\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}$$\begin{array}{ccccc}\\ {\mathrm{cov}}\left( {\hat b_{xy\left( i \right)},\hat b_{xy\left( j \right)}} \right) \approx \frac{r}{{b_{zx\left( i \right)}b_{zx\left( j \right)}}} \sqrt {{\mathrm{var}}\left( {{\hat b}_{zy\left( i \right)}} \right){\mathrm{var}}\left( {{\hat b}_{zy\left( j \right)}} \right)} +b_{xy\left( i \right)}b_{xy\left( j \right)} \\ \left[ \frac{r{\sqrt {{\mathrm{var}}\left( {\hat b_{zx\left( i \right)}} \right){\mathrm{var}}\left( {\hat b_{zx\left( j \right)}} \right)}}}{{b_{zx\left( i \right)}b_{zx\left( j \right)}}} - \frac{{ {{\mathrm{var}}\left( {\hat b_{zx\left( i \right)}} \right){\mathrm{var}}\left( {\hat b_{zx\left( j \right)}} \right)}}}{{b^2_{zx\left( i \right)}b^2_{zx\left( j \right)}}} \right] \end{array}$$\end{document}cov b ^xyi,b^xyj≈rbzxibzxjvarb^zyivarb^zyj+bxyibxyjrvarb^zxivarb^zxjbzxibzxj-varb^zxivarb^zxjbzxi2bzxj2, where subscripts i and j represent SNP i and j, respectively, r is LD correlation between the two SNPs (not available in the summary data but can be estimated from a reference sample with individual-level genotypes). The i-th diagonal 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}$${\mathrm{var}}\left( {\hat b_{xy\left( i \right)}} \right) = b_{xy(i)}^2\left[\frac{{{\mathrm{var}}\left( {\hat b_{zx\left( i \right)}} \right)}}{{b_{zx\left( i \right)}^2}} + \frac{{{\mathrm{var}}\left( {\hat b_{zy\left( i \right)}} \right)}}{{b_{zy\left( i \right)}^2}} - \frac{{{\mathrm{var}^2}\left( {\hat b_{zx\left( i \right)}} \right)}}{{b_{zx\left( i \right)}^4}} \right]$$\end{document}varb^xyi=bxy(i)2varb^zxibzxi2+varb^zyibzyi2-var2b^zxibzxi4. Therefore, we can estimate bxy from all the instruments using the generalized least
