Thread by @jmwooldridge

It seems like every week, if not more frequently, I learn something new about a basic estimation method -- OLS, 2SLS, and offshoots. My students seem skeptical when I tell them this but it's true.

This week: centering before creating squares and interactions.

#metricstotheface

Now, I've taught this in the context of OLS and 2SLS for a long time, and it comes up a lot in my introductory book. It's often needed to give main effects a sensible interpretation -- whether those are exogenous or endogenous, whether it's a pooled method or FE.

But one case where I've been too cavalier is with creating instruments out of squares and interactions of exogenous variables when, say, the structural equation includes w*xj where w is endogenous and xj is exogenous. We can use xj*zh as IVs.

I used to tell my students there's no need to center before creating xj*zh because they only show up in first stage linear projections and the 2SLS estimates are invariant. That claim is true, but there's still value in centering squares and interactions of IVs. Why?

Because we should be looking at the first stage regressions, and centering in creating the IVs makes the pattern of which exogenous variables are most relevant for which endogenous variables much easier to discern.

If I project w onto x1, z1, x1*z1 I could easily get a small and insignificant coefficient on z1, making me think I have a weak IV when in fact it's quite strong. If w*(x1 - x1bar) is also in the model then it will be more informative to project it onto (x1 - x1bar)*(z1 - z1bar).