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For simplicity sake, if stock returns are normally distrusted, would that imply that second moment, variance/volatility, is chi-squared distrusted? If so wouldn't that imply the statistics(employed to in hypothesis testing) in garch modeling are also chi-squared distributed used in a chi-square test?

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Squaring normally distributed variables results chi-square distributions, which (as you imply) is why the chi-square distribution is used in hypothesis tests for the variance.

If you estimate a Garch model and obtain the conditional variance at every point in time, you could use a chi-squared hypothesis test to ask a question like is the variance in a particular period greater than some number. I don't think I've ever bothered to do this, but I suppose it is possible.

Nevertheless, that does not mean that the parameters to a Garch model also require a chi-square test. Garch is often estimated by quasi-MLE and standard errors are calculated using well-known approaches, i.e. not based on a chi-squared distribution. A simpler case is to consider estimating an Arch model, which can be done with OLS on the squared returns (assuming zero mean) and their lags. The distribution of the errors to this equation is non-normal, but OLS is a reasonable estimator when non-normal. You may need to calculate robust standard errors but not necessary. Hypothesis tests on the parameters of the Arch estimation can be conducted as usual.

This is one area where the issue is far more clear in a Bayesian framework. In the Bayesian framework you clearly define the prior on the parameter (often rather the inverse of the parameter for variance). The distribution of the posterior can be used to conduct any relevant hypothesis tests.

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