Estimating Idiosyncratic Volatility from Regression Residuals
Summary
The document explains how idiosyncratic volatility relates to a market-model regression. In that model, the stock return is decomposed into a systematic component explained by the market return and a residual component that the model does not explain. The response identifies idiosyncratic volatility with the dispersion of those residuals, typically measured by their standard deviation over the relevant estimation window.
This addresses the distinction between a time-series residual observation and a cross-sectional volatility measure: a single absolute residual is not itself a volatility estimate, while a standard deviation summarizes residual variation. The source gives no sample frequency, window length, annualization rule, or specific regression design. Those choices depend on the intended study, and the residual-based measure inherits the limitations of the market model and its estimation sample.
Key ideas
- Idiosyncratic volatility represents return variation unexplained by the chosen regressors.
- In a market model, the unexplained component is captured by the regression residuals.
- Estimate volatility from the standard deviation of residuals over a selected sample window.
- A single absolute residual measures one observation’s magnitude, not volatility across observations.
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# Unsystematic/Idiosyncratic/Firm-specific volatility/variance in the market model?
# 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 $ln(\frac{\sum_{i=1}^T \sqrt{e_{it}^2}}{T})$))
## Answer by user6157 (score 2)
https://quant.stackexchange.com/a/8991
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.