international equity markets, a correction for serial correlation of relatively low order (one or two) is often sufficient. This is in line with our expectations, given the way information is likely to be transmitted across markets that are open at different times during the day. In the discussion that follows, we maintain the hypothesis that q = 2 and proceed to estimating the optimal decay rate. For a decay rate to be considered optimal, we must define an objective function whose value changes as 5 changes, and then select a value of 5 that maximizes that function. This is a standard technique in econometrics known as maximum likelihood estimation. In our case, the problem can be approached as follows. The returns in our sample are generated by some distribution. Assume for the moment that the distribution is a multivariate normal with mean zero and unknown covariance matrix, which is fully characterized by the decay parameter 5.11 The likelihood function measures the probability that the data in our sample are generated by a multivariate normal distribution with mean zero and a covariance matrix that varies with 5. Our objective is to find the value of 5 that maximizes the likelihood function or, equivalently, the probability of observing the data in our sample. In performing the optimization of the likelihood function, we use daily data from January 1980 through May 2002. The sample includes 18 equity markets: Australia, Austria, Belgium, Canada, Denmark, France, Germany, Hong Kong, Italy, Japan, Netherlands, Norway, Singapore, Spain, Sweden, Switzerland, United Kingdom, and United States. Therefore, the covariance matrix contains a total of 18 variances and 153 different covariances.12 Our results indicate that, when assuming a serial correlation of order two, the estimated optimal decay rate is 0.10 per month, which implies a half-life of slightly more than six months. Figures 16.4 shows how the maximum likelihood estimates of the U. S. volatility and U.S.-Japan correlation compare to their constant counterparts. Not surprisingly, our findings indicate that there exists significant variation in both volatilities and correlations. COVARIANCE MATRIX ESTIMATION: GENERALIZATIONS The covariance matrix estimator discussed so far has many desirable properties. However, it still fails to address a number of relevant issues. First, it assumes multivariate normality for the joint distribution of international equity returns. As we have argued earlier, this assumption does not appear to be supported by the data. Second, it imposes the same decay rate to all assets and to both volatilities and correlations. One can easily envision scenarios when this assumption is too "Later in this section we relax the assumption of normality. The assumption of zero mean can also be relaxed, and the unknown means can be estimated using maximum likelihood. However, in our case this assumption is fairly innocuous since we are working with daily data. 12The covariance matrix contains all the variances along its main diagonal. The covariances are located off the main diagonal and, since cov(x,y) = cov(y,x), the total number of different covariances in our example is equal to (18 x 17)/2 = 153.