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Advanced Econometrics: Homework 3

December 10, 2019

Instructions:

• Please form groups of three students. If you have trouble finding colleagues, write me an

e-mail, and I will match you with others having the same problem.

• Deadline for submissions is Friday, December 20, 2019, 23:59. Any late submissions

will be awarded zero points.

• Your solution should have a form of Jupyter Notebook with R source-code. Code should be

properly commented, interpretations of results as well as theoretical derivations should be

written in markdown cells. This is the only file you need to send. If you prefer not to write

formulas in LATEX, you can send PDF with your derivations and interpretations in additional

file and R code in Jupyter Notebook.

• Please, be concise, but remember to include and explain all important steps.

• If you have any questions concerning the homework, do contact me by mail and we can set

up a consultation. Do it rather sooner than later, I won’t give any consultation concerning

the homework after December 17.

Problem 1:

(2 points)

Simulate 1000 data points from the linear model

yi = α + xiβ + i,where x ∼ N(20, 9), ∼ N(0, xγ). For each of model parameters, generate a single random value

you will be using throughout the exercise, where α ∈ h0, 4i, β ∈ h0.5, 2i, γ ∈ h0, 3i.

Remember to use set.seed() to make your results replicable. Recall that parameters of normal

distribution are in the form N(µ, σ2), not N(µ, σ).

a) Estimate model y = xβ + on the simulated data using OLS. Interpret the results. Do you

expect any of the OLS assumptions to be violated. If yes, make the corresponding tests, and

interpret the results.

b) Reestimate the model using GLS. State which form the variance-covariance matrix Ω takes

in your case. Also, please state the form of your weighting matrix Ω− 1

2 . Comment on the

results from GLS regression.1

c) Estimate FGLS model for heteroscedasticity of the form σ2i = σ

2xi (recall the food expenditure

example from seminar 8).

d) In the OLS model, estimate standard errors using White heteroscedasticity consistent estimator.

Compare White’s standard errors to those from OLS, GLS, and FGLS.

Problem 2:

(2 points)

Please use the data in wages.csv to answer following questions. Estimate models based on speci-

fication

LW AGEit = β0 + β1EXPit + β2EDit + β3SMSAit + β4F EMit + uit,

where i is indicated by ID variable, t is indicated by Y EAR variable. The dependent variable is

natural logarithm of wage, EXP indicates working experience, ED indicates years of education,

SMSA is a dummy variable for individuals living in urban areas, F EM is a dummy variable

indicating female workers.

Your task is to find out, whether the female variable is a significant determinant of wages. Please

use the standard panel estimation methods (Pooled OLS, Fixed Effects, and Random effects

models), and perform all the necessary tests. For each estimator, state the conditions under which

it is valid.

What are your conclusions? What additional estimation would you suggest?

Problem 3:

(1 point)

Please use the data in wages.csv again, and create a subsample containing all individuals in

just one year of your choice. Hence, you should estimate a model of following specification on a

cross-sectional subsample

LW AGEi = β0 + β1EXPi + β2EDi + β3SMSAi + β4F EMi + ui

.

Test for the group specific heteroscedasticity based on the outcome of SMSA variable. Construct

an FGLS estimator efficient in presence of such relationship.

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