroup$ Investing 100% of allocated funds in the strategy? $\endgroup$ Jul 8, 2021 at 14:02