# Unsystematic/Idiosyncratic/Firm-specific volatility/variance in the market model?

I was asked to use idiosyncratic volatility as a regressor in a cross-sectional regression upon cross-sectional returns as the dependent variable. Returns can be thought of as the raw log stock return over some event. So,

$returns_i = a_i + b_i*X_i + error_i$, where $X$ is the matrix of regressors and idiosyncratic volatility one of the them.

Note that the market model is: $R_{it} = a_i + b_i R_{mt} + e_{it}$.

How do I calculate this? I see papers that use it only in a time series context, i.e. I've seen $I.V._{it} = \sqrt{e_{it}^2}$. But I can't use this, I need a cross-sectional variable.

There's also this Quant.SE thread here but my supervisor asked me to specifically use $e_{it}$. Is $\frac{\sum_{i=1}^T \sqrt{e_{it}^2}}{T}$ wrong?

((Note: I've just found the Pacific-Basin Finance Journal paper "Idiosyncratic volatility, fundamentals, and institutional herding: Evidence from the Japanese stock market" to define it as $ln(\frac{\sum_{i=1}^T \sqrt{e_{it}^2}}{T})$))