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."
What ruins this promising idea? The "large" holding company will still behave like a portfolio of small stocks---it will have their high betas. Thus, only if asset returns depend on how you behave, not who you are---on betas rather than characteristics---can a market equilibrium survive such simple repackaging schemes.
(How would I formalize this concept?)
EDIT: Also, in Financial Asset Pricing Theory by Munk (chapter 10, p.371) it says
It is worth emphasizing that the general theoretical results of the consumption-based asset pricing framework are not challenged by factor models. ... Factor models do not invalidate the consumption-based asset pricing framework but are special cases that may be easier to apply and test. Therefore factors should generally help explain typical indiviudals' marginal utilities of consumption.
(Emphasis added) This helps to understand the role of factors in the ICAPM.