# Rescaling for Imputation under the normal linear model

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 $$\dot{\epsilon}$$?