Input EGARCH model (idiosyncratic volatility)

I have a time-series of historical volatility observations. I want to use an EGARCH model because I believe it is a better representation of the behaviour of these volatilities. Can I estimate an EGARCH model using the observed volatilities without using the underlying returns? I'm using R and I think in the input the program expects returns instead of volatilities.

To be more specific, I'm trying to using the same methodology described in a paper about idiosyncratic volatility. To estimate it, the author run the following regression:

\begin{equation} r_t-rf_t=α_t+b1_t (rm_t-rf_t)+b2_t*SMB_t+b3_t*HML_t+ε_t. equation(1) \end{equation}

\begin{equation} ε_t\thicksim N(0,\sigma_t^2) \end{equation}

\begin{equation} ln⁡(\sigma_t^2)=w+\sum_{i=1}\beta_i ln⁡(\sigma_{t-i}^2)+ \sum_{i=1}c_i\Biggl[\theta \biggl( \frac{ε_{t-i}}{\sigma_{t-i}}\biggl)+\gamma\Biggl[ \left|\frac{ε_{t-i}}{\sigma_{t-i}}\right|-\sqrt{\frac{2}{\pi}} \Biggl] \Biggr] \end{equation}

(I didn't know how to put the sign over the summation; anyway it is from i=1 to p and i=1 to q.)

Idiosyncratic volatility is defined as the standard error of the residuals of the regression in equation(1).

I want to build an EGARCH to have a conditional idiosyncratic volatility. TO do this, I run the regression 1 and took the standard error of the residuals; this is the historical idiosyncratic volatility. Then, when I give this historical idiosyncratic volatilities as input to my program ( I use R and the package rugarch). Does it make sense this procedure or should I do something else? The problem is that in all the application that I view of EGARCH, the inputs are the returns but in my case, if I give returns as input, then I would have an EGARCH for the normal volatility and not the idiosyncratic, which is the one in which I am interested.

• Please give a link to the paper you mention. – Xi'an May 23 '15 at 13:03
• mysmu.edu/faculty/fjfu/default_files/JFE2009_IVOL.pdf pag.3:definition of idiosyncratic risk pag.4 to 6:Estimation of expected idiosyncratic volatility the paper title is:"Idiosyncratic risk and the cross-section of expected stock returns"(Fu,2009) – entusiast_student May 23 '15 at 14:40