# Appropriate way to combine alternative volatility estimates

I have a number of different annualized realized volatility estimates (for the same point in time) that I'd like to combine. Is a simple average over these appropriate? Or should I do this in the variance space? $$\sqrt{(\sum_{i} \sigma_i^2)/ n}$$

Or would a geometric mean be more appropriate?

These volatility estimates are calculated in the typical way from a lognormal price process. Please explain the basic rationale for any answer.

• If you want a proper answer you need to explain to us how the different annualized realized volatilities are calculated and why you thing averaging them gives you a superior estimate than one of the individual estimates. Feb 16 at 22:28
• @phdstudent It's actually a weighted average and the realized vol estimates are computed from different lookback windows. Feb 16 at 22:55

Consider a set of alternative forecasts. The literature on forecast combinations* shows that an average forecast often tends to beat the estimated (ex ante) best forecast from the set. This is because of two reasons. First, different forecasts make different mistakes, and averaging across forecasts tends to make them less influential. The analogy in finance is portfolio diversification. A well diversified portfolio will have lower risk than the average risk of its constituents. Second, identifying the ex ante best forecast is difficult due to issues with estimation precision and with the changing nature of the data generating process. Perhaps stock A gave you great returns for the last 5 years, but there is no guarantee it will continue doing that the next year.

The literature on forecast combinations further tells us that if we are looking for an optimal combination, a simple average is hard to beat. This was known as the "forecast combination puzzle", but it is not a puzzle anymore. A simple average is hard to beat not because better combinations do not exist but because it is difficult to estimate them with good precision. The portfolio analogy is, an equally-weighted portfolio may be your best bet unless you can estimate the expected returns and (especially) the covariance matrix precisely enough to build one that is truly superior. Try that with a time horizon of 100 periods and 1000 stocks to choose from... (The "forecast combination puzzle" has also been studied in the context of volatility forecasting; see Clements & Vasnev (2021).)

Now the caveats. First, some future shocks will be missed by all forecasts. This is the systematic risk of your portfolio; you cannot diversify it away. This is not a drawback of the forecast combination approach, though, as it applies to each and every candidate forecast. Second, if one** of the candidate forecasts is vastly superior than the rest, combining forecasts may be detrimental. (In terms of portfolios, there is probably no alternative in a well functioning market. In an inefficient market, you have a single stock that has an alpha way above any other alphas, so you want to invest in it alone without building a portfolio.) Or if one** of the candidate forecasts is vastly inferior, you do not want it in the combination. The individual performance can be estimated using rolling or expanding windows the way @phdstudent suggests in their answer. The vastly inferior forecast(s) may well be kicked out of the forecast combination, provided that estimation precision is sufficient.

*Summarized in textbooks such as Diebold (2017) and Claeskens & Hjort (2012).
**Or a few.

References

Averaging estimates from different windows has no economic sense. What you should do is estimate volatility using different windows (or better different models) and then compute a metric of how accurate your forecast is (such as a Root Mean Squared Error).

For example Hansen and Lunde (2005) test accuracy of many models of volatility forecast. In this paper they actually compared the accuracy of 330 ARCH-type models and concluded that GARCH(1,1) was superior in their sample.

The paper describes at great length the way to do it. But in a nutshell:

1. Estimate ARCH-type your model in monthly data using a window from $$[t_{start}, t_{end}$$. Compute volatility forecast for month $$t_{end+1}$$.
2. Compute realized volatility during month $$t_{end+1}$$ using daily data (e.g. sum of squared returns).
3. Subtract your forecasted volatility from realized volatility and square it. Do this for several months and check the sum of squared residuals.
• Should we care about economic sense if averaging gives us better estimates/forecasts? The forecast averaging literature is fairly clear about that. Feb 18 at 6:57
• You need to think about over-fitting ... Feb 19 at 4:49
• The forecast combination puzzle is just about that. Equally-weighted forecast average reduces overfitting compared to using a single forecast, while estimated optimal combination may not. And actually, there is some economic intuition here, too. If the data generating process is changing over time, we want to put higher weight on more recent data points. Averaging forecasts based on different lengths of history does exactly that. Feb 19 at 7:43
• Okay. That actually makes sense. Feb 19 at 18:08