# ARMA+EGARCH Model with Negative Omega Parameter

I am using R (version 3.2.5) to model the log-returns on a financial time series. I used the modelfit function from the "rugarch" package to fit an ARMA(0,1)+EGARCH(1,1) with a constant in the conditional mean equation and Normal errors model to the said log-returns.

The modelfit function carries out the log-likelihood maximization on the log of the conditional variance of EGARCH in simultaneous ARMA+EGARCH model estimation. This has two consequences. Firstly, maximization of log-variance instead of variance automatically ensures that the conditional variance of the hybrid model is positive. Secondly, no restrictions are required on the signs of the parameters of the EGARCH variance equation which generally assures a faster and more reliable optimization procedure.

Upon inspection of the signs of the optimal parameters, I found that omega is negative. The plot of the News Impact Curve (NIC) shows a curve which declines the greater positive and negative errors are.

So my questions are:

1. Is there any previous literature evidence of a hybrid ARMA+EGARCH model which displays a negative omega parameter?
2. From a model practical interpretability perspective, is it more reliable to refit the above model by imposing constraints on positiveness omega?
• Model with constant mu... as opposed to nonconstant mu? Also, have you tried looking for references yourself? Did you only find papers with positive omega? How many have you checked? Also, is there now a function modelfit? I cannot find it in the package documentation. Sep 10, 2016 at 9:11
• I just fitted models with constant mu. Do you think refitting them without mu could help me overcome my problem? Then, I read R documentation and other academic papers applying ARMA+GARCH models to financial and nonfinancial time series for fitting and forecasting purposes. But this is the very first case I came across a negative omega. And I do not how to deal with this. Sep 10, 2016 at 9:15
• Did you fit models with constant mu (where there was a choice to have nonconstant mu) or did you fit models that include a constant called mu? For simple GARCH models you cannot have negative omega, of course. But have you checked studies specifically using EGARCH? Sep 10, 2016 at 9:16
• Yes, I did. See for instance "Modelling and Forecasting of Price Volatility: An Application of GARCH and EGARCH Models" by Lama et al. (2015). There the authors apply various ARMA+EGARCH models. But of course omega is always positive. Sep 10, 2016 at 9:20
• As for the constant mu, I mean the constant of the ARMA model equation. I know that it is an option to include it or not in estimation. I found that hybrid models displaying mu perform slightly better in terms of AIC than those which do not. Sep 10, 2016 at 9:26

I don't think there is a problem with negative $\omega$. The conditional variance equation of the EGARCH model as used in "rugarch" is*
$$\sigma_t^2 = \exp (\omega+\dotsc).$$
If we neglect the terms in $\dotsc$ for the moment, we have $\omega<0$ if $\sigma_t^2<e^0=1$. By scaling your data you could always get there. And if the conditional variance is roughly constant and there is little variation around the unconditionl variance $\sigma^2$, neglecting the term in $\dotsc$ can be justified. This is just one example; you could also get negative $\omega$ in other settings if you play with the parameter values of the conditional variance equation.
Negative $\omega$ values are also encountered in applications, see e.g. Chen & Kuan (2002), Table IX.