Part A of Problem Set # 1
Master in Business and Quantitative Methods
Contents
Chapter1. Exercises on endogenous regressors.
1
0.1 Endogeneity of the regressors
Problem 1 For the simple regression model, yi =µ+ui, ui∼N(0, σ2),
prove that the sample mean is consistent and asymptotically normally dis- tributed. Now consider the alternative estimator
ˆ µ =P
iwiyi
wi = (n(n+1)/2)i
= Pi
ii. Note that P
iwi = 1 and P
ii2 =n(n+ 1)(2n+ 1)/6. Prove that this is a consistent estimator ofµ and obtain its asymptotic variance.
Problem 2 In the context of the instrumental variables we have shown that the least squares estimatorβˆ is biased and inconsistent. Nonetheless,βˆ does estimate something. Derive the asymptotic covariance matrix ofβ, and showˆ that βˆ is asymptotically normally distributed.
Problem 3 Prove that the limiting distribution ofβˆIV is
√n(ˆβIV −β)→d N 0, σ2rQ−1ZXQZZQ−1XZ .
and present the asymptotic distribution of βˆIV.
Problem 4 Consider a model for the health of an individual:
healthi=β0+β1agei+β2weighti+β3heighti+β4malei+β5worki+β6exercisei+ui,
where health is some quantitative measure of the persons health, age, weight, height, and male are self-explanatory, work is weekly hours worked, and exercise is the hours of exercise per week.
a. Why might you be concerned about exercise being correlated with the error termu?
b. Suppose you can collect data on two additional variables, disthome and distwork, the distances from home and from work to the nearest health club or gym. Discuss whether these are likely to be uncorrelated with u.
Problem 5 Consider the model:
y∗ =βx∗+ε
Prove that when only x∗ is measured with error, the squared correlation between y and x is less than that between y∗ and x∗. (Note the assumption that y∗=y). Does the same hold true if y∗ is also measured with error?
