Diagnosing Autocorrelation in GARCH Residuals
Summary
The discussion asks whether autocorrelation checks are still needed when GARCH is fitted to first differences of log prices. The accepted response recommends checking residual dependence, while cautioning that small autocorrelation may not make volatility estimates inaccurate. It emphasizes examining autocorrelation in squared residuals, because short-term dependence in the raw residuals can be compatible with the model’s purpose of representing changing volatility.
The answers also describe modeling the conditional mean before the variance: fit an ARMA model, then estimate GARCH parameters from its residuals using maximum likelihood. Another response cautions that differencing is not necessarily the preferred way to address autocorrelation and mentions weighted least squares as an alternative. The thread is informal and does not provide data, diagnostic thresholds, or a worked comparison. Its guidance should be read as model-checking suggestions, not a universal rule that raw residual autocorrelation invalidates a GARCH fit.
Key ideas
- First differencing does not remove the need to diagnose residual dependence in a GARCH analysis.
- Autocorrelation in squared residuals is relevant when checking whether the volatility model captures dependence in variance.
- Small autocorrelation in raw residuals may not imply practically inaccurate volatility estimates.
- A two-step approach can model the mean with ARMA before fitting GARCH to the residuals.
- Differencing may not be the best response to autocorrelation in this setting.
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Full text
# Autocorrelation in the GARCH model residuals # Autocorrelation in the GARCH model residuals 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. ## Answer by Brumder (score 3, accepted) https://quant.stackexchange.com/a/21059 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. ## Answer by John (score 1) https://quant.stackexchange.com/a/21060 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. ## Answer by user29292 (score 0) https://quant.stackexchange.com/a/35732 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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