Step 1 estimates are fixed, and the residual covariances and residual variances of the indicators are freely estimated. Residual variances are estimated in Step 2 by estimating the variances of k residual factors defined by the indicators. This provides an estimate of the discrepancy between the model implied and observed covariance matrices, R = S – Σ(θ), along with the sampling covariance matrix (VR) of R. While the discrepancy between model implied and observed covariance matrices can be computed simply by deriving covariance expectations from the Step 1 model and subtracting the observed covariance matrix, such an approach would not provide the corresponding VR matrix necessary for the calculations below. The VR matrix is expected to be positive semidefinite and, consequently, have no negative eigenvalues. Therefore, the VR matrix has the following eigendecomposition: VR=(P1P0)(E000)(P1′Po′) where P1 is a matrix of principal components (eigenvectors) of VR, and E is a corresponding diagonal matrix consisting of non-zero eigenvalues. P0 reflects the null space of VR. Projecting Ri—a vector of residual covariances estimated from the Step 2 Model—onto P1 and adjusting for corresponding eigenvalues, we have that: E−12P1′RiN(0,Ir)
