# Interpretation of incrementing both variables in an interaction term

I run a regression with an interaction term similar to the form below: $$Y = B_0 + B_1 X_1 + B_2 X_2 + B_3 X_1X_2$$ where $Y$, $X_1$, and $X_2$ are continuous variables. In terms of the interpretation, the effect of a $1$ unit increase in $X_1$ is: $$B_1+B_3X_2$$ Similarly, the effect of a $1$ unit increase in $X_2$ is: $$B_2+B_3X_1$$ But can I conclude about the impact of $X_1$ and $X_2$ from the same regression? Or am I mistakenly using the same result twice?

Is there any way to examine the incremental effects of both: $X_1$ on $X_2$, and $X_2$ on $X_1$?

But can I conclude about the impact of X1 and X2 from the same regression? Or am I mistakenly using the same result twice?

Yes, you can conclude about both X1 and X2 from the same regression, that is one of the main motivations behind multiple regression instead of two simple regressions. And no, you are not using the same result twice, you are cutting the result into pieces that are useful to you.

• +1, note that the conclusions to be drawn about B1 & B2 (ie, X1 & X2) is that they are the slopes of the regression surface along the given dimension when the other variable is exactly equal to 0. Sep 12, 2017 at 13:01

If both $X_1$ and $X_2$ increase by $1$-unit simultaneously, $Y$ will increase by $B_1 + B_2 + B_3 + B_3X_{1i} + B_3 X_{2i}$, where $X_{1i}$ and $X_{2i}$ are the values of $X_1$ and $X_2$ from which you started. The fitted $B$s are automatically scaled to the units of your variables, so the result is just simplifying a mathematical expression.

\begin{align} \hat Y_i &= \hat B_0 + \hat B_1 X_{1i} \quad\quad\ \ + \hat B_2X_{2i} \quad\quad\ \! + \hat B_3 X_{1i}X_{2i} \\ \hat Y_{i'} &= \hat B_0 + \hat B_1 (X_{1i} +1) + \hat B_2(X_{2i}\!+\!1) + \hat B_3 (X_{1i} +1)(X_{2i}\!+\!1) \\ ~ \\ &\text{subtracting the first equation from the second:} \\ ~ \\ \hat Y_{i'} - \hat Y_i &= \hat B_0 - \hat B_0 + \hat B_1 (X_{1i}\!+\!1) - \hat B_1 X_{1i} + \hat B_2(X_{2i}\!+\!1) - \hat B_2X_{2i} + \\ &\quad\ \hat B_3 (X_{1i}\!+\!1)(X_{2i}\!+\!1) - \hat B_3 X_{1i}X_{2i} \\[8pt] \Delta Y &= \hat B_1X_{1i} + \hat B_1 - \hat B_1X_{1i} + \hat B_2X_{2i} + \hat B_2 - \hat B_2X_{2i} + \\ &\quad\ \hat B_3X_{1i} X_{2i} + \hat B_3 X_{1i} + \hat B_3 X_{2i} + \hat B_3 - \hat B_3 X_{1i}X_{2i} \\[8pt] \Delta Y &= \hat B_1 + \hat B_2 + \hat B_3 + \hat B_3X_{1i} + \hat B_3 X_{2i} \end{align}