vN{t) = o2(t)P (20.32) Given the covariance matrix described in equation (20.32), we need to transform (20.30) to the model where the covariance matrix of u?(t) is o2(t)I. This transformation from a heteroscedastic to a homoscedastic model is done as follows. From (20.32) it follows that P-ltllL{t)P-vl = o2(t)I or l(t) = o2(t)P. We transform the original heteroscedastic model into a homoscedastic model R*(t) = B*(t-l)F'(t) + u*(t) (20.33; where R*{t) = P-mR?(t) B*(t-1) =P-mBi{t-l) u*{t) = P~muHt) The specific returns in equation (20.33) are homoscedastic and the least squares estimate of Fi(t), based on equation (20.33), is Ft(t) = \Bi(t-l)T'L(t)-1Bt(t-l)] Bt(t-l)T'L(t)-1Ri(t) (20.34; Fr,(t) is the weighted least squares fWLS) estimate of Fi(t). Given Ff(t), estimates of specific returns are u{t) = R^it) - Be{t - l)Fe(t). Taking a closer look at (20.33), note that for the nxh asset, the transformed regression model is Kit) F*(ty Kit) _ Bfn{t) -fjt) *fjt~) -fijt) (20.35; where B*{t) is a 1 X K vector of exposures for the nth asset. Equation (20.35) shows that the larger (smaller) the scale factors vjt), the less (more) weight is given to the asset returns. So, for example, if we set v (t) equal to one over the log of market capitalization, then we would be weighting large-cap stocks more than small-cap stocks. How do you choose vn{t) in practice? A common specification is to let vjt) be a function of one of the regressors. For example, some empirical research has shown that large-cap stocks have smaller residual volatilities [o2(£)] than small-cap stocks. To reflect this phenomenon-that is, to give more weight to large-cap stocks and less weight to small-cap stocks-we would set v {t) equal the inverse of market capitalization of the wth stock13; refer back to (20.32). Table 20.4 provides a list of potential candidates for the weights. ^Alternatively, we could use 1 divided by the square root of market capitalization as the weight. Whether to use market capitalization or its square root is an empirical issue.