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?