The paper titled, “Answering the Skeptics : Yes, Standard Volatility models do provide accurate forecasts” is a classic paper on volatility modeling by Andersen and Bollerslev.

What’s this paper about ? If you build a volatility model, How do you go about testing it ? This is the key question answered in the paper.

At a daily frequency or intraday frequency, returns do not show serial correlation. However there is a serial dependency amongst them. This needs to get reflected in one way or the other. One of the most popular models in finance are ARCH family of models. These models are characterized by two equations, mean return equation and conditional variance equation. I guess before this paper came out, many researchers had shown excellent in-sample parameter estimates for their ARCH model but failed to show the model’s forecasting power. For validating the ARCH forecasts, the common proxy of unobserved volatility is the returns squared estimate.

The paper goes about answering two questions :

  1. If ARCH family of models have good in-sample estimates, then why do their forecasts not match up return square values in the forecast horizon ?

  2. What’s the right proxy for volatility against which the ARCH/GARCH type of models need to be tested ?

The first question is answered by calculating the theoretical R^2 between return squared and ARCH estimates over any forecast horizon. Straightforward math shows that the explanatory power is bounded above and hence gives a water tight theoretical justification to all the research papers that have reported high forecast errors using ARCH models. The takeaway from this analysis is that forecasting errors of ARCH/GARCH models captured via R-squared is not an anomaly but a direct implication of standard volatility models.

The second question is answered via Continuous time volatility process. The authors start with the diffusion equation that is equivalent to GARCH(1,1) and show that the realized volatility based on intraday data is the right proxy of unobserved volatility. This discretized form of realized volatility is the one that needs to be used while checking the forecast performance of standard volatility models.  Empirical analysis on intraday currency data show that standard volatility models account close to fifty percent of the variability in ex-post volatility.