Thread by @jmwooldridge

I should admit that my tweets and poll about missing data were partly self serving, as I'm interested about what people do. But it was a mistake to leave the poll initially vague. I haven't said much useful on Twitter in some time, so I'll try here.

#metricstotheface

I want to start with the very simple case where there is one x and I'm interested in E(y|x); assume it's linear (for now). Data are missing on x but not on y. Here are some observations.

1. If the data are missing as a function of x -- formally, E(y|x,m) = E(y|x) -- the CC estimator is consistent (even conditionally unbiased).
2. Imputing on the basis of y is not and can be badly biased.
3. Inverse probability weighting using 1/P(m=0|y) also is inconsistent.

4. Missingness based on x is "exogenous selection" to economist and is "not missing at random" to statisticians. To me, this language is difficult to reconcile, but it is "just" language. "Selection on observables" is neutral: observables could be y or x.

5. In economics, it seems odd to assume x is missing based on y rather than based on x. At a minimum, if I use CC and MI and get different answers, I can't prefer MI. They use diff assumptions, and I prefer selection based on x.

6. If one assumes selec based on y -- MAR to statisticians -- then between IPW and MI, I prefer IPW. It requires fewer assumptions for consistency because only P(m=0|y) needs to be specified in addition to E(y|x). And, IPW applies directly to any model without full dist assumps.

7. IPW identifies the params in the linear projection L(y|1,x) if P(m=0|y) is correct. Imputation has no known robustness. This robustness of IPW is the source of certain doubly robust treatment effect estimators based on IPW and regression.