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I am trying to estimate future volatility based on historical stock price data, using (G)ARCH models. I have computed the ACF and PACF of returns and squared returns, and none of them show signs of significant correlation at any lag.

Does that mean that a (G)ARCH model is not applicable here? Is there a different method for forecasting volatility with serially uncorrelated returns and squared returns? Could this apparent lack of serial correlation be an artefact of my chosen sampling frequency (daily)?

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Yes, uncorrelated squared returns suggest a vanilla GARCH model is not appropriate.

You could try a fancier GARCH model that
a) exploits possible autocorrelations of (absolute value of) returns raised to a power $d\in[1,2)$ (though I would not have high hopes for it given your findings so far),
b) incorporates asymmetry and/or
c) incorporates additional explanatory variables.

Alternatively, you could just use the sample's unconditional variance as the prediction of the future variance.

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