# To obtain an accurate mean estimate from the data the

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to obtain an accurate mean estimate from the data, the main thing as far as volatility calculation is concerned is to avoid using extreme sample mean returns that will periodically be produced from short data samples. A corollary of this principle is that if one is interested in volatility, using elaborate models for mean returns, e.g., allowing the risk premium to vary over time, is unlikely to be worth the effort in terms of any improvement in accuracy. Below, we will first
33 adopt the approach of imposing a mean return of zero and then later examine the quantitative effect of the constraint on empirical forecast accuracy in our volatility estimation. Estimating volatility in practice Given that actual securities prices do not come from a constant volatility lognormal diffusion process, computing historical volatility as shown in equations (II.2) - (II.4), is no longer theoretically optimal. But, while the problems we have just mentioned are well-known, option traders, and many academic researchers as well, typically ignore them and calculate historical volatility estimates by the most basic method. The normal (though not necessarily optimal) way most traders deal with the fact that volatility changes stochastically over time is to use only recent observations in the calculation and discard data from the distant past. It then becomes necessary to decide how much past data to include in a historical sample. There is a tradeoff between trying to examine a large sample and trying to eliminate data that are so old as to be obsolete. One consideration in making this choice may be the length of the forecasting horizon. In trying to predict volatility over the next 3 months, it is plausible that one might prefer a short sample of more recent data, perhaps just the last 6 to 12 months, while to forecast volatility for the next 3 years, a longer historical sample might be called for. We examine these issues empirically in the next section. II.4. The Forecasting Performance of Historical Volatility The most common method of producing volatility forecasts from historical data is simply to select a sampling interval and the number of past prices to include in the calculation and then to apply equations (II.2) - (II.4), (making ad hoc adjustments when the procedure appears to be giving inappropriate answers). But the idea that it may be better to adjust the length of the historical sample for different forecasting horizons suggests that it is worthwhile examining the issue empirically.
34 Consider estimating volatility from k past prices in order to forecast the volatility that will be experienced over the next T periods. This might be called, simply, the (k,T) model. 7 The volatility estimate from that procedure is given in equation (II.6) We have used the (k,T) procedure to construct time series of volatility forecasts from monthly data for a large number of financial series, including interest rates, stock prices, and exchange rates. Here we report results for a selection of the most important series: the S&P 500
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