it by the number of business days in the month (p in our notation). However, suppose that positive returns tend to be followed by negative returns (and vice versa), so that daily returns display first order negative correlation. Equation (16.7) suggests that our monthly volatility estimate would be lower than the estimate obtained assuming iid returns. On the other hand, if positive (negative) daily returns tend to be followed by more positive (negative) returns, so that they display first order positive correlation, then equation (16.7) indicates that our monthly volatility estimate would be higher than the estimate under the iid assumption. 84 86 88 90 92 94 96 98 00 02 04 FIGURE 16.3 Annualized U.S. Equity Volatility with Different Corrections for Serial Correlation In Figure 16.3, we plot estimates of the (annualized) monthly volatility for the U.S. equity market using a five-year window of daily data. We consider three alternative estimators that assume a serial correlation correction of 0, 10, and 21 respectively. The plots indicate that the three estimators follow very similar dynamics. However, for the past 15 years in the sample, including a significant correction for serial correlation would have reduced the volatility estimates. Interestingly, as the value of q increases, the estimates display more oscillations around their trends. This is due to the fact that the covariances between returns that are 21 days apart are based on a relatively small number of observations (only approximately 60 observations in a five-year window) and, therefore, are more sensitive to a few extreme observations. As argued earlier, these oscillations often reflect noise rather than real economic signals, and therefore parsimonious corrections for serial correlation are preferable.