Hello everybody!I am currently trying to evaluate volatility forecasting models. I want to construct two models: one GARCH(1,1) model and one B&S at-the-money implied volatility model. I am in possession of daily data, looking like this (we are looking at some out-of-sample data here):The GARCH column equals the output of the eviews GARCH(1,1) forecast, the "B&S-ATM" column contains the observed (no forecast) Black-and-Scholes ATM volatilities at the end of each day (constant 1 month maturity).We are looking at EUR/USD here. As suggested in the literature ( Yu, Lui & Wang (2010) ) I estimate a volatility forecasting model for implied volatility by estimating a simple regression of the form "realizedvolatility = a + b * impliedvolatility", using in-sample data. I then use this model to create my out-of-sample forecasts.For the 1-period-ahead forecast everything is fine, but now I want to evaluate 1-month/21-day-ahead volatility forecasts. GARCH(1,1) forecasts again are fine (I am using the n-step-ahead forecasting method as suggested by Engle & Bollerslev (1986)), but when I estimate my B&S-atm model (using the in-sample data) I experience significant autocorrelation (DW around 0.12) in my model. The RLVOL21 column contains the future realized volatility over the next 21 days, shifted upwards, so it is in the same row as my forecasts. My solution so far is to only use every 21st day to estimate my BS-ATM model, which heavily reduces my data set (I have around 3 years in sample and 1.5 years out-of-sample). Is this corrent and the only valid method?I have checked the literature and could not find anyone explaining properly how they tackle this issue.Thank you all for your reading this.

For US equities, you could use "realizedvolatility = a + b * VIX"; this works for SPX and a half-dozen single names with associated VIX's already computed by theCBOE. The VIX is, more or less, a risk-neutral expectation of the 30-day realized vol. There's generally a risk premium but the regression coefficients will adjust for that. Since your underlying is something else, I would suggest using the "Old VIX" methodology to construct your own 30-day VIX.This only requires close-to-the money options and generally is not too far from the "revised VIX". Details in various CBOE publications.

Well, my problem is not that I don't know which variable to use as the independent one in the regression. I use B&S at the money implied volatilities (on the underlying) for that.My question was more how I should estimate the regression, because when I use daily observations for a 21 day forecast horizon I experience heavy autocorrelation. This is why I wonder if the only way to solve this issue is to use every 21st observation?In addition to the B&S-atm volatility I constructed my own "VIX" as well (I computed the model-free implied volatility using several options of different strike prices). That's not the issue, my issue is the regression estimation.

You should think about the maturity of the options in your last column.

The maturity is one month, so they are in line with the forecast horizon.

How do you achieve that on each day of your table?

OTC data, can't remember if I got it from Bloomberg or British Bankers Association.

I see. Well personally I would probably keep the 21-day forecasts using all the data. Are the GARCH forecast errorssimilarly auto-correlated? If so, and it is horse race of comparing the two methods, maybe it doesn't matter. In the end, the 'optimal' forecast is probably some weighted combo of the two.

The GARCH(1,1) forecasts are computed using daily data. Although they are then (t+1) forecasts they can be "scaled up" into (t + 21) forecasts. There is no autocorrelation (and by the way beautiful normality, at least for some currency pairs) in the forecasts then.However, if I estimate an f = a + b * IV model my model's DW is around 0.12 and my residuals are autocorrelated. I think this is because if for example a new information arrives in two weeks which affects volatility and within the two weeks there has not been any new information (so implied volatility didn't change) all the forecasts of the two weeks will be wrong in the same direction. I think I will stick to the "use every 21st observation" method, unless someone comes up with something else. Thanks a lot for your time Alan, I appreciate it!