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.