While this may be awkwardly-titled, I hope that my question becomes clearer upon reading.
So this is what I gather about Factor Models: they are statistical models set up to explain the returns, ${return}_i$ of a portfolio, based on certain security characteristics (or factors) plugged in as the independent variables, after which the resulting coefficients will be used as the factors' weightings in the realized portfolio.
However, what I do not understand is why the factors are not used in and of themselves, such as in this equation, from page 4 of this article (Clarke, DeSilva, Thorley: Pure Factor Portfolios and Multivariate Regression Analysis, JPM 2017) on factor investing:
$${return}_i = r_M + (r_1 - r_M)b1_i + (r_2 - r_M)b2_i + \ldots + \epsilon_i$$
($r_M$ stands for a "benchmark portfolio return", $r_i - r_M$ together stands for some "one-period return to a pure factor portfolio minus the benchmark return", and $b$ stands for some "factor exposure")
Why isn't this equation instead set up as like a normal linear model? $${return}_i = intercept + \beta_1{factor}_i + \beta_2{factor}_2 + \ldots + \epsilon_i \ \ ?$$
In general, I am quite confused about this set-up, and would extremely appreciate someone taking me step-by-step through the logic of this regression.