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Can we see the beta coefficients in OLS as mean values?

I mean the estimator β alone.

y=Xβ+ε

user2986288
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  • You might say it is the weighted mean value of the slopes. – KittyL Feb 25 '15 at 09:38
  • @KittyL. I do not agree since $$b=\frac{\sum x_iy_i}{\sum x_i^2}$$ – Claude Leibovici Feb 25 '15 at 11:24
  • That is $\frac{x_1^2}{\sum x_i^2} \frac{y_1}{x_1} + \frac{x_2^2}{\sum x_i^2} \frac{y_2}{x_2} + ... + \frac{x_n^2}{\sum x_i^2} \frac{y_n}{x_n}$. Isn't that a weighted average of the $\frac{y_i}{x_i}$'s? – KittyL Feb 25 '15 at 11:39
  • Beta coefficients can be b=ΣxY/Σx^2, after manipulation. Only Y is not a deviated value here. So, it is a weight. But I am not sure if it is a mean. – user2986288 Feb 25 '15 at 11:55

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No, since the coefficient is the slope of line which minimizes the sum of squared residuals. In that sense it cannot be interpreted as the as the mean, you should rather interpret it as the marginal impact of X on Y.

Chiara Tomasi
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