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I wanted to ask, as I've seen this used a couple of times before, about the logic of fitting a GARCH model in absence of estimating ARMA for a series that is clearly an ARMA process (Fitting a GARCH model on an intercept only mean model for data that is an ARMA process).

Would this at all offer anything valuable in terms of investigating conditional variance?

I am asking this as I am currently tasked with investigating conditional variance of a time series. After estimating an appropriate ARMA model, I see that squared residuals are uncorrelated and archlm test for ARCH effects is insignificant at all lags, therefore I cannot fit any GARCH model.

What are my avenues of investigating conditional variance of the series for which I cannot fit a GARCH model?

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Conditional mean
If you are sure your time series has an ARMA structure in its conditional mean and you can estimate it with decent precision, then ignoring it (and going directly for GARCH) does not make sense. But perhaps the ARMA signal is so weak relative to the noise that estimating it as zero (i.e. no ARMA) is more precise than estimating it as some specific instance of ARMA, say, ARMA(1,1)? If so, one could justify approximating the conditional mean with a constant rather than with ARMA.

Conditional variance
If you do not find autocorrelation in squared residuals from your conditional mean model, then indeed there is no point in fitting a GARCH model for the conditional variance. You could however think whether you have any other variables on which you could condition the variance. Perhaps they would offer a basis for a conditional variance model?

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