In a linear model, can the dependent variable be a linear combination containing one of the independent variables

Question

In an OLS linear model framework, I am wondering if there are any problems in applying a linear model such as:

$$ Z - X = \beta_0 + \beta_1 X + \epsilon $$

My general feeling is that having $X$ on the LHS and RHS is problematic but i haven't been able to find anything to confirm this.

Answer 1

There is no problem; you can just move X to the RHS and replace $\beta_1$ with $\beta_1 + 1$. You can then obtain the OLS estimator and subtract $1$ to recover $\beta_1$.