U.S. market was equal to 0.91 percent, with a standard deviation of 0.07 percent. The largest aad was equal to 1.18 percent. How does the evidence from the observed data compare to the simulated histories? The aad for the United States in our sample is equal to 4.46 percent, well outside two standard deviations of the simulated mean aad and, even more striking, well above the largest simulated aad. Considering that we simulated 1,000 histories, one must conclude that there is less than a 0.001 probability of observing the time variation in volatilities that we observe in our sample, if the data were actually generated by a normal distribution with a constant volatility of 16.06 percent. The summary statistics in Table 16.3 confirm that our findings are just as convincing for the other three countries in the sample. Therefore, it is hard not to reject the hypothesis of a constant volatility, at least within our sampling period. Next, we analyze the history of correlations over time. Since we focus on four different markets, we have a total of six correlations. Also in this case, we use both estimators to compute two alternative measures of correlations: One is constant throughout the sample, whereas the other captures time variation through a rolling window of 100 days. Figure 16.2 displays the differences between the two estimators for the six correlations. Following the same approach as in the volatility analysis, we performed a Monte Carlo simulation to determine whether the observed aad's from the constant correlations are a legitimate sign of time-variation in the correlations. The experiment reveals that the observed aad's are larger than the maximum aad's simulated in 1,000 Monte Carlo histories assuming a constant correlation. The summary statistics for this experiment are reported in Table 16.4. To summarize, the evidence from our sample suggests that: II Daily returns appear to be generated by a distribution with heavier tails (a higher probability of extreme events) than the normal distribution. 11 Volatilities and correlations vary over time. These properties of the distribution of daily returns must be kept in mind as we embark in our main task: the identification of a desirable estimator of the covari-ance matrix. The next challenge is to find an estimator that strikes a balanced compromise between statistical sophistication and parsimony. In fact, on one hand we TABLE 16.3 Test of Time Variation in Volatilities     Observed Data         Constant   Standard Deviation of Monte Carlo Data           Standard     Volatility Observed Time Varying Average Deviation Maximum   Estimate aad Estimates aad of aad aad United States 16.1% 4.46% 4.30% 0.91% 0.07% 1.18% Japan 18.4 5.63 4.03 1.04 0.08 1.36 United Kingdom 15.4 3.53 3.35 0.87 0.07 1.14 Germany 18.9 6.29 4.11 1.07 0.08 1.40