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I use publicly available EU-Silc data to estimate the market price of social dwellings (subsidized dwellings). However my X variables are almost perfectly available, my Y variable rent is often missing.

Given observed Y and X variables I have very poor fit. I assume that if I would have completed or almost completed Y variables I would have a better fit. Hence I did some imputation for Y's.

Is there any reason that I should not do that?

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Yes, there are reasons to not do that. Consider three variables $x$, $y$, and $z$. If $y$ is missing, then $x$ and $z$ are used to impute $y$. Then, if a regression equation like $y=x+x$ is used, then of course there will be great fit. I will search for the source, but one method is to do the imputation, but then delete cases where the outcome was imputed.

Edit: A method known as Multiple Imputation with Deletion is recommended by von Hippel (https://onlinelibrary.wiley.com/doi/epdf/10.1111/j.1467-9531.2007.00180.x), but it has its detractors (https://modeling.uconn.edu/wp-content/uploads/sites/1188/2016/05/Don%E2%80%99t-be-Fancy.-Impute-Your-Dependent-Variables.pdf).

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    $\begingroup$ Thank you for your answer @Jay. The links you are referring actually providing conflicting results. But they I enjoyed to read. $\endgroup$ Commented Oct 31, 2018 at 22:43

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