Negative GARCH coefficients Generally, for a strong GARCH(p,q) process it is assumed that all coefficients $\alpha_1, ..., \alpha_q, \beta_1, ..., \beta_p$ are non-negative. However, I estimated a regular GARCH(p,q) model for which the data exhibited strong volatility persistence as one could see in the plot of squared residuals, and I got negative coefficients for some $\beta_i$ and I am wondering if that is okay? Together all coefficients add up to $< 1$, so second-order stationarity is satisfied. But what should I do about the negative coefficients (if anything)? 
 A: So there are two considerations you need to make when inspecting the estimated parameters. First, does my GARCH process possess a strictly stationary distribution, given the parameter estimates and secondly, will the GARCH process be non-negative (since it is a variance), with these same parameters?  
Strict stationarity is a minimum requirement for the maximum likelihood estimator to be consistent, that is, the estimation procedure returning the "true" parameters on average. When restricting the parameters to be positive a sufficient (but not necessary) condition is, as you also point out, that the sum of the $\alpha$'s and $\beta$'s is less than 1. This rule doesn't apply if the parameters are allowed to be negative. 
Non-negativity:
There is no easily checkable rules (to my knowledge) that ensure non-negativity for the general GARCh(p,q) process. But Nelson & Cao (1992) give easy checkable conditions for GARCH(1,q) and (somewhat) manageable conditions for GARCH(2,q). For example the conditions for the GARCH(2,1), $$\sigma_t^2 = \omega + \alpha_1 y_{t-1}^2+\beta_1 \sigma_{t-1}^2 +\beta_2 \sigma_{t-2}^2$$ are
\begin{aligned}\omega & \geq0\\
\alpha_1 & \geq0\\
\beta_1 & \geq0\\
\beta_1+\beta_2 & <1\\
\beta_1^{2}+4\beta_2 & \geq0
\end{aligned}
These conditions only ensure that the $\sigma^2$-process is non-negative, but says nothing about stationarity of $\sigma_t^2$ or $y_t$.
