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Advanced Econometrics I

EMET4314/8014

Semester 1, 2025

Assignment 3

(due: Tuesday week 4, 11:00am)

Exercises

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

1.  Let Z be a random variable with EZ2 < ∞ . Prove that  implies ZN  = Op (1).

2. The pdf of a normal distribution is  exp , for −∞ < y < ∞.

(a)  Derive the moment generating function of a normally distributed random vari- able. Denote it by MY (t; (µ, σ)).

(b)  Take the first two derivatives of MY (t; (µ, σ)) and evaluate them at zero.

(c)  Evaluate the mgf for the standard normal case: MY (t; (0, 1)). (This proves a Lemma from the week 3 lecture notes.)

3.  Let Y = Xβ* +u with dim X = N × K and the usual definitions. Define the projection matrix PX := X(X′ X)—1X′ and the residual maker matrix MX  := IN  − PX . Show that:

(i)  PX Y =Yˆ         (hence the name projection matrix)

(ii)  MXY =ˆu         (hence the name residual maker matrix)

(iii)  MXu =ˆu

(iv)  Symmetry: PX  = PX′ and MX  = MX′

(v)  Idempotency: PXPX  = PX  and MX MX  = MX

(vi)  tr PX  = K and tr MX  = N − K

4. Use a derivation similar to lecture notes 3 to show that  is an unbiased estimator for σ2u.

5.  Consider the asymptotic distribution of  under the assumption of ho-moskedasticity, that is:  where . Note, as usual, .

(a) Derive the asymptotic distribution of  under homoskedasticity.

Justify each step!

(b)  Suggest a consistent estimator for the asymptotic variance of  der homoskedasticity.

(c)  Prove that your estimator from part (b) is consistent.  In your proof, make use of the op(1) and Op(1) notation. Justify each step!