are obtained from time-series regressions of the linear factor model in equation (16.12). Then an estimator is constructed for the covariance matrix of the factors and for the idiosyncratic variances. Finally, the entire covariance matrix for the N assets is estimated using equation (16.13). The parsimonious nature of this approach becomes apparent with an example. Suppose we want to estimate a covariance matrix for the returns in the Russell 3000 universe. Given the symmetric nature of the variance-covariance matrix, we would need to estimate a total of (3,000 X 3,001 )/2 = 4,501,500 different parameters. Assume, however, that a linear factor model with 50 factors satisfactorily describes the returns on the Russell 3000 universe. In this case, once we have estimated the factor loadings in B, we have to estimate a covariance matrix with (50 X 51)/2 = 1,275 different parameters, and the 3,000 volatilities in £ Clearly this is a much easier task. In fact, one can apply the techniques described in this chapter to estimate ~LF and SE, and then construct the appropriate estimator for SR. SUMMARY Covariance matrices are a necessary input to many problems in finance, such as construction of optimal portfolios, optimal hedging, monitoring and decomposition of portfolio risk, and pricing of derivative securities. Investment decisions can be significantly affected by a choice of a particular covariance matrix estimator. Therefore, it is important to identify the main features of financial data that should be taken into account when selecting a covariance matrix estimator: II Volatilities and correlations vary over time. In addition, volatilities and correlations may react with different speed to market news and may follow different trends. II Given the time-varying nature of second moments, it is preferable to use data sampled at high frequency over a given period of time, rather than data sampled at low frequency over a longer period of time. II When working with data at relatively high frequencies, such as daily data, it is important to take into account the potential for autocorrelation in returns, due to different liquidity across assets and asynchroneity across markets. II Daily returns appear to be generated by a distribution with heavier tails than the normal distribution. A mixture of normal distributions often provides a better description of the data-generating process.