# Weighted OLS with weights as a regressor

Suppose one estimates: $$y_i =\beta_0 +\beta_1 x_i +\varepsilon_i$$ with weighted least squares using $$w_i$$ as weights. If one were instead to estimate

$$y_i =\beta_0 +\beta_1 x_i +\beta_2 w_i +\varepsilon_i$$

and still use $$w$$ as weights, are there any fundamental issues? In a few minutes of thinking it seemed like this is still okay, but I wanted to confirm I'm not overlooking any logical error in this.

• Hi: Usually, if one is using weights in an OLS, that means that the variances of the error terms are not assumed constant and are different for easch $i$. . So, by using weights, one gives less weight to observations where the variance is greater and vice versa. So, the second case won't achieve what the first case is doing if that's what you're asking. You won't get any errors when you run the regression but it won't have any meaning besides adding more noise to the original regression because it's not using weights in the correct way.. Sep 21 at 2:51
• I think my question may not be clear. Both regressions are weighting by $w$. In the second, I'm also including $w$ as a regressor. My question was motivated by a Syverson paper regarding plane tickets. Average price was the outcome and the regression was weighted by # of passengers. I was curious if # of passengers could be included in the regression. Sep 21 at 7:28
• Oh. Now it makes more sense and I think that should be okay. Since, you clarified, hopefully someone else will say something but I'm pretty certain it's okay to kind of have it in two places. I never heard of that before but I don't see anything wrong with it. Sep 21 at 13:33
• As sensible as WLS of $y_i = b_0 + b_1 x_i + e_i$ using $x_i$ as weights. Including $w_i$ as a regressor is one thing, doing WLS using one of the regressors is another. They are two totally different things. One is about the mean, the other about the variance. Sep 22 at 14:29
• @chan1142: I agree about variance versus mean in terms of them being totally two different things. But are you saying that makes the second one okay or not okay ? To me, it seems okay but I know that you know your stuff so I'd like to know what you are saying about the validity of the second one. Sep 23 at 3:21