GARCH(1,1) with insignificant constant

I'm fitting a GARCH(1,1) model to some data:

$Y_{t} = \sigma_{t}\epsilon_{t}$ with $\epsilon_{t} \sim t(\nu)$,

$\sigma_{t}^{2} = a_{0} + a_{1}Y_{t-1}^{2} + b_{1}\sigma_{t-1}^{2}$.

Estimating the parameters and standard errors I get a p-value of approx. 0.26 for $a_{0}$. Now, $\hat a_{0}$ is very close to zero. I can't seem to fit the model without the constant in R, so I'm wondering whether it would be alright to proceed working with the model.

Plots of the ACF/PACF of the squared standardized residuals suggest a good fit (the other estimates are significant at the 5% level (one of them barely)).

Cheers!

I can't comment yet so I try a short answer:

You could impose a parameter restriction and set $a_0=0$ to gain efficiency.

E.g. in R using the rugarch package you might use the option fixed.pars = list(omega=0) in the specification.

The model is still meaningful. The long term volatility in this case would be zero, and as long as $a_1$ (and $a_2$) is significant the model exhibits a mean reversion to the long term volatility of zero. Of course this long-term volatility would never be reached in practice since mean reversion is "slow" and new information will come in and push you away from the long-term vola.

• Thanks for your reply! I've since figured out that rugarch can do that. Fitting the model with omega restricted to equal zero yields a larger value of AIC/BIC though. Does this mean that I should stick with the model with the insignificant constant? May 24, 2014 at 10:15
• What about the other diagnostic checks? Have you also tested for autocorrelation in the (squared) standardized residuals (e.g. Bix-Ljung test)? Gof? Sign bias? Out-of-sample performance? These criterions are much more important than the AIC or BIC.
– Joz
May 24, 2014 at 21:00
• Hi, thanks for getting back to me. I've done some checks. The omega != 0 model yield a more satisfactory fit: H0 can't be rejected for the Standardized Residuals at various lags (1,3,7). The same is true for the standardized squared residuals. The null hypothesis for the ARCH LM test can't be rejected for lags 2,5,10. Sign bias test also yields high p-values. The model with omega = 0 fares worse: Testing the standardized squared residuals, H0 gets rejected for lags = 3,7. H0 for the ARCH LM test also gets rejected for lags 2,5,10. This suggests I should stick with the omega != 0 model, right? May 24, 2014 at 21:38
• Yes. But out-of-sample diagnostic checking is also very important if you want to use the model in a predictive manner.
– Joz
May 25, 2014 at 15:03