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.

  • $\begingroup$ 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. $\endgroup$ – Olaf Mar 22 '16 at 14:58
  • $\begingroup$ Unfortunately the statistical software I am using (Eviews) does not allow me to put bounds on $\alpha_1$. $\endgroup$ – Stephan Mar 22 '16 at 18:09
  • $\begingroup$ @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. $\endgroup$ – 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.

  • 1
    $\begingroup$ 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. $\endgroup$ – 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.

  • $\begingroup$ 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? $\endgroup$ – Stephan Mar 22 '16 at 19:45
  • $\begingroup$ 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. $\endgroup$ – RandyF Mar 22 '16 at 19:57
  • 1
    $\begingroup$ 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. $\endgroup$ – John Mar 22 '16 at 21:25
  • $\begingroup$ 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. $\endgroup$ – Stephan Mar 22 '16 at 23:15
  • $\begingroup$ @Stephan Seems reasonable to me. $\endgroup$ – John Mar 23 '16 at 16:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.