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Advanced Econometrics I

EMET4314/8014

Semester 1, 2025

Assignment 4

(due: Tuesday week 5, 11:00am)

Exercises

Provide transparent derivations. Justify steps that are not obvious. Use self sufficient proofs. Make reasonable assumptions where necessary.

1. Prove that the OLS estimator βˆOLS for β in the linear regression model is consistent. In your derivation, make use of the op (1) and Op(1) notation!

2. The linear regression model in matrix format is Y = Xβ + e, with the usual definitions. Let E(e|X) = 0 and

Notice that as a covariance matrix, Σ is symmetric and nonnegative definite.

(i) Derive Var(|X).

(ii) Let  := C ′Y be any other linear unbiased estimator where C is an N × K-dimensional matrix that is based on X. Following the lecture notes, prove that Var

(iii) An oracle tells you Γ. Let  and  Define the generalized least squares (GLS) estimator  This defines the GLS estimator as the OLS estimator of ˜Y on ˜X.

Derive Var(|X). How does it compare to Var(|X) from part (ii)?