In Supplementary Note, we show that the MSE of the MTAG estimates are always weakly smaller than the MSE of the corresponding single-trait GWAS estimates, which equals MSE(β^j,t)≡E[(β^j,t−βj,t)2]=E(εj,t2). Intuitively, this result holds because the MTAG estimates have smaller variance than the GWAS estimates and both are unbiased on average across all SNPs; the MTAG estimates are unbiased on average (despite being biased for particular SNPs when the homogeneous-Ω assumption is violated) because both the true effects βj,t and the MTAG estimates β^MTAG,j,t are mean zero across SNPs.
