# Dependent variable functionally related to independent variable

My question pertains to the case where a dependent variable is functionally related to an independent variable in a multiple linear regression. Here is a fictional scenario suppose that $$Sales_A,Sales_B,Sales_C,Sales_D,Sales_E$$ represent the sales resulting from products A,B,C,D,E and Total Sales = $$Sales_A +Sales_B +Sales_C + Sales_D + Sales_E$$. Consider the following model: $$Sales_D = Sales_I + Total Sales + \text{other control variables}$$

Assuming the assumptions for linear regression are met and there is no multicollinearity between the independent variables. If the goal is to determine how increases in $$Sales_I$$ effect changes in $$Sales_D$$ would the above model work or does the fact that the dependent variable is "part" of one of the independent variables cause problems?

• why not put totalsales - sales_d to the model isntead of total sales to avoid this problem? Sep 29, 2021 at 23:26
• Wouldn't controlling for (totalsales - sales_d) be different than controlling for total_sales? In the latter the interpretation of the coefficient for Sales_I would be holding total sales constant the effect is \beta_1 which is different than holding (totalsales - sales_d) constant. Sep 29, 2021 at 23:35
• also I'm curious if there are any technical reasons as to why it won't work as opposed to a workaround so the question is more theoretical in nature. Sep 29, 2021 at 23:37