Solving normal equation gives different coefficients from using `lm`?

Question

I wanted to compute a simple regression using the lm and plain matrix algebra. However, my regression coefficients obtained from matrix algebra are only half of tho

Answer 1

lm is not performing standardization. If you want to obtain the same result by lm, you need:

X1 <- cbind(1, X)  ## include intercept

solve(crossprod(X1), crossprod(X1,Y))

#           [,1]
#[1,]  4.2222222
#[2,]  1.0000000
#[3,] -0.3333333
#[4,]  1.0000000
#[5,]  0.6666667
#[6,]  2.3333333
#[7,]  1.3333333

I don't want to repeat that we should use crossprod. See the "follow-up" part of Ridge regression with glmnet gives different coefficients than what I compute by “textbook definition”?