Variance of the estimators in Linear Model with manual formulae in R

Question

I'm working on a linear model like:

I'm applying the formulae to the data to calculate the estimators of a and b using R. First I got some data like:

x <- c(10,11,23,24)
y <- c(4,2,7,8)

Then I applied the lm() to have the results:

model <- lm(y~x)          # the model

coefficients(model)       # the coefficients
(Intercept)           x 
 -0.5500000   0.3411765 

sqrt(diag(vcov(model)))   # the s.e.
(Intercept)           x 
 1.69932945  0.09333313 

So I decided to calculate manually the coefficients and the s.e. I used this formulae for the coefficients:

And in R I did something like:

bHat <- cov(x,y)/var(x)
aHat <- mean(y)-(cov(x,y)/var(x))*mean(x)

bHat
[1] 0.3411765
aHat
[1] -0.55

Then I calculate the s.e.. I used those formulae:

And

So I tried those formulae in R.

se.bHat <- sqrt(sigma/sum((x-(mean(x)))^2)) 
se.bHat
[1] 0.211205

 se.aHat <- sqrt((sigma/4)*  (1+4*mean(x)^2)/sum((x-(mean(x)))^2))
 se.aHat
 [1] 3.590497

But the result is not equal to the lm() output. Am I using the wrong formulae in the theory or am I applying them wrongly? Thanks in advance.

EDIT:

I've tried as suggested this as the formula σ^2 = 1/(n-p) Sum(w[i] R[i]^2) with help(summary.lm):

sigma <- (1/(4-2))*sum(residuals(model)^2)
se.bHat <- sqrt(sigma/sum((x-(mean(x)))^2))
se.bHat
[1] 0.09333313

And the result is not ok. I'm going to try also with the intercept.

Answer 1

due to the fact, that i can´t give a comment:

i know a different formula for the s.e of a:

I have it from the book: Applied Regression Analysis, Linear Models, and Related Methods by John Fox (1997).

Using it in R: (maybe a too many brackets :D)

> (se.aHat <- ((sigma*sum(x^2))/(n*(sum((x-mean(x))^2)))))
[1] 2.887721
> sqrt(se.aHat)
[1] 1.699329

Answer 2

The problem is that I used the wrong formula for sigma square. using:

help("summary.lm")

As suggested by Roland (see comments below), you can find that the right formula is as stated by R guide:

the square root of the estimated variance of the random error

σ^2 = 1/(n-p) Sum(w[i] R[i]^2)

where R[i] is the i-th residual, residuals[i].

So