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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?

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  • $\begingroup$ why not put totalsales - sales_d to the model isntead of total sales to avoid this problem? $\endgroup$
    – rep_ho
    Sep 29, 2021 at 23:26
  • $\begingroup$ 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. $\endgroup$
    – query_again
    Sep 29, 2021 at 23:35
  • $\begingroup$ 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. $\endgroup$
    – query_again
    Sep 29, 2021 at 23:37

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I don't see an issue as formulated it looks like the model would look at how additional sales from product I are cannibalizing or complementing sales from product D assuming observations are stores or something similar. Others more well informed in statistics may be able to pinpoint where the model is incorrect.

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