In my 2018 paper with Beth Tipton, published in the Journal of Business and Economic Statistics, we considered how to do cluster-robust variance estimation in fixed effects models estimated by weighted (or unweighted) least squares. We were recently alerted that Theorem 2 in the paper is incorrect as stated. It turns out, the conditions in the original version of the theorem are too general. A more limited version of the Theorem does actually hold, but only for models estimated using ordinary (unweighted) least squares, under a working model that assumes independent, homoskedastic errors. In this post, I'll give the revised theorem, following the notation and setup of the previous post (so better read that first, or what follows won't make much sense!).
In my 2018 paper with Beth Tipton, published in the Journal of Business and Economic Statistics, we considered how to do cluster-robust variance estimation in fixed effects models estimated by weighted (or unweighted) least squares. A careful reader recently alerted us to a problem with Theorem 2 in the paper, which concerns a computational short cut for a certain cluster-robust variance estimator in models with cluster-specific fixed effects. The theorem is incorrect as stated, and we are currently working on issuing a correction for the published version of the paper. In the interim, this post details the problem with Theorem 2. I'll first review the CR2 variance estimator, then describe the assertion of the theorem, and then provide a numerical counter-example demonstrating that the assertion is not correct as stated.
There's lots of linear algebra out there that's quite useful for statistics, but that I never learned in school or never had cause to study in depth. In the same spirit as my previous post on the Woodbury identity, I thought I would share my notes on another helpful bit of math about matrices. At some point in high school or college, you might have learned how to invert a small matrix by hand. It turns out that there's a straight-forward generalization of this formula to matrices of arbitrary size, but that are _partitioned_ into four pieces.
As in many parts of life, statistics is full of little bits of knowledge that are useful if you happen to know them, but which hardly anybody ever bothers to mention.