# Idiosyncratic volatility forecast using EGARCH (Fu, 2009)

I have a time series of historical volatility observations. I want to use an EGARCH model because I believe it is a good 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 follow the methodology described in a paper about idiosyncratic volatility. To estimate it, the author runs the following regression:

\begin{aligned} r_t − r^f_t &= \alpha_t + b_{1,t}(r^m_t−r^f_t) + b_{2,t}SMB_t + b_{3,t}HML_t + \varepsilon_t \ \ \ \ \ \ \ \text{(1)} \\ \varepsilon_t &∼ N(0,\sigma^2_t) \\ \ln(\sigma^2_t) &= \omega + \Sigma_{i=1}^p\beta_i \ln(\sigma^2_{t−i}) + \Sigma_{i=1}^qc_i \left[\theta(\varepsilon_{t−i}\sigma_{t−i}) + \gamma\left( ∣\varepsilon_{t−i}\sigma_{t−i}∣−\sqrt{2\pi}\right)\right] \end{aligned}

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,

1. I run the regression (1) and took the standard error of the residuals; this is the historical idiosyncratic volatility.
2. Then I supply these historical idiosyncratic volatilities as input to my program (I use R and the package "rugarch").

Does this procedure make sense, 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.

• I tried to make the formulas appear OK using $\LaTeX$ typesetting, but you would have to check very carefully whether they are correct. – Richard Hardy Jul 23 '16 at 14:48
• Thanks you Richard, I was wondering if you offer private consulting service. I'm struggling with my code. – fran Aug 4 '16 at 20:01
• Sorry, I don't provide private consulting. – Richard Hardy Feb 20 '17 at 15:04
• I was going through my old answers and noticed this one was not accepted. Do you perhaps need further clarification? – Richard Hardy Feb 20 '17 at 15:05

Let me give you an answer that is a bit more explicit and general than you directly need, but that could nevertheless be useful.

• The true conditional variance $\sigma_t^2$ of a time series is unobservable, just as any other theoretical moment of the data generating process (DGP). $\sigma_t^2$ is a property of the DGP that we try to estimate.

• GARCH models (including EGARCH) assume $\sigma_t^2$ follows a deterministic process where $\sigma_t^2$ is completely determined by the past values of itself ($\sigma_{t-1}^2,\dotsc$) and the past values of shocks ($\varepsilon_{t-1}^2,\dotsc$).
Shocks may or may not coincide with returns depending on the conditional mean specification for the model for returns.
An outcome of an estimated GARCH model will be a series of fitted conditional variances $\hat\sigma_t^2$.

• If you have a series of realized volatilities $\tilde\sigma_t^2$, you should probably not treat them as perfect estimates of $\sigma_t^2$. Realized volatility is probably conditional variance measured with error, so $\tilde\sigma_t^2 \neq \sigma_t^2$.

You are right that the expected input is returns (if the conditional mean model is empty). Therefore, you would not substitute $\varepsilon_t$ or $\varepsilon_t^2$ with $\tilde\sigma_t^2$ directly. Instead, you would have to code the model estimation (an optimization problem) yourself.
If you assume the true DGP is EGARCH and treat realized cond. variances $\tilde\sigma_t^2$ as estimates of the true cond. variances $\sigma_t^2$, then you could specify the optimization problem corresponding to an EGARCH model that uses $\tilde\sigma_t^2$ in place of $\varepsilon_{t-1}$ as inputs. That would be something like errors-in-variables models where the regressand and the regressors (lagged realized cond. variances) would be measured with error.