Diagnosing Negative ARCH Estimates in GJR-GARCH Models
Summary
The document considers a negative, statistically insignificant ARCH coefficient when fitting a GJR-GARCH model to S&P 500 returns. It frames the issue as a model-selection and specification question: a leverage term may affect the estimated ARCH coefficient, but an estimate's sign alone does not settle whether the model is appropriate.
The responses recommend first checking whether volatility data show sign bias. If there is no sign bias, a standard GARCH model may be sufficient; if sign bias is present, the GJR model may be relevant. When fitting produces a negative coefficient, parameter constraints are one possible remedy, while EGARCH is offered as an alternative that models sign effects and avoids negative conditional volatility. Another response recommends comparing overall fit between GARCH and GJR-GARCH. The discussion gives no diagnostic results or formal comparison for the stated sample, and it does not establish that any one remedy is universally suitable; model choice depends on the evidence and available software.
Key ideas
- Check for sign bias before adding a leverage term to a GARCH model.
- A negative, insignificant ARCH estimate should be considered alongside the model's overall fit.
- Parameter bounds may be used when a GJR-GARCH estimate has an unwanted sign.
- EGARCH is presented as an alternative for modeling sign effects while maintaining nonnegative volatility.
- The document provides recommendations but no empirical model comparison for the cited return sample.
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Full text
# How to deal with negative ARCH terms?
# 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:
- Remove the $\alpha_1$ entirely from the model, as it is statistically insignificant. However, I don't think it can be done that easily...
- 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.
## Answer by Neeraj (score 2, accepted)
https://quant.stackexchange.com/a/25037
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.
## Answer by RandyF (score 0)
https://quant.stackexchange.com/a/25031
I would look at the significance of the overall GARCH model compared to the GJR model. Use whichever provides a better fit.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.