more specifically F{t) =   1 Pi(tfR(t) fait)   1 PiitfW) Aw   i -pK(t)Tm A*')   (20.20) The term pK(t)TK(t) represents the &th principal component of returns. Equation (20.20) shows that each estimated factor return is a simple weighted average of the asset returns where the weights are given by its corresponding (scaled) eigenvector. In practice, estimating the principal components over time generates a time series of factor returns. This concludes our discussion of standard PCA; next we explain the asymptotic principal component (APC) method developed by Connor and Koraczyk (1986). Connor and Koraczyk (1986, 1988) apply an asymptotic principal component technique introduced by Chamberlain and Rothschild (1983) to estimate the factors influencing asset returns. The APC method is somewhat different from the typical Wall Street application of principal component analysis. To motivate the asymptotic principal component approach, recall that factors are pervasive in that they relate to all N securities at a point in time. In practice, it is typical to have many more securities than historical observations; that is, N (number of assets) is much bigger than T (number of observations over time) and that the K market factors are not observed. We write the return process for each of the N assets over all T time periods-compare to Equation (20.12)-as R=BF + u (20.21; where R =NxT matrix of excess returns; each row of R represents a time series of excess returns on the "th security B =Nx K matrix of factor loadings F = KxT matrix of factor returns; each row of F represents a time series of factor returns u =NxT matrix of specific returns The asymptotic principal component method is similar to standard PCA except that it relies on large sample (asymptotic) results as the number of cross sections (N) grows large. From a practical perspective, standard PCA and APC differ in how we estimate V{t). In APC we derive factors from the T X T cross product matrix