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I am currently working through Stef van Buuren's book Flexible Imputation of Missing Data and I am currently at Chapter 3.2: https://stefvanbuuren.name/fimd/sec-linearnormal.html.

The setup consists of a univariate imputation target $y$ and predictors that are collected in $X$. $X_{obs}$ contains the subset of $n_1$ rows of $X$ for which $y$ is observed and $X_{mis}$ contains the complementing subset of $n_0$ rows of $X$ for which $y$ is missing. The vector containing the $n_1$ observed data in $y$ is denoted by $y_{obs}$ and the vector of $n_0$ imputed values in $y$ is indicated by $\dot{y}$.

Next to two other methods, the OLS method and the OLS method including noise are described:

  1. OLS: $\dot{y}=\hat{\beta}_0+X_{mis}\hat{\beta}_1$, where $\hat{\beta}_0$ and $\hat{\beta}_1$ are the least squares estimates from the observed data
  2. Stochastic OLS: $\dot{y}=\hat{\beta}_0+X_{mis}\hat{\beta}_1+\dot{\epsilon}$, where $\dot{\epsilon}$ is randomly drawn from the normal distribution $\dot{\epsilon} \sim N(0,\hat{\sigma}^2)$.

Instead of sampling $\dot{\epsilon}$, is there a reason it is not advisable to take $\dot{y}$ and scale it so that its standard deviation matches the increase in the standard deviation due to $\dot{\epsilon}$?

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On having a fresh look at the problem, it seems to me that the main reason is that the scaling also impacts the mean of the imputation in an unintended way and not only the standard deviation as intended, unlike the Stochastic OLS.

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