portfolios (e.g., Frank Russell 3000). 3.   The nonestimation universe represents all the assets that have exposure and return information but do not qualify for the estimation universe. These assets may be excluded on the basis that they have extreme returns. 4.   The proxy universe represents all assets that do not have exposure information or lack other data that are required to estimate factor returns. IPOs are examples of assets that fall in the proxy universe. This information forms the basis for factor return estimation. Note that factor return estimation does not require any portfolio-level information. Cross-Sectional Regressions Our main objective is to use a set of asset exposures to explain the cross-sectional dispersion of asset returns. At a point in time (e.g., a day), factor returns are estimated from the cross-sectional regression model in equation (20.23). Under standard assumptions, uf,(t) is an N-vector of mean-zero specific returns with covariance matrix o2(t)I where I is an N X N identity matrix. Note that we are assuming that the specific returns are homoscedastic-the variances are constant across security returns. The ordinary least squares (OLS) estimate of F*{t) is given by F£(t) = \B£(t-l)TB£(t-l)Y B^t-lfR^t) (20.29) Ordinary least squares estimation assumes that the covariance matrix of specific returns is 02(£)I, and that the variances of specific returns are constant across assets (i.e., returns are homoscedastic). In practice, this assumption is likely to be violated, which would lead to inefficient estimates as described by the OLS estimator. Alternatively expressed, a more reasonable description of the covariance matrix of specific returns, w*(£), is given by W) =   o?(f) 0 0 0 0 G22(t) 0 0 0 0 \ 0 0 0 0 2 Cm (t) of(t)*o2(t) fori*? (20.30) We can transform Z(t) into a homoscedastic covariance matrix, 02(t)I, by making some assumption about the relationship between each asset's specific variance, 02(£), and a "common variance," 02(£). One specification is 02(t) = vJt)o2(t) (20.31) where vjt) is a scalar that captures differences in volatilities across assets. In this case,