This appendix formalizes the importance of orthogonality conditions discussed in Section 3.1. It defines $\tilde{W}_2$ as the residual from a regression of $W_2$ on $\omega^o$ and shows that the estimator inputs are not impacted by redefining $W_2$ as $\tilde{W}_2$. This section proves a proposition that under certain conditions, the coefficient on $X$ from a regression of $Y$ on $X$ and observed controls is the same as the coefficient in a regression of $Y$ on $X$ and the index of these controls multiplied by their true coefficients. The proof involves algebraic manipulation and the use of variance and covariance values. This section provides a detailed proof of the asymptotic bias of the estimator $\hat{\beta}$ and $\hat{R}$. It shows that the estimator inputs are not affected by the way $W_1$ is defined, and it derives the asymptotic bias of $\hat{\beta}$ and $\hat{R}$ in terms of variance and covariance values. The proof involves solving a system of equations to find the asymptotic bias $\Pi$. This section includes tables with summary statistics for the data used in the NLSY wage analysis and early life and child IQ analysis. The tables provide mean values for various variables, such as region, marital status, occupation, and drinking intensity, and highlight the importance of controlling for these variables in the regressions.This appendix formalizes the importance of orthogonality conditions discussed in Section 3.1. It defines $\tilde{W}_2$ as the residual from a regression of $W_2$ on $\omega^o$ and shows that the estimator inputs are not impacted by redefining $W_2$ as $\tilde{W}_2$. This section proves a proposition that under certain conditions, the coefficient on $X$ from a regression of $Y$ on $X$ and observed controls is the same as the coefficient in a regression of $Y$ on $X$ and the index of these controls multiplied by their true coefficients. The proof involves algebraic manipulation and the use of variance and covariance values. This section provides a detailed proof of the asymptotic bias of the estimator $\hat{\beta}$ and $\hat{R}$. It shows that the estimator inputs are not affected by the way $W_1$ is defined, and it derives the asymptotic bias of $\hat{\beta}$ and $\hat{R}$ in terms of variance and covariance values. The proof involves solving a system of equations to find the asymptotic bias $\Pi$. This section includes tables with summary statistics for the data used in the NLSY wage analysis and early life and child IQ analysis. The tables provide mean values for various variables, such as region, marital status, occupation, and drinking intensity, and highlight the importance of controlling for these variables in the regressions.