Advanced Econometrics I

EMET4314/8014

Semester 1, 2025

Assignment 1

(due by: Tuesday week 2, 11:00am)

Exercises

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

1. Consider the space Z = (0, 1] equipped with the metric d(x, y) = |x − y|. Consider the following sequence in Z: xn = 1/n, n = 1, 2,.... Is it a Cauchy sequence? Does it converge?

2. Let X, Y be elements from a Hilbert space. Prove:

(i) Cauchy-Schwarz inequality: |⟨X, Y⟩| ≤ ∥X∥ · ∥Y∥

(ii) Triangle inequality: ∥X + Y∥≤∥X∥ + ∥Y∥

3. Prove: If E (X2) < ∞ and E (Y2) < ∞, then

• |EX| < ∞ and |EY | < ∞;

• |E(XY )| < ∞;

• |Cov(X, Y )| < ∞.

This is useful: to guarantee existence of covariances, we only need fnite second moments. That is why we defne L2 to be the space of random variables with fnite second moments.

Related useful fact (for your enjoyment, no need to prove):

E (|Y | p) < ∞ implies E (|Y | q) < ∞ for 1 ≤ q ≤ p

(by Liapunov’s inequality).

4. Prove: Cov(X, Y ) = Cov(X, E(Y |X))

5. Prove: if X ∈ {0, 1} then Var/Cov(X,Y X ) = E(Y |X = 1) − E(Y |X = 0).

6. Consider the space L2, as defned in the lecture. Let X, Y ∈ L2. Prove that E(XY ) is an inner product.

7. Let X2, X3, Y ∈ L2. Find the projection of Y on sp (X2, X3). (What I’m trying to say here is that you are NOT including the constant in the span.)

Use the following Gram-Schmidt orthogonalization procedure to construct an or-thonormal set:

Lemma 1 (Gram-Schmidt). Let V1, V2, V3,... be a linearly independent sequence in an inner product space. Set U1 = V1/||V1||, and defne recursively:

Then U1, U2, U3,... is an orthonormal sequence with sp(U1, U2,...,Uk) = sp(V1, V2,...,Vk).

8. Let X1,...,XK, Y ∈ L2. Use calculus to derive the following:

where X := (X1,...,XK)′ so that dim X = dim b = K × 1.

This demonstrates that Psp(X1,...,XK)Y can also be obtained by “traditional” methods.