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?

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

1 Answer 1


The difference lies not in the equations but in what they are used for. Whereas in linear regression X is an observed known value, in a linear factor model X is itself a random variable. The linear factor model is a statement about the joint distribution of X, Y and Z. Furthermore, linear factor models are often used to describe time series, for example it could relate a time series Y_t to another time series X_t that in turn is described by another stochastic process.

  • $\begingroup$ Hmm, I think I may be getting an inkling of what you are trying to say. Could you give me an example of a factor model like this that does not fall under regression? $\endgroup$
    – Coolio2654
    Commented Oct 27, 2021 at 2:12
  • $\begingroup$ Check this out. ocw.mit.edu/courses/… $\endgroup$
    – mlofton
    Commented Aug 12, 2023 at 14:18
  • $\begingroup$ Eric always writes quite clearly so this one might be a little bit more user friendly. faculty.washington.edu/ezivot/research/… $\endgroup$
    – mlofton
    Commented Aug 12, 2023 at 14:21

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