Thanks.
I recalled writing up slides on this for a prep provider six years ago, but I couldn’t find it. (I was looking in Equity, not in Portfolio Management.)
The structure of the models looks the same:
R_i = \alpha_i + \beta_{i,1}F_1 + \cdots + \beta_{i,K}F_K
where:
R_i is the return on asset i
F_k is factor k, k = 1, 2, \ldots, K
\beta_{i,k} is the sensitivity of return R_i to factor F_k
A macroeconomic model might have factors such as inflation, GDP growth, and interest rates. We use data for those factors and known returns on asset i to do a regression analysis and estimate the \beta_{i,k}s. So here the factor returns are data, and the sensitivities are estimated at the end.
A fundamental factor model might have factors such as P/E ratio, net profit margin, and so on. For each company i we compute a set of standardized sensitivities. For the P/E ratio, for example, the standardized sensitivity for company i is:
\beta_{i,P/E} = \frac{Company\ i's\ P/E - mean\ P/E}{\sigma_{P/E}}
Using these sensitivities and known returns, we do a regression analysis to estimate the factor returns: the F_ks. So here the sensitivities are data, and the factor returns are estimated at the end.