This question is related to another question here. The question linked to asks for examples of factors commonly used in ICAPM models (intertemporal capital asset pricing model). What I want to know here is this: what is underlying principle that makes a factor a valid factor in factor models such as the CAPM, ICAPM, or APT?
This is a partial answer (see section 5.1, p. 79 of Cochrane's book, "Asset Pricing"). Suppose you wanted to treat the size of the firm as a factor. Stocks of smaller firms typically have higher average returns. You could then form a "large" holding company comprised of smaller firms. The new, large company would then have returns like a smaller company but would in fact be large and "the managers could enjoy the difference."
(How would I formalize this concept?)
EDIT: Also, in Financial Asset Pricing Theory by Munk (chapter 10, p.371) it says
(Emphasis added) This helps to understand the role of factors in the ICAPM.