# Intuition: What is the difference between linear factor models and regular linear regression?

So, I have a very vexing theoretical question that I hope some experienced econometrician can help me with. Being in finance, I have recently been exposed to linear factor models, which are models that help predict an asset's returns, Y, by some chosen independent variables, X: $$Y = a + \beta \, X + Z$$ And I think "Wow, this is simple enough, just linear regression, and finance thinks these are so advanced?", but then, it turns out that they are completely different types of models, because supposedly they are 1) non-parametric, and 2) can incorporate non-linearities (despite looking very much like a linear equation).

The closest references I can find to both facts are here, and here.

Despite having some advanced experience in statistics, the intuition behind these models completely eludes me. The difference between them and OLS cannot be either

1. The distributions of the errors, Z, since that is what Generalized Linear Models (GLMs) are, or
2. The structure of the equation, since it seems identical to regular OLS.

Could someone please explain what fundamentally separates linear factor models from regular regression?

• I edited your tags. The question is not related to factor analysis or pca Commented Oct 27, 2021 at 9:26