How to simulate a new variable in regression analysis Suppose my current regression model is $y=\beta_0+\beta_1x_1+\beta_2x_2$
Now I want to add a new variable $x_3$ (normally distributed). All I know is $x_3$'s mean and standard deviation.
Is it a sound methodology to randomly generate $x_3$, according to its mean and sd, and include it in the regression (and maybe bootstrap it a few hundred times)?
 A: The only way to learn about $x_3$ is to collect data about it. Simulating random data is circular because it only tells you information about your randomization procedure.
In particular, you'll need to know about the tuples $(y, x_1, x_2, x_3)$, that is, how all of your $x$s vary with respect to the $y$ for each observational unit.
A: There are routines for generating random Normal deviates, which could be employed to add a noise variable to a regression model.
While adding such a variable has no value from the point of prediction, it can actually, per sources, unfortunately introduce bias into the regression model.
Further, the amount of the bias can be demonstrated to be proportional to the magnitude of a constructed auxiliary variable (see, for example, discussion in Some Notes on Misspecification in Multiple Regressions).
As such, while there is seemingly no potential value in investigating the impact of introducing a so-called 'irrelevant variable' in the regression analysis, it could nevertheless provide some insights on conceivable bias arising from model misspecification.
