Section 3

Question 1

Key point about normally distributed variables?

Answer 1

A linear transform of a normally distributed variable is also normally distributed

Question 2

Show that the OLS estimator is normally distributed?

Answer 2

See page 1 my notes

Question 3

What does the normality of the error term directly imply?

Answer 3

Normality of the OLS estimator

Question 4

What are the 4 steps of obtaining an estimator of the error variance?

Answer 4

1) define RSS, e’e as: e’e=ε’M’Mε=ε’Mε
2) take trace tr(ε’Mε)=ε’Mε (it’s a scalar tf trace of the function will equal itself)
3) take expectation of e’e -> σ^2tr(M)
4) find tr(M)

Do proof, is in my notes page 1

Question 5

Steps to find the distribution of the error variance? And proof?

Answer 5

1) rewrite RSS as a function of the error variance: RSS=e’e=ε’Mε
2) normalise error distribution to get standard normal

Question 6

Show M is symmetric and idempotent?

Answer 6

See proofs bottom of page 3 booklet

Question 7

Find an estimator for the error variance AND show it is unbiased?

Answer 7

See page 1 sides 1 and 2 of notes

Question 8

Steps to find the distribution of the error variance? and proof?

Answer 8

See page 1 side 2 of notes

Question 9

What is distributional result 3.2?

Answer 9

For a standard normal variable x~N(0,I), and symmetric and idempotent matrix A with rank r, then:

x’Ax~χ^2
(r)

Question 10

For symmetric and idempotent matrices…

Answer 10

The rank is equal to the trace

Question 11

When does zero covariance imply independence?

Answer 11

When the variables are ALSO normally distributed

Question 12

What is distributional result 3.3?

Answer 12

If x is a SNV and y is χ^2 with r DofF, and x and y are independent of one another, then the variable t shown below has a t-distribution with r degrees of freedom:

t=x/(y/r)^0.5 ~t(r)

Question 13

Show that t(j)=(b(j)-β(j))/s.e.(b(j)) ~t(n-k)?

Answer 13

See 3.2.1 in my notes

Question 14

Explain how we can test for single linear combinations of regression parameters?

Answer 14

Using DR3.1, can show Rb-r has following normal distribution:
Rb-r ~ N(Rβ-r, σ^2R(X’X)^-1R’)
tf allows us to test if Rβ-r=0 or if Rβ=r

R is 1xk row vector and r is scalar, β is kx1 vector as normal

Question 15

Key point to remember about normal and t distributions?

Answer 15

Populations that are normally distributed will have a t-test done on the samples chosen from them

Question 16

Explain how we can test for multiple parameter restrictions?

Answer 16

Using, again: Rb-r ~ N(Rβ-r, σ^2R(X’X)^-1R’)

By making R a pxk matrix (p=no. linear combos), Rβ-r is now a column vector and with Rβ-r=0 now a vector of zeros

Then, using the F statistic (see equations) we can test for multiple restrictions (try prove the F stat too!)

~F(p,n-k) (DofF)

Question 17

What is p?

Answer 17

Number of linear combinations to test; ie. if β2=β3=0 then p=2 (number of = signs)

Question 18

See

Answer 18

end of section 3 for examples

Question 19

Learn DR 3.4 and 3.5!

Answer 19

and proof of F!!!