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I have estimated an EGARCH model of the following form.

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According to the literature, $|\omega_3|<1$ ensures stationarity of the conditional variance. However, my $\omega_3$ coefficient in front of the logged GARCH term is close to one (about 0.95 to 0.98).

But there is no remaining ARCH effect in the residuals and the squared residuals are not significantly correlated up to 20 lags.

Should I discard the estimated model or is it still okay to keep?

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On stationarity

It is not uncommon to find highly persistent volatility in GARCH models. Sometimes it may be caused by neglected structural changes or other types of model misspecification, but at least the finding is generally not surprising.

I do not know much about the estimation of EGARCH models, but could it be that $\omega_3$ is restricted (by the estimation procedure) to be below one? If so, a value close to one could indicate that the coefficient might have been higher absent the restriction. In other words, the estimation procedure tries to squeeze the features of the data into a predefined feasible range when these features (such as persistence of volatility) do not quite fit in there.

On residual diagnostics

ARCH patterns being absent from residuals is encouraging; from this perspective you could keep your model.

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