Why does $\hat{\epsilon}'\hat{\epsilon}$ of a factor model measure risk?

Question

$\hat{\epsilon}'\hat{\epsilon}$ from the market model: $R_{it} - \hat{\alpha} - \hat{\beta}R_{mt} = \hat{\epsilon}$, or from a factor model such as the Fama-French 3 factor model, is often used in the literature to capture the idiosyncratic risk of stock $i$.

What risk is this measuring? Who cares if the disturbances are high variance? All this means is that you've excluded relevant factors from your specification of the returns generating process, right? Or does its interpretation as "risk" come from an a priori assumption that the returns generating process has been correctly specified?

Answer 1

I think one should look at the problem from two different angles to get an answer to this.

Firstly, you can look (as you said you did) look at $\hat{\epsilon}$ in terms of a disturbance like you said, meaning the returns $R_{it}$ are depending linearly on the $R_{mt}$ - the market or factor returns. Then you can figure there is some regression involved an the theory of linear regression assumes the model like you stated it above where $\hat{\epsilon}$ is some disturbance with a normal distribution with mean $0$. So in order to find your true parameters $\hat{\alpha}$ and $\hat{\beta}$ you take a look at the disturbed data and fit a line through it so that the vector of the remaining disturbances (residuals) is minimized with respect to its sum of squares ($\ell^2$-norm). So the more your returns $R_i$ resemble your market returns $R_m$, the smaller the disturbances are according to your model.

Secondly, you can look at the problem from a more practical viewpoint. We say that the asset returns $R_{it}$ are some returns of a markets assets. Take a stock which is a constituent of a stock index with the stock index returns being $R_{mt}$. Now one wants to know which part of the variance corresponds to the market risk and which part of the variance corresponds to the stocks individual properties (idiosyncratic risk caused by earnings quality, debt ratios or whatelse you can think of - just not your other factors ;-) ). Since one often assumes market risk and idiosyncratic risk to be uncorrelated you can decompose the stocks variance: $$ \sigma_{i}^2 = \sigma_{m}^2 + \sigma_{id}^2 $$ where $\sigma_{id}^2=\hat{\epsilon}^\prime\hat{\epsilon}$. The more the $R_i$'s resemble your market ($R_m$'s), the smaller the idiosyncratic risk will be. One speaks of the idiosyncratic risk as being diversified away when this happens.