large number of applications in finance require measures of volatilities and correlations. A well-known example is the portfolio optimization problem originally developed by Markowitz (1952), in which an investor forms a portfolio of assets from a given universe by maximizing the expected return on the portfolio subject to a risk constraint. Risk in this case is measured by a weighted sum of the variances and covariances of all assets. More generally, risk measures are needed to solve problems such as optimal hedging, pricing of derivative securities, decomposition of risk for a given portfolio, and so on. When dealing with multiple assets, measures of risk are typically organized in a variance-covariance matrix, which is a square array of numbers that contains variances along its main diagonal and covariances between all pairs of assets in the off-diagonal positions. Unfortunately, although it is a necessary input to many problems in finance, the true covariance matrix of asset returns is not observed and, therefore, must be estimated using statistical techniques. Having established the need for estimation, one may still be skeptical about the need for an entire chapter on this topic. After all, variances and covariances can often be estimated using fairly basic methods. For example, suppose that our objective is to estimate the variance-covariance matrix of monthly returns for a given set of assets, and assume that we have access to 10 years of monthly data (120 monthly observations). We could estimate variances and covariances using the well-known formulas for sample moments: 120 varf r- (m )1 = - M !\ 120 and