Week 5

Question 1

How to calculate the 95% confidence interval for beta.j?

Answer 1

(beta hat.j - 2.09 sqrt(s^2v.jj), beta hat.j + 2.09sqrt(s^2v.jj) Note in this case n = 20, replace 2.09 with the correct students t-value.

Question 2

How to describe confidence intervals?

Answer 2

We cannot state that beta.j has a probability of 95% to be in the interval, we can state that: “The 95% confidence interval is (…, …)”

Question 3

How do we derive the 95% prediction interval?

Answer 3

If we have predictor y* hat = X beta* hat of y*, we know that y* hat - y has mean zero and variance: Var(y* hat - y*) = sigma² (X*(X’X)^-1X*’ + I.m).

If we have a squared normal random variable it follows the chi-squared distribution, we get that (y* hat - y*)’ Var(y hat* - y*) (y* hat - y*)/sigma² is chi-squared distributed.

If we replace sigma² by s² we get an F-distribution (where m denotes the compontents in y*). (y* hat - y*)’ Var(y hat* - y*) (y* hat - y*)/ms² ~ F(m, n - k).

In the special case that m = 1: (y hat * - y*)/(s sqrt(x’_*(X’X)^-1x* + 1))

Question 4

What do n - k usually mean in this course?

Answer 4

n: number of data points
k: number of regressors

Question 5

How to test a linear restriction on the beta?

Answer 5

1. First rewrite the restriction to a linear equation. w is the vector of the equation.
2. Then we test the following:

t_w = (w’ beta hat - r)/(sqrt(s²w’(X’X)^-1w)) ~ t(n -k)

Question 6

How is the linear restriction derived?

Answer 6

It’s known that w’beta hat ~ N(w’beta, sigma²w’(X’X)^-1w), thus if we write:

(from now on d = ) (w’beta hat - w’ beta) / (sqrt(sigma²w’(X’X)^-1w)) ~ N(0, 1)

Then we can state:

d/sqrt(s²/sigma²) = (w’beta hat - w’beta)/sqrt(s²w’(X’X)^-1w)

Which follows a students t-distribution, we can test for the restriction using:

t_w = (w’beta hat - r) / sqrt(s²w’(X’X)^-1w) ~ t(n - k)

Question 7

In what formula is that includes s² is it chi-squared distributed?

Answer 7

(n - k)s²/sigma² is kai-squared distributed to degree n -k

Question 8

When is students t-distribution often found?

Answer 8

When you divide a normal distribution by an chi-squared distribution.

Question 9

How to test several linear restriction at the same time?

Answer 9

Because we have: R beta = N(R beta, sigma²R(X’X)^-1R’), we can rewrite this as:

(beta hat - beta)’R’(R(X’X)^-1R’)^-1R beta (beta hat - beta) / sigma ² ~ chi squared(m).

Thus if we rewrite to include s², we have:

(beta hat - beta)’R’(R(X’X)^-1R’)^-1R beta (beta hat - beta) / m s² ~ F(m, n - k)

Question 10

What is the formula of the adjusted R²?

Answer 10

R bar² = 1 - (e’e/(n - k))/(SST/(n - 1)) = 1 - s²/(SST/(n - 1))

Question 11

What is model averaging?

Answer 11

It joins two models together by assigning them both some weight.