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    $\begingroup$ Since your model is $$E[Y] = Z_r\beta_{Z_r} = (XV_r)\beta_{Z_r} = X(V_r\beta_{Z_r}),$$ the connection between the PCR coefficients $\beta_{Z_r}$ and the coefficients of $X$ (on the right hand side) seems particularly clear and simple. $\endgroup$ Commented Oct 23, 2016 at 16:36
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    $\begingroup$ @whuber thanks a lot, this makes sense. I guess I missed the forest for the trees. $\endgroup$ Commented Oct 23, 2016 at 17:22