Miles Kimball: Adding a Variable Measured with Error to a Regression Only Partially Controls for that Variable https://blog.supplysideliberal.com/post/2019/10/10/adding-a-variable-measured-with-error-to-a-regression-only-partially-controls-for-that-variable: "Compare the coefficient estimates in a large-sample, ordinary-least-squares, multiple regression with (a) an accurately measured statistical control variable, (b) instead only that statistical control variable measured with error and (c) without the statistical control variable at all. Then all coefficient estimates with the statistical control variable measured with error (b) will be a weighted average of (a) the coefficient estimates with that statistical control variable measured accurately and (c) that statistical control variable excluded. The weight showing how far inclusion of the error-ridden statistical control variable moves the results toward what they would be with an accurate measure of that variable is equal to the fraction of signal in (signal + noise), where “signal” is the variance of the accurately measured control variable that is not explained by variables that were already in the regression, and “noise” is the variance of the measurement error....

#noted #2019-10-14