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Applied Econometrics Assignment The assignment is divided in three parts: 1) Interpreting real world events in statistical language

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Applied Econometrics Assignment
The assignment is divided in three parts: 1) Interpreting real world events in statistical language. 2) Simulation study of econometric theory 3) Empirical study using econometric tools. There is a data set I will send after getting the quote. Number 2 on the pdf regarding "Cheating in the sumo world" may not be a part of it

1 Practicalities
Assignment for the Econometrics class. Due on Wednesday 28 December 2022, 23.59h. Assignments can be made in groups of maximum 2 people. Discussions between groups regarding
general content and programming techniques is allowed (actually encouraged). However, copypasting of material (code or report) is NOT allowed and will be reported. You are expected
to hand-in a report with discussion of the questions and results, tables and figures. No code
should be present in the report. This should be provided separatedly in a working .py (or .r)
file. I need to be able to run that file if the data set is in the same folder as the .py file. DO
NOT LINK TO folders like ’C:\Dropbox\blablabla\econometrics\assignment\data’
For the report, you are expected to clearly translate your technical findings into plain English.
Just reporting tables with estimatess and graphs is NOT enough. If you are unsure, what is
expected, you might want to watch the plain_english.mp4 (again) and put yourself in the
role of the analyst (though you might want to leave out the part about me being a golden
retriever).
2 Cheating in the Sumo world
1. Comment on the scenes regarding cheating in sumo.mp4. Use your knowledge of econometrics, statistics and probability in order to explain what is described. Use terms such
as randomness, random sample, statistical significance, selection bias, etc.
3 Simulation study
You will investigate the distributions of the OLS estimator and the corresponding t-values, and
the model test.
yi = β0 + β1x1i + β2x2i + εi, with IE[εi] = 0 and V [εi] = σ2 (1)
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Table 1: Simulation combinations
n = 100, εi ∼ N 0, σ2 n = 100, εi ∼ Laplace(0, σ/√2)
n = 10, εi ∼ N 0, σ2 n = 10, εi ∼ Laplace(0, σ/√2)
In order to have different experiments for each group, all groups need to use a different seed.
Select as the seed the product of your birthdays (dd/mm) when you concatenate the day and
month identifiers.
E.g. 01/01 and 31/12 becomes 101 * 3112 = 314312.
# f i r s t b i r t h d a y
bd_1 = 3112
# s e c o n d b i r t h d a y
bd_2 = 3112
group_seed = bd_1 ∗ bd_2
# s e e d t h e random numb er g e n e r a t o r
rng = np.random.default_rng(group_seed)
We are interested in the distribution of βb, t and F (the model test).
Consider the 4 situations in Table 1.
1. Generate the variables x1 and x2 for all observations one time. Assume that x1 ∼
N (5, 1) and x2 ∼ N (0, 2) and ρ(x1, x2) = 0.5. Create a matrix X that includes both
the realized data x1 and x2 in addition to a constant term. Note: You will only create
one copy of this matrix. Using this matrix you will simulate multiple vectors for the
dependent variable.
2. Set the true values of β = (10, 0.5, 2.2) and σ = 0.25. Generate error terms for each
observation for each simulation. This means generate a matrix ε with n rows (number
of observations) and S columns (number of simulations). Hence, generate
E =
264
ε
(1)
1 · · · ε
(S)
1
...
.
.
.
...
ε
(1)
n · · · ε
(S)
n
375
(2)
for each of the combinations in Table 1. Use 216 simulations. Use rng.normal for
the normal random number generation and rng.laplace for the laplace random number
generation.
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3. Using the linear model
yi = β0 + β1x1i + β2x2i + εi
generate the dependent variable y1, ..., yn for each simulation.
Y =
264
y
(1)
1 · · · y
(S)
1
...
.
.
.
...
y
(1)
n · · · y
(S)
n
375
(3)
4. Create a plot of the error terms for both distributions. Do they look alike?
5. For each simulation calculate βb, the OLS estimate, the t-tests (vs the true values) and
the model test (always vs zero values for the parameters). You should have S OLS
estimates, S t-tests for each explanatory variable (including) the constant and S values
for the model test.
6. Create histograms for each of the quantities you found in the previous question.
7. For the t-tests plot the density of the appropriate t- distribution in the graph. Similarly,
for the model test plot the appropriate F distribution. Do they seem accurate in all cases?
8. Instead of the t-distributions, now plot the density of the normal distribution in the
graph. Similary, for the model test, plot the appropriate χ2 distribution. Do they seem
accurate in all cases?
9. Explain the consequences of your results. Which of the four (sample size, distribution)
combinations is problematic?
4 Empirical investigation
In this part you are asked to perform an empirical analysis on the return on education. It
considers the wage as a function of a number of explanatory variables among which whether
you go to junior college (2-years) or university (4-years). The data is in data.csv and the
variable descriptions are in Table 2. You will use a subset of this data. However, everyone will
use a different subset depending on your group seed.
# r e a d t h e f u l l d a t a s e t
data_full = pd.read_csv(’data.csv’)
num_obs = 6000
# s e l e c t 6 0 0 0 o b s e r v a t i o n s r a n d om l y ( t h e r n g u s e s y o u r s e e d )
observations = rng.choice(len(data_full), num_obs ,
replace=False)
# s e l e c t on t h e o b s e r v a t i o n s f o r y o u r gr o u p
data = data_full.iloc[observations , :].copy()
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4.1 Earnings from schooling
The dependent variable is log(hourly wage) = lwage. Hence, we have
lwagei = β0 + β1x1i + ... + β2xKi + εi (4)
You are encouraged to go beyond the questions asked here, but please motivate what you
are doing.
1. Investigate the descriptive statistics and note any peculiarities (if any).
2. Which of the non-linear models we saw in class is used here?
3. Use as explanatory variables, jc, univ and exper (in this order). Estimate the model
using OLS. Which variables are economically significant, and which variables are statistically significant? Explain.
4. Now test whether β1 (jc) is equal to β2 (univ). Explain which test you use, the
distribution and whether or not you can reject the null hypothesis.
5. In the test above you tested for equality. What if you wanted to test if β2 > β1?
This cannot be done in the usual manner. However, if you write θ = β2 - β1 we
have β1 = β2 + θ. Plug this into your regression equation and collect terms. Now
re-estimate the model using totcoll = jc + univ instead of univ using OLS. Compare
your estimates to the results above. Can you do a one-sided test now? If so, can you
reject the null-hypothesis?
6. The variable phsrank is the persons high school percentile. (A higher number is better.
For example, 90 means you are ranked better than 90 percent of your graduating class.)
Add phsrank to your original model and report the OLS estimates in the usual form. Is
phsrank statistically significant? How much is 10 percentage points of high school rank
worth in terms of wage?
7. Does adding phsrank to the original model substantively change the conclusions on the
returns to two- and four-year colleges? Explain.
Continue to use the model above for the next questions.
8. Test if the overall model makes sense. Write down the null hypothesis, the test statistic
and its distribution (under the null) and the test results.
9. Compute the confidence and prediction intervals for all observations. That is, construct,
confidence and prediction intervals around yb. Provide a plot (y vs yb with the confidence
and prediction intervals.)
10. There is the potential for heteroskedasticity. Perform a test for this and if needed adjust
the standard errors to heteroskedasticity consistent [HC] standard errors. Write down
the null hypothesis. Do you still find the same level of significance?
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Table 2: Description of the variables in the dataset
Variable Description

11. Try to improve the model by adding additional variables in the dataset (or combinations
of them). Explain your choices and results.
12. Before handing in, read the last line of Section 1 again and make sure that the code
runs from a clean (freshly started) Python or R session. You can do this by restarting
the kernel or (if you want to be absolutely sure) the whole of Python / R.
Best of luck!
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