Population regression coefficients
It is well known that the slope coefficient in a bivariate regression equals the covariance between the dependent and independent variable divided by the variance of the independent variable. But did you know that the slope coefficients in a trivariate regression can also be expressed in terms of variances and covariances? See here.
For a long time, I struggled to find a satisfactory explanation for why the regression approach to difference-in-differences (DiD) recovered the correct DiD parameter. I finally found the time to sit down and work it out. This document gives the answer and along the way demonstrates what the least squares coefficients recover when all variables vary only at the group level.
Two-way fixed effects
In this document, I show how to prove what a regression on individual-level and time fixed effects recovers.