TD2 Properties of Least-Squares

TOULOUSE SCHOOL OF ECONOMICS MASTER 1 - FALL 2016

INTERMEDIATE ECONOMETRICS

TD2 Properties of Least-Squares

Exercise 2.1

Take the modelyi =x⁰_1iβ1+x⁰_2iβ2+εi withE(εi|xi) = 0. Suppose that β1 is estimated by regressing y onx1 only. Find the probability limit of this estimator. In general, is it consistent forβ1? If not, under what conditions is it consistent?

Exercise 2.2

We have a sample of law schools. For each school, we observe the median starting salary of its graduates. We model this variable by:

log(salary) =β₀+β₁LSAT+β₂GP A+β₃log(libvol)+β₄log(cost)+β₅rank+u, where LSAT is the median LSAT score for the graduating class, GP Ais the median college GPA for the class, libvol is the number of volumes in the law school library,costis the annual cost of attending law school, andrankis a law school ranking (with rank = 1 being the best).

1. Explain why we expectβ5≤0.

2. What signs do you expect for the other slope coecients? Justify your answers.

3. The estimated equation is

log(salary) = 8.34 +d .0047LSAT+.248 GP A+.095 log(libvol) +.038 log(cost)−.0033rank

n= 136, R²=.842.

What is the predicted ceteris paribus dierence in salary for schools with a median GPA dierent by one point? (Report your answer as a percent.) 4. Interpret the coecient on the variablelog(libvol).

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5. Would you say it is better to attend a higher ranked law school? How much is a dierence in ranking of 20 worth in terms of predicted starting salary?

Exercise 2.3 The model is

y_i=x_iβ+u_i E(u_i|x_i) = 0 V ar(u_i|x_i) =σ²_u wherex_i is real and x6= 0. Consider the three estimators

βb= Pn

i=1xiyi

Pn i=1x²_i βe= 1

n

n

X

i=1

yi

x_i

ee β =

1 n

Pn i=1yi 1

n

Pn i=1xi

1. Find the conditionnal expectation and variance of these estimators.

2. Under the stated assumptions, are these estimators consistent forβ? 3. Find the asymptotic distribution ofβbande

βe.

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