GLS

Question 1

If εi is spherical, then it is …, and use …

Answer 1

homoskedastic and serially uncorrelated, then use the Standard Linear Model (SLM)

Question 2

If εi is nonspherical, then it is …, and use …

Answer 2

heteroskedastic and autocorrelated, then use the Generalized Linear Model (GLM)

Question 3

GLM assumptions:

Answer 3

1: y=Xβ+ε (Linearity)
2: E(ε|X)=0 (exogeneity, regressors contain no information on the derivation of Yi from its conditional expectation)
3: Var(εi|X)=σ^2ωi (heteroskedaticity), Note: when E(εi|X)=0, the variance-covariance matrix is Var(ε|X)=σ^2Ω
4: rank(X)=rank(X’)=rank(XX’)=rank(X’X)=k
5: ε|X~N(0,σ^2In)
6: {(Yi,Xi):i=1,…,n} are independent and identically distributed

Question 4

Properties of βOLS^ in GLM:

Answer 4

1: β^ still unbiased if E(ε|X)=0
2: β^ is not the best unbiased estimator, it is not asymptotically efficient
3: β^ is multivariate normal
4: β^ is consistent
5: β^ is asymptotically normal

Question 5

Heteroskedasticity-consistent estimator (HCE).

Answer 5

HCE=(X’X)^-1Σ(εi^)^2xixi’(X’X)^-1
Interpretation:
Under homoskedasticity, all elements are the same across the diagonal
Under heteroskedasticity, all elements are different across the diagonal

Question 6

When do we use the Generalized Least Squares (GLS) estimator?

Answer 6

When there is correlation between the residuals (εi is heteroskedastic, but not autocorrelated)

Question 7

GLS estimator form:

Answer 7

βGLS^=(X’P’PX)^-1X’P’Py
If Ω^-1 is known, βGLS^=(X’Ω^-1X)^-1X’Ω^-1y

Question 8

Properties of GLS estimator:

Answer 8

1: GLS is unbiased if E(ε|X)=0
2: GLS is the best unbiased estimator
3: GLS is consistent
4: GLS is normal
5: GLS is asymptotically normal

Question 9

When do we construct βFGLS^ and what is its form? (Feasible GLS)

Answer 9

If Ω depends on unknown parameters, we cannot compute βGLS^. However, Ω depends on a small set of unknown parameters, we can estimate them, construct Ω^:
βFGLS^=(X’(Ω^)^-1X)^-1X’(Ω^)^-1y

Question 10

Properties of FGLS:

Answer 10

1: FGLS is consistent
2: FGLS is asymptotically efficient and normal, under regularity conditions

Question 11

When do we use the White test?

Answer 11

When we have heteroskedasticity of unknown form: H0: σi^2=σ^2 vs. H1: σi^2≠σ^2 (homoskedastic vs heteroskedastic)

Question 12

How to perform the White test?

Answer 12

1: Compute β^ and ε^
2: square each term of vector ε^ and make this the dependent variable
3: Carry out an OLS regression of εi^2 on X. If xi contains a dummy, the square of that variable is also a dummy, so do not add this in the test.
4: Compute LM =nR^2~χ_(J-1)^2, where J is the number of regressors including the column of ones. Reject H0 if LM>χ_(J-1)^2
5: (Instead of step 4) Carry out a model F-test