The main problem in volatility forecasting is that the variable of interest is unobservable, which complicates not only the construction of forecasts but also their comparison. This article challenges the common practice of using only proxy-robust loss functions, which have the nice property that they lead to the same ranking of forecasts regardless whether the unobservable true volatility is used or some unbiased proxy. It is shown that two proxy-robust loss functions need not necessarily produce similar rankings but may even produce completely contradictory rankings. Two likelihood-based loss functions are proposed instead, which are not exactly proxy-robust but are still robust in the classical sense. The first is based on a t-distribution and is meant for daily data. The second is based on an F-distribution and is meant for high-frequency data. In the latter case, the squared error loss function may also be used when a logarithmic transformation is applied to the realized variances in order to achieve approximate normality. An alternative transformation is proposed which allows the adaptation to the degree of non-normality. The forecasting procedures that are compared by the different loss functions include GARCH, HAR, HARQ, and MIDAS models as well as nonparametric techniques. Finally, the economic relevance of choosing the right forecast is illustrated with the problem of establishing the intertemporal risk–return tradeoff. All theoretical arguments are backed up with empirical evidence obtained from daily data as well as from high-frequency data.

波动率预测的主要问题是感兴趣的变量是不可观察的，这不仅使预测的构建而且使它们的比较变得复杂。本文对仅使用代理-鲁棒损失函数的通用做法提出了挑战，该函数具有很好的属性，无论使用不可观察的真实波动率还是使用一些无偏的代理，它们都会导致相同的预测排名。结果表明，两个代理-鲁棒损失函数不一定需要产生相似的排名，甚至可能产生完全矛盾的排名。相反，提出了两个基于似然性的损失函数，它们不完全具有代理鲁棒性，但在经典意义上仍具有鲁棒性。第一个基于t分布，用于每日数据。第二个是基于F分布，用于高频数据。在后一种情况下，当对数转换应用于实现的方差以实现近似正态性时，也可以使用平方误差损失函数。提出了另一种变换，该变换允许适应非正常程度。通过不同损失函数进行比较的预测过程包括GARCH，HAR，HARQ和MIDAS模型以及非参数技术。最后，通过建立跨期风险-收益权衡的问题来说明选择正确的预测的经济意义。所有理论论据均以从日常数据以及高频数据中获得的经验证据作为后盾。