# How to deal with negative ARCH terms?

Lately I have been trying to fit a GJR-GARCH(1,1) model to fit against the S&P 500 returns over 1985-2015 but I have ran into some problems I can't quite figure out. The GJR-GARCH(1,1) model I am trying to run is specified as follows: \begin{align} &R_{t} = \mu + \eta_t \\ &\eta_t = \sigma_{t-1} \epsilon_t, \epsilon_t \sim (0,\sigma^{2}_{\epsilon}) \nonumber \\ \sigma^{2}_{t} &= \alpha_0 + \alpha_1\eta^{2}_{t} + \beta_1 \sigma^{2}_{t-1} + \gamma_1 \eta^{2}_{t} I_{\eta < 0}(\eta_{t}) \end{align}

However, the parameter $\alpha_1$ appears to be negative (-0.058767) and also statistically insignificant (p value of 0.3952), whereas in a normal GARCH(1,1) model the ARCH parameter does not seem to have this problem. It seems to me that the leverage parameter $\gamma_1$ is affecting $\alpha_1$ in a "bad" way. My question now is how do I deal with the $\alpha_1$ parameter? the things I can come up with are:

1. Remove the $\alpha_1$ entirely from the model, as it is statistically insignificant. However, I don't think it can be done that easily...
2. Stick to the GARCH(1,1) model where $\alpha_1$ is statistically significant and also positive and do not deal with leverage effects.

But yet I can't find a conclusive answer to this problem. Thanks.

• You should run a fitting scheme where you can put bounds on your parameters. In such a constrained optimization you might find an $\alpha_1$ which is positive and statistically significant. – Olaf Mar 22 '16 at 14:58
• Unfortunately the statistical software I am using (Eviews) does not allow me to put bounds on $\alpha_1$. – Stephan Mar 22 '16 at 18:09
• @Stephan Many GARCH programs impose constraints on GARCH coefficients. Sometimes they will transform parameters in order to ensure above 0. You could try call R or Matlab versions of these functions from Eviews. – John Mar 22 '16 at 21:29

Before fitting GJR-GARCH model, first ensure that volatility exhibit sign bias. If there is no sign bias (only ARCH effect), then there is no need of fitting GJR-GARCH model. Also look at this answer: The test for misspecification of GARCH model.

If your data has sign bias and parameter of GJR-GARCH model is coming out to be negative, then you can put bound on your parameters as pointed by @Olaf. But such option is not available in E-Views. As an alternative you can use E-GARCH model (E GARCH is available in E-Views). E-GARCH model also considers sign bias in the volatility and at the same time excludes the possibility of negative volatility, irrespective of sing of parameters of the model.

• I've been looking at a few other related problems where also wanted to put constraints on eviews. It seemed that indeed the EGarch approach was used a lot to counter the non-negativity constraint. Finally, I was adviced by someone else to use data with a higher frequency, since in practice he said that data with a low frequency using the GJR model "almost always" exhibits a negative ARCH term. – Stephan Mar 23 '16 at 14:41

I would look at the significance of the overall GARCH model compared to the GJR model. Use whichever provides a better fit.

• Thank you for your reply. However, I don't quite understand why you stated that it doesn't matter if the intercept is insignificant. The parameter in question is the ARCH $\alpha_1$ parameter and not the $\alpha_0$. If the ARCH parameter in the GJR model would be insignificant, would this imply that I should use the standard GARCH(1,1) instead since the model is badly specified? – Stephan Mar 22 '16 at 19:45
• You're right. I clearly did not read the post closely enough. The last two sentences still hold, so I kept those. If the variable is insignificant, I'd try to remove it if possible. It shouldn't have an impact on the overall significance of the model - that part of my original post still holds as well. I will note, though, that I am not familiar with the GJR model outside of knowing what it is. My comments are strictly based on my understanding of statistics and time series as a whole. – RandyF Mar 22 '16 at 19:57
• I would recommend instead of the significance, to look at something like the AIC. Even if the ARCH to is not significant, it still may make sense to fit the GJR-GARCH model. – John Mar 22 '16 at 21:25
• Taking a look at your suggestion, I found that the normal GARCH(1,1) had an AIC value of -3.538903 and the GJR GARCH(1,1) model had an AIC value of -3.538184. On the basis of that, I wouldn't say the GJR model fits that much better than the first model, meaning that I am somewhat tempted to still choose the normal GARCH model. – Stephan Mar 22 '16 at 23:15
• @Stephan Seems reasonable to me. – John Mar 23 '16 at 16:00