如何在R中手动获得最小二乘估计

Question

I can't figure out how to get the least squares estimates (beta 1 hat and beta not hat) by hand using formulas instead of using functions.

I have tried the formula below, I feel like I need to use a for loop but I just can't figure out how to do it. Once I get beta 1 I should be able to get beta not.

x <- toluca$lot 
y <- toluca$work

beta1hat <- (sum(x[i]-mean(x)*sum(y[i] - mean(y))/sum(x[i]-mean(x)^2)

It says I is not defined but I don't know what I would define I as because it is a specific value within x.

Answer 1

You can code the matrix form of the equation for the ordinary-least squares estimator in R. Here is an example:

set.seed(123)

x <- 1:10
a <- 2
b <- 3

y <- a*x + b + rnorm(10)

xm <- matrix(c(x, rep(1, length(x))), ncol = 2, nrow = 10)
ym <- matrix(y, ncol = 1, nrow = 10)

beta_hat <- MASS::ginv(t(xm) %*% xm) %*% t(xm) %*% ym

This gives exactly the same output as summary(lm(y ~ x)) . Note, however, that the lm function is NOT a simple one liner because calculating the inverse of a matrix is a numerically challenging problem, so some precautions must be taken.