variables to test for causality. We call this method a generalized SMR (GSMR) analysis. The basic idea of GSMR is that if x is causal for y, any SNP associated with x will have an effect on y, and the expected value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b_{xy(i)}$$\end{document}b^xy(i) at any SNP i will be identical in the absence of pleiotropy. Let m be the number of GWS top SNPs associated with x after clumping. We have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\mathbf b}}_{xy} = \left\{ {\hat b_{xy\left( 1 \right)},\hat b_{xy\left( 2 \right)}, \cdots, \hat b_{xy(m)}} \right\}$$\end{document}b^xy=b^xy1,b^xy2,⋯,b^xy(m) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b_{xy(i)} = \hat b_{zy(i)}/\hat b_{zx(i)}$$\end{document}b^xy(i)=b^zy(i)∕b^zx(i), and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\mathrm b}}_{xy}\sim N\left( {{\bf 1} {\bf b}_{xy},{\mathbf V}} \right)$$\end{document}b^xy~N1bxy,V where 1 is an m × 1 vector of ones and V is the variance-covariance matrix of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\mathbf b}}_{xy}$$\end{document}b^xy. We have derived previously that the ij-th element of V is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{ccccc}\\ {\mathrm{cov}}\left( {\hat b_{xy\left( i \right)},\hat b_{xy\left( j \right)}} \right) \approx
