Description
Part 1 Assumptions of OLS Regression
Recall that in our first lecture on regression, we talked about the Gauss-Markov Assumptions. If all these
assumptions are met, the OLS estimator is the Best Linear Unbiased Estimator (BLUE). In a simple
bivariate case, if the “true” data-generating process is Y = β0 + β1X + .
The Gauss-Markov Assumptions
can be state as the following:
(a) Linearity: A linear relationship between X and Y hold in the sample.
(b) Exogeneity of Predictors: The conditional mean of the error term, given the predictor, is zero
(x = [x1, x2, …, xn]
> is the value vector of X):
E[i
|x] = 0, for all i = 1, 2, …, n.
(c) No Perfect Collinearity: Explanatory variables cannot be perfectly correlated.
(d) Homoskedasticity:
• No Heteroskedasticity: The conditional variance of the error term, given the predictor, is constant:
V ar[i
|x] = σ
2
, for all i = 1, 2, …, n.
• No Autocorrelation: Conditional on the predictor, the error terms are uncorrelated across the
observations: Cov[i
, j |x] = 0 , i 6= j.
1. [15pts] For each of the assumptions, discuss what will go wrong when the assumption is violated. Be
brief in your answers. Note: In addition to class materials, you can learn more about these assumptions in
the Wikipedia article on the Gauss–Markov theorem, particularly the “Gauss–Markov theorem as stated in
econometrics” section. (You can skip all the mathematical proofs and remarks.)
[Your Answer Here]
2. [5pts] Let β0 = −0.25, β1 = 1.2, X ∼ Γ(5, 4), and ∼ Normal(0, 1). Here, Γ(α, ψ) denotes the Gamma
distribution with shape parameter α and rate parameter ψ. (You can search how to use R to simulate from
this distribution.)
Simulate a dataset of size n = 3, 000 from this process in which all of the assumptions you’ve discussed above
hold. Estimate a OLS model and plot regression diagnostics of this model.
# Your code here
Bonus Question [10pts]: From assumption (a), (b) and (d), choose one assumption and simulate a data
that violates that assumption (all other assumptions should be satisfied). Create a plot which illustrates how
the violation of the assumption affects the regression results. This can be a scatterplot with both the “true”
and “false” OLS lines, a sampling distribution of the OLS estimator (comparing your estimate model results
with actual simulations), or anything that shows how the violation leads us to false decisions if we assume
the assumption is true. (The point is to demonstrate a contrast between the “true” and the “false”, not just
diagnostics of the “false”.)
When simulating data, you don’t have to use the parameters set in the previous problem.
Hint: You can search how to use + stat_function() to plot a nonlinear line when plotting with ggplot(),
or search how to use the base R functions such as plot() and curve().
# Your code here
Part 2 Causality
A study on COVID-19 constructed a “COVID risk factor” score based on the COVID infection rate of a
given area (defined by zip code).
A researcher wants to estimate the effect of having a vaccination center in the area on that area’s COVID
risk factor score. She compiled a dataset that contains each area’s COVID risk factor score and whether the
area has a vaccination center. She then estimated the effect of having a vaccination center using the “naive
estimator” we discussed in class.
You noted that the quality of information residents have about COVID and the vaccine can be a confounding
variable that affects both the area’s infection rate and whether there is a vaccination center in the area.
Assume that you are able to estimate the relationships this “informedness” confounder (info) and the original
“vaccination center” predictor (vaccine) have with the COVID risk factor score (covid_risk), which can be
simulated using the following code (n is sample size):
set.seed(1234) # set the same seed to ensure identical results
e = rnorm(n, 0, 0.5)
covid_risk = rescale( 0 – 7*vaccine – 2*info + e, to = c(0, 100))
1. [5pts] Import the data covid.csv, according to the counterfactual framework, constructing a counterfactual
“risk factor” in the dataframe.
# Your code here
2. [10pts] Fill out the table below (round to 1 decimal points):
Group Y
T Y
C
Treatment Group (D = 1) E[Y
T
|D = 1] =? E[Y
C |D = 1] =?
Control Group (D = 0) E[Y
T
|D = 0] =? E[Y
C |D = 0] =?
3. [15pts]Estimate the following:
(a) The Naive Estimator of ATE
(b) Treatment Effect on the Treated
(c) Treatment Effect on the Control
(d) Selection Bias
4. [15pts] Write a non-technical, short summary reporting your results in response to the above mentioned
researcher who used the naive estimation. Imagine that you are explaining this to an audience who may not
be familiar with the specific terminologies of the counterfactual framework (such as ATE or Treatment Effect
on the Treated), but is interested in your substantive findings.
[Your Answer Here]
Part 3 Linear Probability Model and Logistic Regression
admin.csv contains a dataset of graduate school admission results with the following variables:
Variable Name Variable Detail
admit Admission Dummy (Admitted is 1)
gre GRE score
gpa GPA
rank Institution Tier (Tier 1 to 4)
1. [10pts] Import admin.csv to your R environment. Estimate (a) a linear probability model and (b) a
logistic regression model to predict the probability of being admitted based on the applicant’s GRE, GPA,
and institution tier. Display the two modeling results in a table.
# Your code here
2. [10pts] In one or two paragraphs, summarize your modeling result for each model.
[Your Answer Here]
3. [15pts] Plot the predicted probability of admission based on one’s GPA percentile and institution rank
(holding GRE at the mean) for the logistic regression model. For the purpose of this exercise, please set the
value of gpa to range from 1 to 4. Make sure to add appropriate title and labels to your figure.
# Your code here
Part 4 (Not Graded) Final Replication Project
At this point, you should complete most of the data cleaning and start replicating the descriptive tables and
figure. You can submit an additional PDF file if you have made progress in replication Table A1a, Table
A1b, and Figure 1.
