Changes in slope when deflated by a third variable

Question

Assume you observe $(y_{i},x_{i})$, for $i=1...N.$ They are linearly related by: $$ y_{i}=mx_{i} $$ Now, assume we deflate both $y_{i}$ and $x_{i}$ by another variable, $c_{i},$ which varies across $i.$ Call $y_{i}/c_{i}=\tilde{y_{i}}$ and $x_{i}/c_{i}=\tilde{x}_{i}$. I ran on OLS regression and obtained, again a slope parameter of $m,$ as $$ \frac{\frac{y_{i}}{c_{i}}}{\frac{x_{i}}{c_{i}}}=\frac{y_{i}}{c_{i}}=m $$ Assume now that we have a constant term, $b$, such that: $$ y_{i}=mx_{i}+b $$ Now, when I run a regression of $\tilde{y_{i}}$ on $\tilde{x_{i}},$ I no longer get a slope of $m.$ If $c_{i}$ were a constant $c,$ then yes, I would obtain: $$ m=\frac{cy_{2}-cy_{1}}{cx_{2}-cx_{1}}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} $$ . However, given that $c_{i}$ is itself variable, I get $$ \tilde{m}=\frac{c_{2}y_{2}-c_{1}y_{1}}{c_{2}x_{2}-c_{1}x_{1}}\neq\frac{y_{2}-y_{1}}{x_{2}-x_{1}} $$ unless specific circumstances hold. Now, although I understand what is happening algebraically, what I fail to understand is why, intuitively, just the inclusion of an intercept makes all the difference. The fact that $c_{i}$ is a variable does not affect the calculation of a slope when there is no intercept, but does in the latter case. What is going on? Why is the intercept so impactful?

Answer 1

The intuitive explanation is that when the intercept is constant, the change in rise will be directly proportional to the change in run, as both data points will lie on the same line.

Once you introduce a variable factor in the constant, you are evaluating the slope where each data point lies on a separate line, which immediately alters the value of the gradient.

We can see this clearly by writing $y_i=mx_i+b$ as $\tilde y_i=m\tilde x_i+bc_i^{-1}$ on dividing by $c_i$. The gradient is now $$\frac{\tilde y_j-\tilde y_i}{\tilde x_j-\tilde x_i}=\frac{m\tilde x_j+bc_j^{-1}-(m\tilde x_i+bc_i^{-1})}{\tilde x_j-\tilde x_i}=m+\boldsymbol{\frac{b(c_j^{-1}-c_i^{-1})}{\tilde x_j-\tilde x_i}}$$ where the term in bold indicates the "offset" between the two points. See my notes here for details.