Statisticians very often control for proxies when the actual causal variable can’t be measured; for instance, party affiliation might be used as a proxy for political beliefs. Because Z isn’t a perfect measure of M, some of the influence of X on Y might “leak through” if you control for Z. Nevertheless, controlling for Z is still a mistake. While the bias might be less than if you controlled for M, it is still there. For this reason later statisticians, notably David Cox in his textbook The Design of Experiments (1958), warned that you should only control for Z if you have a strong prior
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