Given a linear model like: $y = \alpha + \beta_{1}X1 + \beta_{2}X2 + \beta_{3}X3$

And if I want to interpret the marginal effects of $\beta_{1}$, then I am wondering about its interpretation.

I would calculate its slope along X1 values of interest by doing something like:

Marginal Effect: $ \mu_{i} = \alpha + \beta_{1} * X_{i}1 $

The uncertainty that I have is; I calculated the value of $\beta_{1}$ conditional on the $\beta_{2}$ and $\beta_{3}$; so the parameters were updated together. The marginal effect, however, does not take into account the covariates explicitly, although like I said, $\beta_{1}$ was optimized given $\beta_{2}$ and $\beta_{3}$.

So, I am wondering if I would still say something like:

A one-unit increase in X1 is associated with a n change in $\mu$, on average, after controlling for $\beta_{2}$ and $\beta_{3}$. Or, should no claim be made to the other variables?


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