Section 4

Question 1

What does section 4 consider?

Answer 1

Violation of var(ε)=σ^2I(n)

ie. violating assumption of errors being homoskedastic and not autocorrelated

Question 2

What does var(ε)=?

and so how is this when not violating A2?

Answer 2

E(εε’)= nxn matrix with variances down main diagonal and autocovariances in all other elements (see notes)

Under homoskedasticity, all of the main diagonal elements are the same σ^2, and all the autocovariances are equal to 0 tf var(ε)=σ^2I(n)

Question 3

What does var(ε) equal when violating A2?

Answer 3

var(ε)=σ^2Ω, where Ω is not equal to I(n)

Question 4

What are non-spherical disturbances? (2)

Answer 4

If in the var(ε) matrix, either:
a) the main diagonal elements aren’t constant (heteroskedasticity)
or
b) the off diagonal elements aren’t all 0s (autocorrelation)

Question 5

Is the OLS estimator still unbiased when A2 is violated?

Answer 5

Yes - see S2.3 for proof (doesn’t use A2 in it)

Question 6

Is the OLS estimator still efficient when A2 is violated? And note?

Answer 6

No - see S2.3 derived variance matrix of b, but used A2 to derive it
Furthermore, when proving variance of estimator was smallest amongst all unbiased estimators used A2 aswell

Question 7

Derive the heteroskedastic and AC consistent variance estimator?

Answer 7

Same as before up til:

var(b)=(X’X)^-1X’E(εε’)X(X’X)^-1 then just simplify (see notes for proof)

Question 8

What is a more efficient estimator than the heteroskedastic and AC consistent variance estimator?

Answer 8

GLS estimator

Question 9

Explain how a GLS estimator works?

Answer 9

Works by transforming the regression model into one with no heteroskedastic or autocorrelated errors.
The new model is then estimated by OLS like normal

Question 10

Given var(ε)=σ^2Ω, what do we know about Ω? (4)

Answer 10

nxn (square), +ve definite, invertible and symmetric

Question 11

Given var(ε)=σ^2Ω and its properties, what does this therefore mean? How does this help us develop GLS?

Answer 11

Means there is an nxn matrix P such that:
Ω^-1=P’P
This is called a Cholesky Decomposition and P is essentially equivalent to the square root of the matrix Ω^-1
By multiplying the regression model by P, we can eliminate the HTK and AC issues

Question 12

ε*=?

Answer 12

Pε - doesn’t violate A2!

Question 13

Prove ε* doesn’t violate A2?

Answer 13

See notes page 1 side 2

Question 14

Prove the formula for b(GLS)?

Answer 14

See notes page 1 side 2

Question 15

Show that b(GLS) is unbiased?

Answer 15

See notes page 1 side 2

Question 16

Derive var(B(GLS)) from equation before?

Answer 16

See notes page 1 side 2

Question 17

Derive var(B(GLS)) from first principles?

Answer 17

See S4 page 7

Question 18

Prove that the variance of the GLS estimator is smaller than that of the OLS estimator (use updated equation not original!)? (ie. prove GLS is more efficient)

Answer 18

See notes for putting it into form D=σ^2AΩA’ where:
A = (X’X)^-1X’-(X’Ω^-1X)^-1X’Ω ^-1

tf since Ω is a variance matrix tf is positive definite by definition. By pre and post multiplying it by A and A’, it is essentially the same as squaring it, and since σ^2 must be positive too since it is variance, by multiplying all these parts together can see that it must be positive definite aswell tf GLS is more efficient!

Question 19

See

Answer 19

page 8 summary of A2 being violated

Question 20

How to apply GLS for heteroskedastic errors?

Answer 20

See notes page 2 side 1 for WHY
The GLS transform is to divide the model by the square root of the variable in the error variance formula (ie. the root of the variable the error is a function of)

ie. doing PY since P = diagonal matrix with 1/sq.root of variable

SEE NOTES

Question 21

How to apply GLS for autocorrelated errors?

Answer 21

see notes aswell page 10

Question 22

What is the Prais-Winsten formula used for?

Answer 22

When doing GLS for autocorrelated errors, PY column represents the quasi-differences of the variables; apart from the first observation. The first observation is different because when calculating the quasi-difference of the first observation you need to know Y0 which you do not have (Y1 is first obs.) tf econometricians use the PW formula that avoids the loss of the first quasi-differenced observation

Question 23

Why do we need a different equation sometimes for feasible least squares (FGLS)?

Answer 23

Since often we don’t know Ω, for example, in the autocorrelation case we do not know the coefficient of autocorrelation ρ. Tf there needs to be an intermediate step in the estimation procedure that finds Ω(hat) and tf finds ρ(hat), such that the FGLS estimator becomes (see equations)