Patton (2011) Volatility Forecast Comparison Using Imperfect Volatility Proxies

Summary

Patton (2011) establishes theoretical conditions under which forecast comparison rankings are preserved when an imperfect (noisy) volatility proxy replaces the unobservable true conditional variance. The key result is that the mean squared error (MSE) and QLIKE (quasi-likelihood) loss functions are robust — they give the same ranking of forecasters whether evaluated against the true variance or any conditionally unbiased proxy — while mean absolute error (MAE) is not robust.

Key Claims

• A loss function $L(\hat{h}, h)$ is robust if for any two forecasts $\hat{h}_1$, $\hat{h}_2$ and any conditionally unbiased proxy $\tilde{h}$ (i.e., $E[\tilde{h} \mid \mathcal{F}_{t-1}] = h$): $E[L(\hat{h}_1, \tilde{h})] \leq E[L(\hat{h}_2, \tilde{h})]$ iff $E[L(\hat{h}_1, h)] \leq E[L(\hat{h}_2, h)]$.
• MSE: $L = (\hat{h} - \tilde{h})^2$ — robust.
• QLIKE: $L = \tilde{h}/\hat{h} - \log(\tilde{h}/\hat{h}) - 1$ — robust.
• MAE: $L = |\hat{h} - \tilde{h}|$ — not robust; rankings can reverse under noisy proxies.
• Provides a theoretical foundation for using realized volatility (or other imperfect proxies) as a benchmark in forecast horse-races.

Concepts Introduced or Extended

• Volatility Forecast Evaluation
• Realized Volatility

Entities Mentioned

• Andrew Patton

Quotes

"The use of an imperfect volatility proxy in place of the latent conditional variance introduces estimation error that may distort forecast comparisons."

My Take

A clean result with immediate practical impact: it validates industry practice of evaluating volatility forecasts against VIX$^2$ or realized variance as a proxy. The QLIKE criterion in particular has become standard in empirical volatility work precisely because it is both robust and convex.