This is regarding the simple case of y=mx+b. It's my understanding that the OLS estimator must necessarily pass through the mean of X and Y. How do you prove that this is always so?
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1Please add the self-study tag. – Michael R. Chernick Jan 02 '20 at 16:24
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Done, albeit very late :) – Hexatonic Jun 02 '20 at 00:05
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Because the parameter estimate for the intercept term is: $$\hat b=\bar y-\hat m\bar x$$ which leaves OLS with one choice: going through the mean.
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Interestingly, as the mean is sensitive to outliers, so apparently is an OLS's parameter estimate of intercept and slope. Least Absolute Deviation regression, per my experience, can pass through one of the observed points (or, for the one variable case, the robust median!). – AJKOER Jan 05 '20 at 14:14