Independent variable correlated with dependent variable We are doing a regression to predict APR for auto loans.  Among other independent variables, can I use Dealer mark-up?  The issue is that Dealer mark-up is a part of APR: Dealer Mark-up = APR-Note Rate.
 A: Rearranging your equation so that the response variable is on the left-hand-side you get:
$$\text{APR} = \text{Dealer Markup} + \text{Note Rate}.$$
If Dealer_Markup is a known explanatory variable in your model then the regression is essentially just a regression for the Note_Rate.  The former variable serves only as a fixed "offset" variable in the regression model.  Given a set of explanatory variables that can be used as predictors of the latter, you could model this using either of the following equivalent models in R:


*

*Model 1: APR ~ offset(Dealer_Markup) + Explanatory_Var1 + ... + Explanatory_Varm

*Model 2: Note_Rate ~ Explanatory_Var1 + ... + Explanatory_Varm
In the first model we directly model APR and we use Dealer_Markup as an offset variable in the regression, so that its value enters the model with a fixed coefficient of one.  In the second model we ignore  APR and Dealer_Markup and go straight to modelling the Note_Rate.  In both cases the models will give the same outputs for the estimated coefficients on the explanatory variables, and will have the same goodness-of-fit statistics.
As you can see, using Dealer_Markup in a regression of APR is essentially equivalent to working directly with a regression on Note_Rate.

What is the purpose of your model? As a broader level, if you want to know if this is a sensible model you need to take a step back and as yourself the purpose of your regression.
If the purpose is predictive (i.e., you want to predict APR from other variables) then you should consider what information you have available at the time of prediction, and in particular, whether you have knowledge of Dealer_Markup in that prediction problem.
If the purpose of the model to to make causal inferences then things get a little trickier.  In this case you must be aware of the possible presence of confounding variables and collider variables.  The Dealer_Markup could be a collider variable in this case, and you should have a deeper think about the plausible causal structure of your problem, and the goal of the regression.
A: Shouldn't be a problem. If you don't have anything else in the model your regression coefficient would just be 1 (assuming that formula in your question is the whole formula for Mark-up)
In general with regression, you should believe your dependent variable is correlated in some way with your independent variable(s). Those correlations don't necessarily have to be linear, but there should be some relationship there. Otherwise your regression model won't be any good!
