sample: ^s =SBB,s(m)-B'sSAAj(m)Bs Third, compute the covariance matrix for the assets with the longer history, using their entire history. Again, this can be done by applying equation (16.6) to the first set of assets. Let SAAT(m) indicate the estimator that uses the entire history. Fourth, construct all the covariance estimates by exploiting the information collected so far: SAA(m) = SAAJ(m) SBA{m) = SBA/m)-B's\^AAJm)-SAA{m)^ (16.10) SBB{m) = SBBsS(m)-B's[SAAsS{m)-SAA{m)\Bs When the researcher faces more than two subsets of assets with histories of different lengths, the same methodology can be applied recursively, starting from the shortest history common to all assets and moving back in steps until the entire set of available data is used. ALTERNATIVE COVARIANCE MATRIX ESTIMATION METHODS The estimation technique that we have described in the previous sections has the appealing feature of capturing most of the empirical regularities of financial data, while being easy to implement when applied to large sets of assets. In this section, we briefly review some alternative covariance matrix estimators that have been proposed in the literature and discuss how they relate to our framework. GARCH Processes Since the work of Engle (1982) and Bollerslev (1986), generalized autoregressive conditionally heteroscedastic (GARCH) processes have become one of the most popular methods to estimate volatility in financial markets. These processes were originally designed to capture the tendency for volatility to cluster over time: Periods of high (low) volatility tend to be followed by more periods of high (low) volatility. Formally, a univariate GARCH(1,1) process for the daily volatility on a generic asset can be written as: varT+1[r(i)| = co + avarT[r(i)l + pVj (16.11) In words, the volatility for the asset at time T + 1 depends on the volatility of the asset at time T and on the squared return on the asset at time T. The coefficient a captures persistence in volatility; the closer a is to 1, the larger the persistence. The coefficient p reflects the tendency for volatility to adjust in reaction to market