<p>Suppose one estimates:
<span class="math-container">$$y_i =\beta_0 +\beta_1 x_i +\varepsilon_i$$</span>
with weighted least squares using <span class="math-container">$w_i$</span> as weights. If one were instead to estimate</p>
<p><span class="math-container">$$y_i =\beta_0 +\beta_1 x_i +\beta_2 w_i +\varepsilon_i$$</span></p>
<p>and still use <span class="math-container">$w$</span> 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.</p>

CC BY-SA 4.0
2021-09-21T02:51:21.293Z
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..
mark leeds
CC BY-SA 4.0
2021-09-21T07:28:13.530Z
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.
Michael Gmeiner
CC BY-SA 4.0
2021-09-21T13:33:19.130Z
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.
mark leeds
CC BY-SA 4.0
2021-09-22T14:29:36.457Z
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.
chan1142
CC BY-SA 4.0
2021-09-23T03:21:31.063Z
@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.
mark leeds
CC BY-SA 4.0
2021-09-23T03:33:37.950Z
I think the answer depends only on whether you want to control for $w_i$ (independently of $w_i$ being used as weights). And then the question of whether to use OLS or WLS follows. Not the other way around. So, if you believe that controlling for $w_i$ (in the mean equation) is sensible, then it is OK. Once your model is determined, you select on estimation method. If you decide to use WLS, you need to decide on the weight variable, when the weight being a regressor or not does not matter. A regressor can be used as a weight variable for WLS if you believe the error variance depends on it.
chan1142
CC BY-SA 4.0
2021-09-23T03:39:08.987Z
Say, Step 1 is the mean specification (the model), and Step 2 is error variance specification. We don't begin with step 2. Whether $y_i = \beta_0 + \beta_1 x_i + \beta_2 w_i + \varepsilon_i$ is OK or not is a matter of Step 1, not Step 2. Once Step 1 is done, you move on to Step 2. Not the other way around. Don't let your model (Step 1) be determined by Step 2. That's what I want to say.
chan1142
CC BY-SA 4.0
2021-09-23T11:02:01.023Z
yes. I agree. Predictor variables should be chosen first. Then, once that is done,  estimation method. good point.
mark leeds
<econometrics>
Weighted OLS with weights as a regressor
Michael Gmeiner
# Weighted OLS with weights as a regressor

(Asked by: username_0 on 2021-09-20)

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.

**Tags:** econometrics
> **Comments:**
> **username_1**: 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..  
> **username_0**: 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.  
> **username_1**: 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.  
> **username_2**: 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.  
> **username_1**: @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.  
> **username_2**: I think the answer depends only on whether you want to control for $w_i$ (independently of $w_i$ being used as weights). And then the question of whether to use OLS or WLS follows. Not the other way around. So, if you believe that controlling for $w_i$ (in the mean equation) is sensible, then it is OK. Once your model is determined, you select on estimation method. If you decide to use WLS, you need to decide on the weight variable, when the weight being a regressor or not does not matter. A regressor can be used as a weight variable for WLS if you believe the error variance depends on it.  
> **username_2**: Say, Step 1 is the mean specification (the model), and Step 2 is error variance specification. We don't begin with step 2. Whether $y_i = \beta_0 + \beta_1 x_i + \beta_2 w_i + \varepsilon_i$ is OK or not is a matter of Step 1, not Step 2. Once Step 1 is done, you move on to Step 2. Not the other way around. Don't let your model (Step 1) be determined by Step 2. That's what I want to say.  
> **username_1**: yes. I agree. Predictor variables should be chosen first. Then, once that is done,  estimation method. good point.  

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## 0 answers

