# 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.

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.

• Thank you Skoestimeier sir. 1. In some research papers Idiosyncratic volatility= standard deviation of residuals x square root of trading days. It means that while calculating standard deviation in numerator there is squared demeaned residuals and divide by n − 1. So is this correct if I will use this formula. Denominator is same as you mentioned “ Frequently, researchers will omit the subtraction of k from the denominator of the calculation, or simply use k=1” My question in first post is still unanswered – Priya Dec 11 '19 at 8:06
• Ok, it seems like i misunderstood your question. 1.: Your are fine to assume $k=1$ if you accept that $RSE$ represents an unbiased estimate of the residuals standard deviation. 2.: I think you are asking for a minimum limit of $n$, i.e. the number of observations. Well, the more the better, but these answers from Cross Validated 1 and 2 suggest a minimum of 10 observations, so applying the Fama/French three factor model (with a loss of four degrees of freedom) leads us to a min. of $n =10+4=14$. – skoestlmeier Dec 12 '19 at 17:33