value of βj,t that minimizes a weighted sum of the squared deviations from the moment conditions, with deviations weighted more heavily if they are estimated more precisely. In the s, we show that the solution to the minimization problem in equation (2) is: β^MTAG,j,t=ωt′ωtt(Ω−ωtωt′ωtt+∑j)−1ωt′ωtt(Ω−ωtωt′ωtt+∑j)−1ωtωttβ^j.Standard asymptotic properties of GMM relate to T→∞. In the s, we show that for fixed number of traits T, as the sample size for the GWAS of any trait t becomes large, the MTAG estimator β^MTAG,j,t is consistent and asymptotically normal.
