Advanced Econometrics I

EMET4314/8014

Semester 1, 2025

Assignment 2

Exercises

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

1. Let X2, X3,Y E L2. Use calculus to derive the following:

Provide explicit and fully derived solutions for β2 and β3 Do not use linear algebra! Compare your results to the projections of Y on sp (X2, X3) (from assignment 1) and of Y on sp (1, X2, X3) (from the lecture).

2. Let Y, E L2 for i =1,...,N be a scalar random variables with independent and identical distribution with py :=EY and o :=VarY, < oo. The sample average is defined as YN :=ΣNi=1Y/N. Derive EYN and VarYN.

3. Using the same definitions as in exercise (2), define

Derive VarZN.

This result illustrates that, if the limit distribution of Zy exists, it will be non-degenerate. That is, it does not just collapse to a point.

Remark:

You may correctly conjecture, based on the CLT, that the limit distribution is standard normal under mild conditions. Proving this requires some not too difficult manipulations of moment generating (or characteristic) functions.

4. The definition of the OLS estimator using matrix notation is:

where dim X = N x K and dim Y =N x 1. Derive oLs using calculus.

The following tools from matrix calculus may be helpful:

Lemma 1.

5. Let A be some real number. Show that A/N=o,(1).

6. Let Y=XB+u, where E(Xu)=0, where dim X=Kx1 and dim Y=1x1. Define

where λ > 0 and Ix is the K-dimensional identity matrix.

Derive the probability limit of θ. Is consistent for β*? In your derivation, make use of the op,(1) and Op,(1) notation!