# Linear Factor Model vs. Linear Regression Model

I've been reading some literature that discusses 'linear factor models' which appear to describe the general equation often used in OLS regression. When people refer to a 'linear regression model' are they essentially just referring to a linear factor model? Where does the term linear factor model fit in in statistics and why, if at all, is it necessary to make the distinction?

Until now, I've only heard of linear regression models(LRM) as opposed to linear factor models(LFM). It looks like these are interchangeable terms, though different uses of the word 'factor' can be misleading here.

Here's two links calling the same generic form of the model by different names:

Regression: http://en.wikipedia.org/wiki/Linear_model

From briefly searching the web, it looks like 'factor models' are used to describe economic and financial areas. Factor, here, means an independent variable in the model. An additional factor means another column (or row) in the model's matrix representation. Factors can be considered continuous, numeric variables since the Stanford article lists macro-economic variables and returns on portfolios as examples of factors down the page.

On the other hand, programming in the statistics language R, a factor variable or factor (also known as categorical variable) can only assume a limited set of different values. Like the months of the year or simply the set {1,2}.

Finally, as of this writing, both meanings of 'factor' are currently used in academic papers.

• An LFM contains noise (is probabilistic), while a simple LRM does not. See also deeplearningbook.org/contents/linear_factors.html: "A linear factor model is deﬁned by the use of a stochastic linear decoder function that generates x by adding noise to a linear transformation of h." Jan 7, 2018 at 13:50