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Suppose I run a simple linear regression of an outcome variable on a predictor variable. If I save the fitted values from this regression and then run a second regression of the outcome variable on the fitted values, what will I get?

Welcome to the first installment of the Econometrics Puzzler, a new series of shorter posts that will test and strengthen your econometric intuition. Here’s the format: I’ll pose a question that requires only introductory econometrics knowledge, but has an unexpected answer.

If you study enough econometrics or statistics, you’ll eventually hear someone mention “Stein’s Paradox” or the “James-Stein Estimator”. You’ve probably learned in your introductory econometrics course that ordinary least squares (OLS) is the best linear unbiased estimator (BLUE) in a linear regression model under the Gauss-Markov assumptions.

By the end of a typical introductory econometrics course students have become accustomed to the idea of “controlling” for covariates by adding them to the end of a linear regression model.

Here’s a slightly unusual exercise on the topic of Bayes’ Theorem for those of you teaching or studying introductory probability. Imagine that you’re developing a diagnostic test for a disease.