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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?

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  • $\begingroup$ 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. $\endgroup$
    – Richard Hardy
    Apr 4, 2017 at 19:37
  • $\begingroup$ 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? $\endgroup$
    – Albe
    Apr 4, 2017 at 19:44
  • $\begingroup$ 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? $\endgroup$
    – Richard Hardy
    Apr 4, 2017 at 19:56
  • $\begingroup$ 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? $\endgroup$
    – Albe
    Apr 4, 2017 at 20:13
  • $\begingroup$ 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? $\endgroup$
    – Albe
    Apr 4, 2017 at 20:17

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What would be the best way to then remove the remaining ARCH effects? This is according to the Ljung box Q statistic of residuals squared and ARCH-LM test

First, the ARCH-LM and likely the Ljung-Box tests are not directly applicable on standardized residuals from a GARCH model as the null distributions of the tests statistics are different than the standard ones (those that apply for raw data rather than model residuals); see e.g. Wooldridge (1991) or Francq & Zakoian (2011), Section 8.4. What you could use instead is the Li-Mak test (Li & Mak, 1994); it was developed specifically for the standardized residuals of a GARCH model.

Second, if the remaining ARCH effects are genuine, try a different specification of the GARCH model: either change the lag order of the vanilla GARCH or try GJR-GARCH or other modification.

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?

You would like to have the assumptions of the model satisfied. A GARCH model assumes the standardized residuals are i.i.d., so there should be no ARCH effects in them. Now if the violations are small and the model is sufficiently simple, it might still do well. There is a fine line between underfitting (simple model, some assumptions violated in sample) and overfitting (complicated model, all assumptions satisfied in sample).

References:

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