and correlations.16 We applied this approach to our sample of 18 equity markets and found interesting results. Assuming, as before, a correction of order two for the serial correlation in the data, and a mixture of normal distributions, the maximum likelihood estimates for the decay parameters are equal to 47 percent for volatilities and 4 percent for correlations. This suggests that volatilities are mostly affected by very recent observations, since the half-life of the volatility estimator is only slightly longer than one month. However, correlation estimates use a considerably longer history of data, with a half-life of almost 17 months. Since individual observations have a much larger weight in the estimation of volatilities relative to correlations, the implication is that volatilities tend to respond much faster than correlations to market surprises. Next, we compare the values of the likelihood functions for the two different specifications of the risk model. Clearly, the model with two different decay parameters is less constrained. If the evidence supported the model with a single decay parameter, then we should expect the two likelihood functions to be very close in value. Otherwise, the model with two different decay parameters should generate a larger value of the likelihood. The difference in our case leads to a strong rejection of the model with a single decay parameter. We conclude this section by pointing out the strong potential of this last specification of the covariance matrix estimator. For example, when working with different asset classes, as we do, one can accommodate different decay rates for the volatilities in different asset classes, and a different decay rate for the correlation matrix. Even more generally, one could specify a different volatility process for each asset, and estimate those processes separately, and then estimate the correlation matrix for all the assets using a different model.17 ESTIMATING COVARIANCE MATRICES WITH HISTORIES OF DIFFERENT LENGTHS So far we have worked in a fairly ideal scenario in terms of data availability. In fact, in all our examples we assume that daily data are available for the entire sampling period for all the assets in our universe. Although this may be true in some applications, most practitioners know too well that this is not usually the case. Even for such widely used data as daily equity returns in developed markets, the available history can be considerably shorter for some of the smaller markets. The problem becomes even more extreme when dealing with data from emerging markets. How should we deal with histories of different lengths?18 One easy but definitely suboptimal answer is to disregard part of the longer series and start the analysis at a lsWe use the symbol A to identify estimators that use the weights w, and the symbol - to identify estimators that use the weight v. 17For some interesting applications of this approach, see Engle (2002). 18This section requires familiarity with regression analysis and some tolerance for rather heavy formal notation. However, in our opinion, the benefits for the researcher who faces this kind of problem outweigh the cost of reading through this section.