Do I still need to include variables used to generate weights in a regression model when applying inverse probability weighting in estimation? Say, I would like to run a regression like this: $y = x_1+x_2+x_3+x_4+x_5$.
And I used $x_3$, $x_4$ and $x_5$ to generate a inverse probability weight($\pi$) for the availability of $y$.
I generate the $\pi$ with a logistic regression of the availability of $y$ (observation=0 or 1) as below:
obs = 1 if $y$ is not NA
obs = 0 if $y$ is NA
missingmodel: glm(obs ~  $x3$ + $x4$ + $x5$,
                    data=data, family=binomial)
$\pi$ = 1/missingmodel$fitted.values
If I include this $\pi$ to my main analysis, do I still need to include $x_3$, $x_4$ and $x_5$ in the regression?
i.e. 
should my model be  
(1) $y = x_1+x_2+x_3+x_4+x_5$ with $\pi$ used in estimation or
(2) $y = x_1+x_2$ with $\pi$ used in estimation. 
Thank you.
About inverse probability weighting (IPW):
In the IPW approach, a missingness model is specified, i.e. a model for the probability that an individual is a complete case. The analysis model is fitted only to complete cases, but more weight is given to some complete cases than others.
Shaun R Seaman and Ian R White. Review of inverse probability weighting for dealing with missing data. Stat Methods Med Res. 2013 Jun;22(3):278-95
 A: This really depends on your research question. There's no "rule"; it's a theory question. Let's say that one of your variables in $\pi$ is the number of adults in the household ($adults$). This would be fairly common variable to include in an inverse probability weight, because the more adults there are in a household, if only one adult is chosen to complete the survey, the probability of any given adult in the household being selected is lower. Now, let's say that your outcome of interest is whether the household is above or below a poverty threshold. You would want to include $adults$ in your model, because (a) more adults could mean more working members contributing to greater household income and thus lesser probability of household poverty or (b) more adults could mean more mouths to feed with one household income, and thus a greater probability of household poverty. You would want to be sure to have $adults$ in your model, so you could test to see which of these hypotheses (or both) are supported by your data. However, if you had reason to think that more adults would be, for example, associated with both greater non-response and greater poverty, then you would also want to review the literature on endogenous selection, to see how best to specify your model. So like many econometric questions, your answer depends on your theory and not just your data.
