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I have the following model: $$y=\beta_{0}+\beta_{1}x_1+\beta_{2}x_2+\beta_{3}x_1x_2+u$$

where

$y$=dependent variable

$x_1$=standardized independent variable

$x_2$=dummy (binary) independent variable

$x_1x_2$=interaction term of $x_1$ and $x_2$.

The interpretation of standardized variable is a bit tricky. If we give (ceteris paribus) an additional one unit of standard deviation to $x_1$, $y$ will differ by $\beta_1$. What is the correct interpretation of the interaction term in this case? Does $\beta_3$ mean an additional unit or an additional unit of standard deviation (c.p.) affecting $y$?

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Notice that, if $x_2=0$, then $$y= \beta_0 + \beta_1 x_1 + u$$ and that, if $x_2=1$, then $$y= (\beta_0+\beta_2) + (\beta_1+\beta_3) x_1 + u,$$ so $(\beta_1+\beta_3)$ is how much one would expect $y$ to change for a unitary change in $x_1$ if $x_2=1$ and $\beta_1$ is how much one would expect $y$ to change for a unitary change in $x_1$ if $x_2=0$.

That is, the expected change in $y$ depend on whether $x_2=0$ or $x_2=1$.

In that context, one way to think of $\beta_3$ is as an additional (positive or negative) expected change in $y$ when $x_2=1$ beyond the expected change $\beta_1$. It is possible even that $\beta_1$ and $\beta_3$ cancel out one another, in which case $y$ is constant w.r.t. $x_1$ when $x_2=1$.

Since you mention no other transformation besides standardization, both $\beta_1$ and $\beta_3$ must be thought of as the expected change in $y$ in case of observing an increase of one standard deviation in the independent quantitative variable in its original scale.

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