exposition we maintain the standard factor model. We assume that A{t) is small enough to be ignored, so that V(t) = B(t - l)B(t- 1[ (20.14) In the PCA approach, we first need to estimate the exposures matrix. A simple sample estimator of V(t) is T-l T Vit^-^Rit-JM-})1 ;=0 (20.15; We find the exposures matrix, B, by decomposing V(t) in terms of its eigen- system1 v(t) = p(tm)P(ty (20.1( where P(t) =NxNmatrix of eigenvectors with each eigenvector stacked columnwise; that is, P(t) = [pt(t) I p2(t) I ... I pN(t)] and pjt) represents the Kth column of P(t) &(t) =NxN diagonal matrix with the eigenvalues 0 (t) (n = 1, . . . , N) as its elements. (f) = \(t) 0 0 0 0 &2(t) 0 0 0 0 0 QN(t) (20.17) It follows from (20.14) and (20.16) that BBT = P(t)G(t)P(t)T and the factor loading matrix B is determined by the K largest eigenvalues and their corresponding eigenvectors; that is, =pe~'2 = p1(t)\jQ2p2(t)\...\JeKpK(t) (20.18; Equation (20.18) says that each column of the factor loading matrix, B, consists of an N X 1 eigenvector scaled by its corresponding eigenvalue. Given our estimate of B{t - 1), we can estimate the factor returns, F{t), by regressing R(t) on B. The regression yields: 12Factor models suffer from what can be referred to as "rotational indeterminacy," meaning that the parameters of the factor model are determined only up to some nonsingular matrix.