RDP 2018-08: Econometric Perspectives on Economic Measurement Appendix B: Loosening Strict Error Exogeneity Introduces Another Avenue for Weights to Generate Inconsistency

Consider weights that are exogenous, being functions only of the regressors. Retain the strict error exogeneity assumption. Using the law of iterated expectations, Equation (A3) becomes

So there is no inconsistency. But without strict error exogeneity, Equation (B3) does not follow from Equation (B2). Consistency is not guaranteed, even with well-behaved weights.