Normal Returns, Chi-Square Variance Tests, and GARCH Inference

Summary

The note distinguishes inference about variance from inference about GARCH parameters. Squaring a normally distributed variable produces a scaled chi-square quantity (with one degree of freedom when the mean is zero), which helps explain chi-square tests for variance under their assumptions. If a fitted GARCH model supplies conditional variance estimates, a chi-square-based test might address whether variance in a period exceeds a specified value, although the response presents this as a possibility rather than a routine procedure.

That relationship does not make GARCH coefficients chi-square distributed or require chi-square tests for them. The response describes quasi-maximum-likelihood estimation and standard error methods, and gives ARCH estimated by ordinary least squares on squared returns and their lags as a simpler illustration. Non-normal equation errors do not automatically rule out OLS, though robust standard errors may be appropriate. Bayesian analysis offers another route: specify parameter priors and use posterior distributions for hypotheses. The discussion is conceptual and does not set out a complete testing procedure or its assumptions.

Key ideas

• Squaring a zero-mean normal variable gives a scaled chi-square variable, relevant to variance tests.
• A chi-square test on estimated conditional variance is conceptually distinct from testing GARCH coefficients.
• GARCH coefficients can be tested using standard inference methods after quasi-maximum-likelihood estimation.
• ARCH can be estimated by OLS on squared returns and their lags, with robust standard errors considered when appropriate.
• Bayesian parameter posteriors can support hypothesis tests when priors are specified.

Tags

# Volatility Return Distribution/Garch Modeling


# Volatility Return Distribution/Garch Modeling












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?

## Answer by John (score 4)

https://quant.stackexchange.com/a/8201

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.