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I am estimating GARCH model for volatility calculation and as a data input I have used log first difference data (ln(a)-ln(b)). Usually I would check for autocorrelation in residuals(to check the model), but since my input was already in the form of first difference, is this check still necessary?

The reason I am not sure is that one of the solutions for autocorrelation is the first difference which I have already applied in the first step and when I did the test I got the autocorrelation for some of my datasets.

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You should check for autocorrelation. However, its presence does not necessarily mean your model will produce inaccurate figures. The ARCH family of models were developed to help analyze the volatility of a time-series. This data is assumed to display a degree of heteroskedasticity. Using the GARCH model, small amounts of auto-correlation (not of practical importance) can cause large p-values unless your sample-size is massive.

You should test the squared residuals of your model for autocorrelation rather than the standard method of t vs (t-1), since significant (short-term) autocorrelation in this data may actually be appropriate.

EDIT: Good insight from @John as well. Would comment on his answer but don't have the rep. First-differencing can theoretically cause inaccurate GARCH residuals and is really not the preferred method for dealing with autocorrelation in this instance. John's suggestion or weighted OLS estimators is the better way to go in this case.

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To add to @Brumder's answer, people typically take a two-step approach when concerned about both Garch and autocorrelation: first fit some sort of ARMA(p,q) model, and then second use maximum likelihood on the residuals of the first step to estimate the Garch parameters.

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It is apparently true that one of the majors ways of clearing the presence of serial correlation in the residuals is to either increase the lag lengths or to run a difference model as well as running a regression model, through the origin (ignoring the intercept terms) or better still run an auto-regressive distributed lag model, but what most scintillating researchers in economics and other related disciplines fail to understand is that we have different types of econometric techniques and each of them has its own underlying assumptions. Succinctly speaking, the condition of no residual serial correlation or no autocorrelation is often one of the underlying assumptions of the method of ORDINARY LEAST SQUARE(OLS) which is an econometric technique for modelling mainly LINEAR MODELS not for Nonlinear regressions. Therefore, discerning from the above, GARCH and other forms of ARCH family models have their own underlying assumptions.In fact, if you are trying to estimate any of the above models and you find absence of heteroscedasticity for instance in your model, you should just start crying immediately because your model is not likely to have ARCH EFFECT and GARCH model wouldn't run in the absence of that because the residuals of the model wil not be conditionally heteroscedastic. In the absence of all these you should just cry and go and change to any other modelling technique but not within the ARCH models. So I think that since no autocorrelation is only one of the asumptions of OLS, we should not have to worry much about it whenever we are estimating Nonlinear regression models(CHINONSO'S ARGUMENT 2017)

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