Genetic correlation is (asymptotically) proportional to Mendelian randomization estimates. If we use a genetic instrument gi:=∑j∈SXijβj to estimate the effect b12 of y1 on y2, the 2SLS estimate is b̂2SLS:= gTy2/gTy1 [59]. The expectations of the numerator and denominator are E[gTy2]=ρS(y1, y2) and E[gTy1]=hS2(y1). Thus, plimN→∞b^2SLS=rS(y2,y1)hS2(y1)/hS2(y2). If we use the same set S of SNPs to estimate b12 and b21 (e.g., if S is the set of all common SNPs, as in the genetic correlation analyses in this paper), then this procedure is symmetric in y1 and y2.
