Same variable on both sides of a regression model I am sketching a regression model for examining the effects of multiple variables on the difference between a perceived value $B_i$ and predicted value $\hat{B}_i$ at any given  timepoint $i$. The $\hat{B}_i$ values are predicted by a non-statistical model.
$Y_i = \beta_0 + \beta_1(A_i) + \beta_2(\hat{B}_i) +\ ...$
where $Y_i = B_i - \hat{B}_i$.
In particular, I am interested in examining the effect of $A_i$ on $Y_i$, but I want to control for the non-statistical model predictions ($\hat{B}_i$), which I suspect deviate systematically from the perceived values ($B_i$) in my dataset. However, is it problematic to have the $\hat{B}_i$ variable on both the LHS and RHS in this way?
Alternatively, I could model $B_i$ as the response variable and continue to include $\hat{B}_i$ as a fixed effect, but modeling the difference between the two values seemed more interpretable for the particular theory I am evaluating. But, perhaps modeling $B_i$ is preferable?

Your model reduces tor typical linear regression
$$\begin{align}
B_i - \hat{B_i} &= \beta_0 + \beta_1(A_i) + \beta_2(\hat{B}_i) +\ ... = \\
B_i &= \beta_0 + \beta_1(A_i) + (1+\beta_2)(\hat{B}_i) +\ ...
\end{align}$$
Having the difference on the left-hand side doesn't make it more interpretable, as you can always re-arrange the terms back to the previous form.
