# 2-dimensional represent of a multiple regression function Y_i=beta_1+beta_2X_(2i)+beta_3X_(3i)+u_(i)

Hunter Shah 2022-10-13 Answered
2-dimensional representation of a multiple regression function
Supposing I have a multiple regression population function of the form:
${Y}_{i}={\beta }_{1}+{\beta }_{2}{X}_{2i}+{\beta }_{3}{X}_{3i}+{u}_{i}$
with ${X}_{3i}$ a dummy variable (only takes values $0$ and $1$).
I am given a sample of points. Although the latter takes place in 3 dimensional space, the question states "its results can be represented in $Y$ vs ${X}_{2}$ space". I don't understand how graphing $Y$ vs ${X}_{2}$ will give us a 2 dimensional representation of our population regression function. Isn't ${X}_{3i}$ being completely omitted?
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scranna0o
Assuming that I properly understand your problem, you have
$Y={\beta }_{1}+{\beta }_{2}{X}_{2}+{\beta }_{3}{X}_{3}$
where ${X}_{3}$ is a binary variable.
This means
${X}_{3}=0\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}Y={\beta }_{1}+{\beta }_{2}{X}_{2}$
${X}_{3}=1\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}Y=\left({\beta }_{1}+{\beta }_{3}\right)+{\beta }_{2}{X}_{2}$
and then, the two dimensional representation of the function (two parallel lines).