I think equation 8 is correct. Their proof in the appendix checks out. The SD correction is for simple correlations, but they're computing a correction for a beta. If you take the formula for correcting a correlation for attenuation, and you convert both sides (correction and observed) to betas using r= (sd(x)/sd(y))B, then you recover equation 8.
I feel like if you could just collaborate with a statistician or econometrician who deeply knows the theory of errors-in-variables models, and would just need to get up to speed on the sibling regression model, the two of you could derive a theoretical correction factor for the data generating process that you assume here.
I think equation 8 is correct. Their proof in the appendix checks out. The SD correction is for simple correlations, but they're computing a correction for a beta. If you take the formula for correcting a correlation for attenuation, and you convert both sides (correction and observed) to betas using r= (sd(x)/sd(y))B, then you recover equation 8.
And yet it's the wrong equation, since the correction is done using sqrt(reliability).
I feel like if you could just collaborate with a statistician or econometrician who deeply knows the theory of errors-in-variables models, and would just need to get up to speed on the sibling regression model, the two of you could derive a theoretical correction factor for the data generating process that you assume here.