may spike during periods of extreme market distress, they appear to move considerably more slowly than volatilities over time.15 Therefore, it would be useful to construct a covariance matrix estimator that can accommodate the different dynamics in volatilities and correlations. Luckily, this task is easily accomplished within our framework. We start from the relationship between covariance and correlation for a generic pair of daily returns: cov[r (d),r (d)] = corr[r (d),r (d)] X std[r (d)] X std[r (d)] In words, the covariance between the two daily returns is equal to the correlation between those returns, multiplied by the product of their volatilities, as measured by the standard deviations. Since a covariance matrix is nothing else than a collection of covariances and variances (squared volatilities), we can apply the same decomposition to the entire covariance matrix. If £ is a covariance matrix for a set of N assets, then we can write: 2 = DQD' (16.1 where D is a diagonal matrix of return volatilities (and so is D'), and Q is a correlation matrix with Is along its main diagonal, and all pairs of return correlations off the diagonal. The covariance matrix decomposition in equation (16.8) may appear obvious. However, it has a powerful implication for our task: One can estimate volatilities and correlations using different assumptions on their dynamics, and still preserve the positive semidefinite nature of the covariance matrix. For example, the following specification allows for a different weighting scheme (decay rate) for volatilities relative to correlations: lT\';\ 2>^) - t=\ I' and c6rrT[f,(i),f;(i)] = 2X\^)v!/2r;,^) t=\ stdTMd) xstdT\tj<   14These features of volatility have been extensively documented since the work of Engle (1982). 15See, for example, De Santis and Gerard (1997).