We recommend using the in-sample estimate of ρ (denoted ρ̂), rather than the population value of ρ. Under unbiased sampling ρ̂ is consistent for ρ with O(1/N) variance, so in this case, the distinction between ρ and ρ̂ is not of great importance. Under biased sampling (as discussed in the previous section), the expected LD Score regression intercept depends on the expected sample correlation E[yi1yi2|s=1] (which is estimated consistently by ρ̂), not population ρ. Thus, we advise to use ρ̂ rather than ρ when constraining the intercept.
