The K factors are given by the first K eigenvectors of V. That is, each eigenvector represents a time series of a particular factor. Note, however, as in the case of standard principal components there is an indeterminacy issue. Connor and Koraczyk show that factors can be determined only up to some nonsingular linear transformation. This concludes our discussion of principal component analysis. A DETAILED LOOK AT THE LINEAR CROSS-SECTIONAL FACTOR MODEL_________________________________ In this section we explain the linear cross-sectional factor model, which forms the basis of estimating risk. In order to estimate risk, we need to generate a time series of factor returns. Estimation of factor returns begins by assuming that each asset has an exposure to one or more factors. These exposures to factors are measurable and may be industry classifications, investment style exposures (e.g., book-to-price), or something else. Given the exposures, returns on individual securities are regressed, cross-sectionally, on the factor exposures. The estimates from this regression are the one-period factor returns. Repeating this process over time generates a time series of factor returns. Local Framework The local linear factor model posits a relationship between a cross section of returns and asset exposures, returns to factors, and specific returns. Specifically, the model describes the cross section of N (n = 1, . . . , N) asset returns as a function of K (k = 1, . . . , K) factors plus N specific returns. Mathematically, we have Ri(t) = Bl(t-l)Fi(t) + ui(t) (20.23) where R*(t) = N-vector of local excess asset returns (over the [local] risk-free rate) from time t - 1 to t. We take t as the current date. B*{t- 1) = NxK matrix of exposures that are available as of t- 1. In practice, the factor exposures may not be updated at the same frequency as the asset returns. In this case the information in the matrix B will be dated earlier than t - 1. F*(t) = K-vector of factor returns. The return period is from t - 1 to t. u?{t) = N-vector of mean-zero specific returns, from t - 1 to t, with covariance matrix o2{t)I where I = NxN identity matrix 02(£) = Variance of i/{t) at time t The security returns in (20.23) are computed as follows. Let R^(t) represent the ?zth asset of R?{t). R*n(t) is defined as: