Page 180 - Applied Statistics with R

180 CHAPTER 9. MULTIPLE LINEAR REGRESSION

coef(lm(y ~ x1 + x2, data = sim_data))

## (Intercept) x1 x2
## 7.290735 -2.282176 5.843424

Also, these values are close to what we would expect.

c(beta_0, beta_1, beta_2)

## [1] 5 -2 6

We then calculated the fitted values in order to calculate , which we see is the

same as the sigma which is returned by summary().

y_hat = X %*% beta_hat
(s_e = sqrt(sum((y - y_hat) ^ 2) / (n - p)))

## [1] 4.294307

summary(lm(y ~ x1 + x2, data = sim_data))$sigma

## [1] 4.294307

So far so good. Everything checks out. Now we will finally simulate from this
̂
model repeatedly in order to obtain an empirical distribution of .
2
̂
We expect to follow a normal distribution,
2
̂
2
∼ ( , ) .
22
2
2
In this case,
̂
2
∼ ( = 6, = 16 × 0.0014534 = 0.0232549) .
2
̂
2
∼ ( = 6, = 0.0232549) .
2
Note that 22 corresponds to the element in the third row and third column
since is the third parameter in the model and because R is indexed starting
2
at 1. However, we index the matrix starting at 0 to match the diagonal
elements to the corresponding .