Appendix A: Proof of Equation (18)

The asymptotic variance of ŷ_{1,OLSb ME} can be approximated as follows:



where all terms were defined in the main text. Thus, the associated non-centrality parameter is:



where the last equality holds using the definition of λ. Note that the partial squared correlation, R_y,M|T² can be expressed as follows:



where R_y,T² is the squared population correlation between y_ijk and T_i. Thus, solving (A.3) for (1 -R_M,T²) and inserting this expression into (A.2) yields:



Finally, (18) can be obtained from (A.4) using the two relations: (1) σ² =σ_y²(1 -R_y,M^Obs,T²), where R_y,M^Obs,T² is the total regression R² value; and (2) (1 -R_y,M^Obs,T²) =(1 -R_y,T²)(1 -λR_y,M|T²).