0.6 - 1 - 0.4   \ i K i 0.2 ■ ^Av ■^jJKk^j^^j^*   84 86 83 90 92 94 96 98 00 02 04 U.S.-Japan Correlation FIGURE 1 G.4 Volatility and Correlations: Comparison betweem Constant and Maximum Likelihood restrictive. For example, when building covariance matrices that include different asset classes (e.g., equity and fixed income) it may be desirable to allow for a different weighting scheme for each asset class. In addition, even when working with a single asset class, it may be desirable to use different decay rates for volatilities and correlations. In fact, it is often argued that although volatilities tend to change quickly, correlations are more likely to move slowly over time. In this section, we discuss how to generalize our covariance matrix estimator to incorporate these desirable features. Mixture of Normal Distributions The evidence of heavy tails in the return distribution suggests that extremely large (positive or negative) returns occur more often than predicted by a multivariate normal distribution. Therefore, assuming normality when writing the likelihood function can be problematic. In fact, the maximum likelihood approach tries to find a value of the decay parameter that maximizes the probability of observing the data in our sample, while maintaining the hypothesis that the data are generated by a normal distribution. If the sample contains enough extreme observations, the estimate of the decay parameter will be affected by the need to accommodate those extreme observations within a normal distribution. To capture the heavy tails, we assume that at each point in time returns can