normal distribution. Sometimes, it is also assumed that this distribution is stationary over time, which implies that means, volatilities, and correlations do not change over time. Here, we argue that these assumptions are usually incorrect and, therefore, should not be maintained when constructing a covariance matrix estimator.

As a first step, we analyze the distribution of realized daily returns for the equity indexes of four of the largest markets in the MSCI universe: the United States, Japan, the United Kingdom, and Germany. We focus on daily returns from January 1997 to December 2001, for a total of 1,935 observations. A well-known property of the normal distribution is that, relative to its mean, 95.4 percent of the observations are within a two standard deviation interval, and 68.3 percent of the observations are within a one standard deviation interval. Given the size of our sample, if the returns for each market were normally distributed, then we should expect only 89 observations to fall outside a two standard deviation range relative to the long-term average return, and 1,322 observations to be within one standard deviation of that average. Table 16.2 shows that neither condition is satisfied by the data. In fact, for all the countries in our set, we find that the number of observations outside the two standard deviation range is considerably larger than what is predicted by a normal distribution, and so is the number of observations concentrated around the long-term average. Although this is not a formal test of the hypothesis of normality, the consistency of the evidence across the four markets suggests that daily returns follow a distribution with heavier tails than the normal (so-called leptokurtic distribution).

Next, we address the issue of stationarity. Again, we use daily data for the United States, Japan, the United Kingdom, and Germany. The sample starts in January 1980 and ends in May 2002, for a total of 5,850 observations. We use two different estimators for the covariance matrix. The first estimator assumes that the moments of the distribution are constant throughout the sample, and therefore uses the entire history of data and assigns the same weight to each observation. The second estimator is based on a popular technique used by many practitioners to capture time variation in second moments. At each point in time, volatilities and correlations are estimated using only the most recent data, contained in a moving window of prespecified length. In our case, the window contains the most recent 100 observations. Each day, we update the estimates by adding the most recent return observations and deleting the observations that are now 101 days old.

We start with an analysis of the volatilities. Figure 16.1 displays the estimates obtained from the two methodologies for each of the four equity markets. Visual inspection suggests that the estimates obtained from a rolling window of data oscil-

TABLE 16.2 Empirical Distribution of Daily Equity Returns

Sample Period January 1997 to December 2001 Sample Size 1,935

N(0,1) Germany Japan U.K. U.S.

Number of returns > 2std 89 162 132 127 128

Number of returns < lstd 1,322 1,330 1,375 1,407 1,404