Third, we estimate the mixed fourth moments E(α1α23) using the following equation: (6)E(α^1α^23∣α1,α2)=α1α23+3α1α2σ22+α22σ12+3σ12σ22, where σ22 is the LD score regression intercept for trait 2 (normalized by dividing by s22) and σ12 is the cross-trait LD score regression intercept (normalized by dividing by s1s2). For simulations with no LD, we fix σ22=1∕s22 and σ12 = 0. Thus, we obtain an estimate κ^k of κk=E(αk2α1α2)−3ρg by computing the weighted average of α^1α^23 over SNPs (with weights given by equation 5), and subtracting 3α^1α^2σ22+α22σ12+3σ12σ22.
