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What are the options for loss functions, when trying to compare the volatility (sigma) forecasts from different GARCH models? I was thinking about the Qlike function but am not sure if this would give the loss on volatility or the mean values.

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In principle, you should always use a loss function which is strictly consistent for the property you are trying to predict, in your case the variance/standard deviation.

As explained in my answer to this question such loss functions do not exist for the variance, unless you are willing to make the assumption that the mean of your target is zero. Under this assumption, the variance is simply an expectation (it is the second moment of the distribution) and strictly consistent loss functions for expectations can be used, e.g. the QLIKE you mentioned.

You can have a look at Patton's Volatility forecast comparison using imperfect volatility proxies, who uses the zero-mean assumption and discusses many loss function candidates for evaluating volatility forecasts.

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