# Remaining heteroskedasticity even after GARCH estimation

This is according to the Ljung-Box $Q$ statistic of residuals squared and ARCH-LM test. Both suggest there are ARCH effects remaining after lag 1 even after I have estimated my GARCH (1,1) model. I have no AR or MA terms in my conditional mean specification, just a constant, and all autocorrelation has been removed as per the $Q$-statistics of the residuals. What would be the best way to then remove the remaining ARCH effects?

Is there even a need to remove all ARCH effects after the GARCH estimation (given GARCH models are iid), if one wishes to forecast volatility?

• Are you inspecting the standardized residuals from the GARCH model? Note that the Ljung-Box statistic may not have its regular null distribution when applied on standardized residuals from a GARCH model; an appropriate alternative is the Li-Mak test. – Richard Hardy Apr 4 '17 at 19:37
• Yep, the standardised residuals squared from a GARCH model. So I should disregard the results for the Ljung box and ARCH-LM, are they inaccurate? Unfrotunately I have no access to the Li-Mak test via Eviews. But, theoretically, could I still feasibly use the model though even with remaining ARCH effects, if they are not that significant, or are there other ways to remove the heteroskedasticity? – Albe Apr 4 '17 at 19:44
• ARCH-LM is definitely inappropriate as probably shown in the original paper introducing the Li-Mak test. The latter is available in R. It is difficult to justify the use of a model if its assumptions are violated (standardized errors as proxied by standardized residuals should be i.i.d.). Have you tried changing the lag order, i.e. using GARCH(p,q) instead of GARCH(1,1)? Or using some modification of the GARCH model, e.g. GJR-GARCH? – Richard Hardy Apr 4 '17 at 19:56
• GJR seems to to lead to significant terms in the residuals squared, indeed. EGARCH and GARCH-M both lead to some insignificant lags after lag 2 and up to lag 10 (much like GARCH (1,1)) for the Ljung-box Q-statistic. I have not tried higher order GARCH models as my professor simply wants me to use a GARCH(1,1) in addition to GJR, EGARCH and GARCH-M. I really have no experience and little time within which to learn and do it on R, unfortunately. Perhaps I could get away with showing the ARCH-LM for 1 lag for all of these? At 1 lag using the ARCH-LM, the p-value is significant, what do you think? – Albe Apr 4 '17 at 20:13
• Yep, I just checked again, the issue seems to be with the Ljung-box Q-stats, specifically in lags 2-5, and lags 2-10 for two of the index returns. After lags 5 and lags 10, it becomes significant again, and starts increasing. The ARCH-LM is significant for all models at lag 1 but becomes insignificant at higher lags. Based on that, I think it should be ok if I present the ARCH-LM with one lag, and maybe just not present the Ljung-box Q stats? – Albe Apr 4 '17 at 20:17