Linear regression with 3 evenly spaced points

Question

I have a set of three measurements $(x_i, y_i)$ where the x-values are equally spaced. I am interested in extracting the linear slope between these three points (while the intercept has no physical meaning, as there is an unknown offset).

Nevertheless, a least-squares linear fit of these three points would result in the slope being computed simply as $m = \frac{y_3-y_1}{x_3-x_1}$ because $x_2 = \bar{x}$.

Is there any method that include the information from $x_2$ in the computation of the slope?

Answer 1

If your only goal is to somehow let $x_2$ matter, you could run a regression on $x$ and $x^2$, like in

x <- 1:3
y <- rnorm(3)
summary(lm(y~x))
summary(lm(y[c(1,3)]~x[c(1,3)]))
summary(lm(y~poly(x,2, raw=T)))

I am not suggesting this is a good idea from a subject-matter or specification point of view, though.

As regards the comments, $x_2$ does indeed matter for assessing uncertainty, as, in a simple regression (when assuming heteroskedasticity) $$ Var(m)=\frac{\hat\sigma^2}{\sum_i(x_i-\bar{x})^2} $$ So, while $x_2-\bar{x}$ again drops out in the denominator of $Var(m)$, the error variance estimator $\hat\sigma^2$ will generally (i.e., unless $y_2$ is also equal to its sample mean) receive a contribution from the residual $$ y_2-\hat{y}_2=y_2-\bar{y}, $$ where the equality sign is, e.g., explained here.