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In OLS regression, do the control variables need to have linear relationship to the dependent variable (linearity assumption) or just the independent/explanatory variables? Thank you.

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Ordinary Least Squares (OLS) regression assumes a linear relationship between each independent (or explanatory) variable and the dependent variable$^1$. Typically, control variables are not contrasted with independent variables, as mathematically they are the same$^2$. You seem to be referring to certain independent variables as control variables because they are of less (or no) substantive interest. This is fine, just remember that control variables are still independent variables, despite not being of primary interest. Therefore, a linearity assumption is still being made.

For more information on regression and its assumptions, I highly recommend Miles & Shevlin (2001) and Osborne & Waters (2019).

Also, thanks to @Richard Hardy for the suggestion, the assumption of linearity pertains to the linear relationship between the dependent variable $y=(y_1,...y_i,...,y_n)$ and each independent variable $x=(x_1,...,x_j,...,x_J)$. This means that the model predicts the dependent variable as a linear combination of the independent variables, or more specifically $y_{i} = \beta_0 + \beta_1x_{i1}+\dots+\beta_{iJ}x_{iJ} + \epsilon_i$. The linearity assumption is with respect to the coefficients of the model (i.e., $\beta$) and not necessarily to the form of the variables. Thus, and as @Richard Hardy points out, transformations of the independent variables (e.g., $x^2$, $\log(x)$, $\cos(x)$) are still permitted.

$^1$ Note that this is also the case for multivariate regression, where there are more than 1 dependent variable.

$^2$ Thanks to @Dave in the comments for suggesting I clarify this.

References

Miles, J., & Shevlin, M. (2001). Applying regression and correlation: A guide for students and researchers. Sage.

Osborne, J. W., & Waters, E. (2019). Four assumptions of multiple regression that researchers should always test. Practical Assessment, Research, and Evaluation, 8(1), 2.

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    $\begingroup$ +1 “Control variable” just describes where the primary interest is (or isn’t). The math really couldn’t care less what we call a “control” variable. $\endgroup$
    – Dave
    Mar 2 at 22:40
  • $\begingroup$ @Dave Thanks for suggesting I clarify this. I was trying to speak to the language brian used in their question, though I think I improved my response. $\endgroup$ Mar 2 at 23:15
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    $\begingroup$ $y=\beta_0+\beta_1 x+\beta_2 x^2+\varepsilon$ is (still) a linear regression. You may want to phrase your answer to account for cases like this. $\endgroup$ Mar 3 at 11:39
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    $\begingroup$ The way you phrase it sounds internally contradictory to me, but it is good that you draw the reader's attention to that. $\endgroup$ Mar 4 at 6:42

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