# 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:

$$r_t-rf_t=α_t+b1_t (rm_t-rf_t)+b2_t*SMB_t+b3_t*HML_t+ε_t. equation(1)$$

$$ε_t\thicksim N(0,\sigma_t^2)$$

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

(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

In ugarchspec method function, put a matrix of 3 factors in external regressors in the mean equation and in variance equation choose garch order c(1,1) as below. Then you can use returns as input and the resulting volatility will be idiosyncratic.
sp1<-ugarchspec(variance.model = list(model = "eGARCH", garchOrder = c(1, 1),submodel = NULL, external.regressors = NULL, variance.targeting = T),mean.model = list(armaOrder = c(0,0), include.mean = T, archex = FALSE,external.regressors = as.matrix(ts[,1:3])))