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?
 A: 
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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*Francq, Christian, and Jean-Michel Zakoian. GARCH models: structure, statistical inference and financial applications. John Wiley & Sons, 2011.

*Li, W. K., and T. K. Mak. "On the squared residual autocorrelations in non‐linear time series with conditional heteroskedasticity." Journal of Time Series Analysis 15.6 (1994): 627-636.

*Wooldridge, Jeffrey M. "On the application of robust, regression-based diagnostics to models of conditional means and conditional variances." Journal of Econometrics 47.1 (1991): 5-46.

