# 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 $ln(\frac{\sum_{i=1}^T \sqrt{e_{it}^2}}{T})$))

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## 1 Answer

Idiosyncratic volatility is NOT included in the regressors, so it should not be and actually cannot be part of your matrix X. Idiosyncratic volatility is the volatility (of Y) your matrix X (explanatory variables) cannot explain (i.e. remaining unexplained part), so it is the error term of your regression equation.

Just compute the standard deviation of your residuals; it is what you are looking for.

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