A Good Instrument is a Bad Control

Here’s a puzzle for you. What will happen if we regress some outcome of interest on both an endogenous regressor and a valid instrument for that regressor? I hadn’t thought about this question until 2018, when one of my undergraduate students asked it during class. If memory serves, my off-the-cuff answer left much to be desired. 1 Five years later I’m finally ready to give a fully satisfactory answer; better late than never I suppose! The Model We’ll start by being a bit more precise about the