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)?

  • $\begingroup$ If I remember correctly, you could get away with negative coefficients as long as certain inequalites are satisfied (the cofficients cannot be "too negative"). Tsay "Analysis of Financial Time Series" has some note on it, I think. But I may be wrong. $\endgroup$ Apr 6, 2016 at 13:23
  • $\begingroup$ Nope, I just checked, there is nothing in Tsay "Analysis of Financial Time Series". $\endgroup$
    – Taufi
    Apr 6, 2016 at 14:43
  • $\begingroup$ Sorry then. Perhaps that was for multivariate models in Lütkepohl's "New Introduction to Multiple Time Series Analysis", but that is irrelevant. $\endgroup$ Apr 6, 2016 at 14:50

1 Answer 1


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 $y_t$.


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