The appendix provides a detailed mathematical foundation for the consistent partial least squares (PLS) path modeling approach. It begins with the variance-based SEM, where path coefficients are derived from the construct score correlation matrix \( \mathbf{R} \). The matrix \( \mathbf{R} \) is partitioned into submatrices to extract the necessary information for estimating the regression coefficients \( \hat{\mathbf{\beta}} \). In Appendix B, the basic design of PLS is introduced, assuming a factor model with \( N \) i.i.d. column vectors of observed scores \( y_1, y_2, \ldots, y_N \). Each vector is standardized, and the vectors are partitioned into \( M \) subvectors. The relationship between the observed scores and the latent variables is described by linear equations, and the covariance matrix of the measurement errors is diagonal. The appendix then discusses the Mode A algorithm, a numerically stable iterative fixed-point algorithm used in PLS. It explains how the algorithm converges to an estimated weight vector \( \hat{w} \), which is used to define sample proxies for the latent variables. The paper also derives the probability limit of the estimated weight vectors, showing that they tend to be proportional to the true loadings. Finally, the appendix provides a method to estimate the proportionality factor \( \hat{c}_i \) and the squared correlations between the proxies and the latent variables, ensuring consistency in estimation. It concludes with four observations, emphasizing the practical implications and extensions of the approach.The appendix provides a detailed mathematical foundation for the consistent partial least squares (PLS) path modeling approach. It begins with the variance-based SEM, where path coefficients are derived from the construct score correlation matrix \( \mathbf{R} \). The matrix \( \mathbf{R} \) is partitioned into submatrices to extract the necessary information for estimating the regression coefficients \( \hat{\mathbf{\beta}} \). In Appendix B, the basic design of PLS is introduced, assuming a factor model with \( N \) i.i.d. column vectors of observed scores \( y_1, y_2, \ldots, y_N \). Each vector is standardized, and the vectors are partitioned into \( M \) subvectors. The relationship between the observed scores and the latent variables is described by linear equations, and the covariance matrix of the measurement errors is diagonal. The appendix then discusses the Mode A algorithm, a numerically stable iterative fixed-point algorithm used in PLS. It explains how the algorithm converges to an estimated weight vector \( \hat{w} \), which is used to define sample proxies for the latent variables. The paper also derives the probability limit of the estimated weight vectors, showing that they tend to be proportional to the true loadings. Finally, the appendix provides a method to estimate the proportionality factor \( \hat{c}_i \) and the squared correlations between the proxies and the latent variables, ensuring consistency in estimation. It concludes with four observations, emphasizing the practical implications and extensions of the approach.