# 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!

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.