\usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {\hat b_{zy},\hat b_{zx_i}} \right) = \rho _{x_iy}r_{p(x_i,y)}\sqrt {{\mathrm{var}}\left( {\hat b_{zx_i}} \right){\mathrm{var}}(\hat b_{zy})}$$\end{document}covb^zy,b^zxi=ρxiyrp(xi,y)varb^zxivar(b^zy) where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{x_iy}$$\end{document}ρxiy is the proportion of sample overlap between xi and y and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{p(x_i,y)}$$\end{document}rp(xi,y) is the phenotypic correlation between xi and y. In special cases, if y and x are observed in the same sample, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{var}}\left( {\hat b_{zy}{\mathrm{|}}{\hat{\mathbf b}}_{xy}} \right) = {\mathrm{var}}\left( {\hat b_{zy}} \right) - {\hat{\mathbf b}}_{xy}^t{\mathbf{V}}_{zx}{\hat{\mathbf b}}_{xy}$$\end{document}varb^zy∣b^xy=varb^zy-b^xytV zxb^xy, and if there is no sample overlap between y and x, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{var}}\left( {\hat b_{zy}{\mathrm{|}}{\hat{\mathbf b}}_{xy}} \right) = {\mathrm{var}}\left( {\hat b_{zy}} \right) + {\hat{\mathbf b}}_{xy}^t{\mathbf{V}}_{zx}{\hat{\mathbf b}}_{xy}$$\end{document}varb^zy∣b^xy=varb^zy+b^xytV zxb^xy. More generally, if there is a sample overlap between y and x, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{x_iy}r_{p(x_i,y)}$$\end{document}ρxiyrp(xi,y) can be approximated by the intercept of a bivariate LDSC analysis between xi and y (ref. 30). Vzx is the sampling variance-covariance of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\mathbf b}}_{zx}$$\end{document}b^zx with the ij-th element \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {\hat
