Advanced Econometrics I
EMET4314/8014
Semester 1, 2025
Assignment 6
Exercises
Provide transparent derivations. Justify steps that are not obvious. Use self sufficient proofs. Make reasonable assumptions where necessary.
1. Let with E(ei Xi ) ≠ 0. You have available Zi with E(ei Zi ) = 0 and dim Zi = L ≥ dim Xi = K. Consider the estimator
where dimP = K × L. Different choices for the matrix P result in different estima- tors. For example, the IV estimator for the exactly identified case sets P = I.
It can be shown that another choice, namely results in an estimator, bP * , with minimal asymptotic variance.
Notice, however, that bP * is an infeasible estimator because you do not observe P*. Replace resulting in the feasible estimator .
(i) Prove that is consistent.
(ii) Derive the asymptotic distribution of
Provide the asymptotic variance in terms of P* . Then plug in the right hand side of and see how the result simplifies considerably.
You may assume to make things easier.
Use the following nomenclature for brevity:
This implies and
(iii) Suggest a good for P* such that = P* + Op (1).
2. Read the paper “A Practical Guide to Weak Instruments” by Keane and Neal, pub- lished 2024 in the Annual Review of Economics.
Try to read the whole paper and make sense of as much as you possible. I don’t expect you to understand everything, but hopefully you will be able to follow the main points. The objective is to prepare you for the week 8 lecture in which we will discuss the paper in more detail.