Section 5

Question 1

When doing Restricted LSE, how do we go about finding the β(hat) that minimises the restricted equation?

Answer 1

Use Lagrangian method, with:
Objective function: (Y-Xβ(hat))’(Y-Xβ(hat))
constraint: Rβ(hat)=r

Question 2

What is lamda in the lagrangian method thing?

Answer 2

px1 vector or Lagrange multipliers

Question 3

Method for deriving b(RLS)?

Answer 3

1) Differentiate Lagrange wrt β(hat) and λ, then set=0
2) Pre multiply (2) (dL/dβ(hat)) by (1/2)(X’X)^-1
3) Simplify by subbing in b giving (4)
4) Pre-multiply by R
5) Put in form λ=…
6) Substitute λ into (4)
7) Substitute β(hat)=b(RLS) and b=b(OLS) and rearrange into form b(RLS)=…

Question 4

When is the RLS estimator equal to the OLS estimator?

Answer 4

When Rb(OLS)=r

Question 5

Show that e(RLS)=e(OLS)+XW. What is W?

Answer 5

see notes

Question 6

Show that the restricted residual sum of squares is equal to unrestricted sum of squares+W’X’XW?

Answer 6

see notes side 2

Question 7

Show that W’X’XW is equal to the numerator of the F statistic defined in section 3? Hence show that the F statistic defined in section 3 is the same as the F statistic defined in ECON206A?

Answer 7

See notes side 2