MTAG is a generalized method of moments (GMM) estimator. To obtain the key moment conditions we will use, we consider the best linear prediction of the GWAS estimate for trait s, β^j,s, from the SNP’s true effect on trait t, βj,t. We use a first-order condition of this best linear prediction as the moment condition for trait s: E(β^j,s−ωstωttβj,t)=0, where ωst is the (s,t)th element of Ω. There are T such moment conditions for s=1,2,…,T, giving us the vector of moment conditions: (1)E(β^j−ωtωttβj,t)=0. where ωt is a vector equal to the tth column of Ω. Although βj,t is a random effect, we aim to estimate its (unknown) realized value. The efficient GMM estimator for βj,t based on the vector of moment conditions in equation (1) solves (2)β^MTAG,j,t=argminβj,t(β^j−ωtωttβj,t)′WQ(β^j−ωtωttβj,t) where WQ=[Var(β^j−ωtωttβj,t)]−1=(Ω−ωtωt'ωtt+∑j)−1 is the efficient weight matrix. Intuitively, the GMM estimator chooses the 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
