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Advanced Econometrics: Homework 1

October 23, 2019

Instructions1:

• Submit one file for each problem.

“AdvEcox HW1 2018: Group surname1, surname2, surname3 ”.

• Form groups of three yourself.

• As a solution, provide 3 Jupyter Notebooks with R source-code. Code should be properly commented,

interpretations of results as well as theoretical derivations2

should be written in markdown

cells.

• Use “set.seed()“ function, so I can replicate your results.

• Be concise (no lengthy essays please). Although, be sure to include all important things as I

cannot second-guess your work.

• The empirical problems do not necessarily have a unique solution in terms of numbers, you are

assessed based on execution of the analysis not on the right numbers that you should get from

output. The emphasis is put mainly on meaningful presentation and extent of your knowledge.

• The problem set is due on 6th November. Late submission automatically means 0

points.

Problem 1. For this problem, use data in file hw1 data.csv.

Columns containing Y in their names correspond to dependent variables (4) and columns containing

X are independent variables (2). Using following pairs of independent and dependent variables,(X1,Y1),

(X1, Y2), (X1,Y3) and (X2,Y4) do the following:

1. Estimate beta coefficients using OLS and MLE. Compare estimates from both methods, explain

differences (if any). Discuss validity of necessary assumptions in individual cases.

2. Show diagnostic plots (for OLS) and interpret them.

1The contact person for this homework is Martin Hronec, the same mail as for submission of homeworks.

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.

3. Use LAD to estimate conditional median and compare it with the estimate of conditional mean

from OLS for each of the pairs. Explain differences between them.

4. Use quantile regression to estimate conditional quantiles (for tau 0.05,0.25,0.5,0.75,0.95) for

each of the pairs and discuss differences across quantiles.

Problem 2. On the second seminar, we have talked about heteroskedasticity a lot and had an example

where data was generated according to the equation

incomei = α + β · educi + educi,

where ∼ N(0, 200), α = 4000, β = 200.

1. Illustrate theoretically that this heteroskedasticity implies varying slopes in quantiles and elaborate

how to distinguish this kind of quantile dependency that arises purely from heteroskedasticity.

2. Simulate data from such model and show empirically, that your finding holds.

Hint: Derive the quantiles of income for given education first, i.e. qτ (income|educ) and then

think how this relates to slope of quantile regression.

Problem 3.

1. Simulate 42, 168 and 672 data points from exponential distribution (λe−λx) with λ of your choice.

2. Using all 3 samples, fit the λ parameter using MLE and exponential distribution as your assumed

distribution.

3. Then, again using all 3 samples, fit the gamma distribution ( βαΓ(α)xα−1−βx) using MLE again.

Report both estimates and their standard errors.

4. Finally, as the exponential distribution is a special case of gamma distribution, use the three

likelihood-based tests you’ve seen during the lecture as well as seminar to test the null-hypothesis

that the data come from exponential distribution (again using all 3 samples).

5. Compare results of your tests across all 3 samples and discuss whether they match your expectations.

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