I was reminded of the issue of centering IVs when creating an example using PDS LASSO to estimate a causal effect. The issue of centering controls before including them in a dictionary of nonlinear terms seems it can be important.
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#metricstotheface
The example I did included age and income as controls. Initially I included age, age^2, inc^2, age*inc. PDSLASSO works using a kind of Frisch-Waugh partialling out, imposing sparsity on the controls.
But as we know from basic OLS, not centering before creating squares and interactions can make main effects weird -- with the "wrong" sign and insignificant. This means in LASSO they might be dropped.
When I did the example, age was dropped but age*inc was kept. Then I centered to instead use (age - agebar)*(inc - incbar) (and also in the squares). Now both level terms are kept. The causal effect estimate changed nontrivially.
I couldn't find anywhere where this is discussed with LASSO generally. I'd appreciate a citation if anyone has one. I haven't done a simulation, but this one example and my intuition tell me centering controls in nonlinear functions when using LASSO might be important.
With OLS and forcing all controls to remain, it doesn't matter whether we center the controls or not; the estimated causal effect is invariant when assumed constant. But LASSO is different because it's a selection method.
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