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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 ...

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Yes, you can use Multivariate GARCH model to estimate the volatility of a portfolio. For example, the Constant Conditional Correlation(CCC) GARCH model. In the CCC GARCH model, it says there is a constant correlation between portfolio and the model is defined as: Once you have estimated the correlation matrix, the the composed volatility can be computed by ...

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If you want to use GARCH to estimate past local volatility of the portfolio you can do but, but you'd use GARCH to model the portfolio returns, not prices. Then you will be able to build a range of possible volatilities in the futures given a certain confidence level and you would have a local volatility $\sigma_t$ for each historical point.

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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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