Estimating Idiosyncratic Volatility from Fama–French Residuals
Summary
The note describes estimating a stock’s idiosyncratic volatility from a time-series regression of its periodic excess returns on the Fama–French market, size, and value factors. The regression includes an intercept and three factor loadings. Its residual standard error is calculated from the sum of squared residuals divided by the observation count minus the number of estimated parameters; for this model, that count is four.
To express the residual volatility annually, multiply the standard error by the square root of the number of return periods in a year, and convert decimal returns to percent if desired. The answer explains that some studies omit the parameter adjustment or use a different denominator convention, which changes the estimate. It does not establish a universal minimum sample size: the degrees-of-freedom adjustment describes the calculation, while the required observations depend on the research design and estimation needs.
Key ideas
- The three-factor regression estimates an intercept and three factor loadings.
- Residual standard error uses the observation count less the number of estimated parameters in its denominator.
- Annualize periodic residual volatility by multiplying by the square root of periods per year.
- Researchers may use denominator conventions that omit or alter the parameter adjustment.
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# Minimum degree of freedom required for Fama french three factor model
# Minimum degree of freedom required for Fama french three factor model
I want to run Fama/French three factor model each month on daily returns for each securities as I want to calculate idiosyncratic volatility with the help of residuals. It means there are four parameters, i.e. intercept and three betas of risk factors.
My question is that how many minimum degree of freedom is require in this case? In some research papers I found that authors used 17 observation means 13 degree of freedom, I do not understand why they had used only 17 no of observations.
## Answer by skoestlmeier (score 1)
https://quant.stackexchange.com/a/50207
Idiosyncratic volatility is measured as the residual standrad error from a time-series regression of periodic excess stock returns on the returns of factor-mimicking portfolios.
#### Preliminary
Using the Fama/French three-factor model, you run the regression
$$r_{i,t} = \alpha_{i}+\beta_{MKT,i}MKT_{t}+\beta_{SMB,i}SMB_{t}+\beta_{HML,i}HML_{t}+\epsilon_{i,t}$$
where $r_{i,t}$ is the excess return of stock $i$ during period $t$, and $MKT_{t}$, $SMB_{t}$ and $HML_{t}$ are the period $t$ returns of the market, size, and book-to-market factors, respectively.
The residual standard error $RSE$ from the regression above is then calculated as $$RSE_i = \sqrt{\frac{\sum_{j=1}^n{\epsilon_{i,j}^2}}{n-k}}$$ where $n$ is the number of data points that are used to fit the regression and $k$ is the number of parameters estimated by the regression.
#### Degree of Freedom
When the Fama/French three-factor model is used, there are four parameters estimated by the regression ($\alpha_i$, $\beta_{MKT,i}$, $\beta_{SMB,i}$ and $\beta_{HML,i}$), and thus in this case $k=4$. When the CAPM model is used, there are two parameters ($k=2$), and when the Fama/French/Carhart Model is used, there are five parameters ($k=5$).
Frequently, researchers will omit the subtraction of $k$ from the denominator of the calculation, or simply use $k=1$, which statistically assumes that the parameter estimates are exact, and therefore that $RSE$ represents an unbiased estimate of the standard deviation of the residuals.
#### Measure Idiosyncratic Volatility
Idiosyncratic volatility is then calculated by multiplying the residual standard error by $\sqrt{m}$ (with $m$ as the number of return periods in a year) so that it represents an annualized value. If the periodic excess returns used in the regression are represented in decimal form, the annualized residual standard error is frequently the multiplied by 100 so that idiosyncratic volatility ($IdioVol_i$) is measured in percent:
$$IdioVol_i = 100 \cdot RSE_i \cdot \sqrt{m}$$
#### Reference
Bali/Engle/Murray (2016), Empirical Asset Pricing: The cross-section of stock returns, Wiley.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.