exposures (to market cap) above the average. That is, these positions are in large-cap assets. II The short positions-where wjt - 1) < 0-correspond to positions in assets that have exposures (to market cap) below the average. That is, these positions are in short small-cap assets. Extending our analysis to the multivariate framework, the least squares estimate of factor returns is p'(f) = [.B'(f-l)T.B(f-l)| B£(t-T)TR£(t) = WFR£{t) (20.39) WF = {Bz B^B* is a K X N matrix of portfolio weights where each row represents a set of portfolio weights corresponding to a particular factor-mimicking portfolio. For example, the first row of WF may correspond to portfolio weights that comprise the mimicking portfolio for the market size factor. The second row may contain the weights of the portfolio that mimics the value factor, and so on. Note that WFBf{t - 1) = I (where I is the identity matrix). This means that each factor portfolio-that is, each row of WF-has a unit exposure to its factor (the weighted average of exposures is equal to one) and zero exposure to all other factors. In summary, we have shown that the least squares estimator of a cross-sectional regression of asset returns on size exposures is the return on a portfolio that is long large-cap assets and short small-cap assets. Therefore, factor returns represent returns on factor-mimicking portfolios. Predicted Factor and Specific Return Covariance Matrices Each cross-sectional regression generates one set of factor returns at a particular point in time. Repeating the cross-sectional estimation each day over a period of time, say two years, we generate a time series of factor and specific returns. Then we would have the TxK factor return matrix F*{T) where the fth row is a row vector of K elements representing the K factor returns at time t. In addition we would have 3. T X K specific return matrix, U^(T), where the rth row is a vector of K specific returns at time t. We begin (again) with the linear factor model for asset returns as shown in equation (20.23). In order to compute predicted tracking error, we need a forecast of the covariance matrix of asset returns, Rf(t), which we denote by Vf{t). Taking the variance of Re(t), as specified in (20.23), yields Vi(t) = Bt(t-l)t(t)Bi(t-l)T +hUt) (20.40) where 1^(t) = K xK forecast factor return covariance matrix, which we estimate from the TxK matrix of factor returns F?(T) Af{t) =N xN diagonal matrix with specific return variances along the diagonal that are estimated from the data in UZ{T)