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Advanced Econometrics I

EMET4314/8014

Semester 1, 2025

Assignment 10

Exercise

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

You have Y,...,YN iid with pdf

Notice that this is the pdf of the normal distribution with mean μ and variance 1.

(i) Derive the log likelihood function In L(u).

(ii) Derive μML. Is it unbiased?

(iii) Derive Var(μML).

(iv)Derive the score function S(ylμ) as it was defined during the lecture.

(v) Derive the Fisher information I(μ) via E(S(Ylμ)2). Suppose you have an unbiased estimator T(Y1,...,YN) for μ, what is the Cramér Rao lower bound for its variance?

(vi) Confirm the information equality.

(vii) Does μML attain the CRB?

(viii) If μML does indeed attain the CRB, then the average score function can be decomposed as suggested by the Theorem in the subsection Minimum Variance Unbiased Estimators of the week 10 slides. Derive that decomposition and provide a(μ).

(ix) State the asymptotic distribution of μML. (No need to derive!)