How to compare volatility models?

What are the most popular ways to compare volatility models?

Suppose I wanted to compare the forecasting accuracy of a GARCH(1,1) model with the historic 30 day volatility. What method should I use?

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Compare future forecast volatility with future realized volatility...done.... –  Matt Wolf Oct 29 '13 at 23:05
Duplicate of quant.stackexchange.com/questions/8056/… –  user2763361 Oct 30 '13 at 2:37
@user2763361 Not duplicates; this is asking how to compare models, while that other question asks whether explicit models have been compared. –  Shane Oct 30 '13 at 2:47
@Shane After a sensible (non literal) parsing of the other question it is clear Jase wanted a description of any such comparisons as well as references (I take this to be true since if the answerer took the "whether" question literally the answer would be utterly useless) –  user2763361 Oct 30 '13 at 3:00
@user2763361 Others might disagree with me, but I view these questions as entirely different. One is about what models exist (both the question and answer) and the other is about how to compare the existing models. –  Shane Oct 30 '13 at 3:04

There is no one right answer to this question, but a common starting place is to compare the bias and variance of the forecast vs. the realized variance.

Take your forecasted variance $\hat y$ and regress them against the realized variance:

$y = \beta_0 + \beta_1 \hat y + \epsilon$

A few things that you want to see:

• The forecast should be unbiased, meaning that $\beta_1 \approx 0$. If this is not the case, then you will consistently over or under-realize your prediction.
• The error should not have any clear structure. So $\epsilon \mapsto N(0, \sigma)$; plot the residuals and look at them.
• You can view the goodness-of-fit of your model by looking at things like the $R^2$ and the RMSE.
• Normalizing your returns by your forecast should result in a normal distribution: you can review this by using a qqplot.

Make sure that you evaluate your model on the time horizon that is relevant to you. For instance, if you're rebalancing monthly, then it won't be especially important to predict the next day's volatility as much as if would if you were rebalancing daily.

As with any forecast, make sure that you have a sufficient sample size for significance, and use out-of-sample data to evaluate any parameters that you might fit if you use a structural model.

Some useful references:

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Thanks for the tips! –  user847663 Oct 31 '13 at 20:11