A quadratic OLS regression with two predictors is defined as: $Z = b_0 + b_1X + b_2Y + b_3X^2 + b_4XY + b_5Y^2 + e$ (1)
If this regression surface is plotted, it can look like this ($b_0=10, b_1=-.15, b_2=.82, b_3=.15, b_4=.33, b_5=-.30$):
Now I want to derive the two-dimensional functional form of several intersections with a vertical plane. For two special cases, the slopes of this intersection can be easily calculated:
- For the solid black line in the first Figure (where X = Y): the linear slope of this line is $a_1 = b_1 + b_2$, and the quadratic component is $a_2 = b_3 + b_4 + b_5$
- For the dashed black line (where X = -Y): the linear slope is $a_3 = b_1 - b_2$, and the quadratic component is $a_4 = b_3 - b_4 + b_5$.
Question: How can I derive the functional form of the intersection with arbitrary vertical planes?
Specifically, I am interested of the intersection at a particular plane. Consider following contour plot of the same regression surface:
The solid black line is the first principal axis, which is defined by following equations ($p_{10}$ = intercept, $p_{11}$ = slope):
$p_{11} = (b_5 - b_3 + \sqrt{(b_3 - b_5)^2 + b_4^2}) / b_4$ $p_{10} = [(b_1 b_4 - 2 b_2 b_3) / (4 b_3 b_5 - b_4^2)] - p_{11} [(b_2 b_4 - 2 b_1 b_5) / (4 b_3 b_5 - b_4^2)]$
In the specific example, $p_{10} = 1.23$ and $p_{11} = 0.33$.
Now, what is the form of the surface along this line?
I am interested in a solution for the first principal axis, as well as a general solution for arbitrary vertical planes.
Any vertical plane can be described by an equation of the form $ax+by = c$. Dividing $a,b,c$ by some common number doesn't change the plane. So, after doing this division, if necessary, we can assume that $\sqrt{a^2 + b^2} = 1$.
Points on this plane can be described by parametric equations $x=ca + tb, y=cb - ta$. You can check that points $(x,y)$ given by these equations satisfy the plane equation for all $t$.
If you substitute $X=ca + tb, Y=cb - ta$ into your surface equation, you get a formula for $Z$ as a function of $t$, which represents the curve of intersection.
The curve will be a quadratic function of $t$, actually, which means that it's a parabola.