[SOLVED] Socga2332 – problem set 2

Instructions
1. Submit two files for each problem set. The first is a R Markdown (.Rmd) file that can be run without error from start to end. The second is a PDF rendered from your R Markdown file or created using LATEX.
2. Name your files following this convention: [Last Name]_ps2.Rmd and [Last Name]_ps2.pdf (for example, Jiang_ps2.Rmd).
4. You are encouraged to discuss the problems with your classmates. But the R Markdown and PDF files that you submit have to be created on your own. Please do not ask for solutions from students in earlier cohorts.
5. Comment on your code wherever possible and explain your ideas in detail. You will get credits for showing the steps you take and for explaining your reasoning, even if you do not get the correct final result.

knitr::opts_chunk$set(echo = TRUE) ## load packages here
Part 1: Assumptions
Consider the “true” data-generating process (DGP):
Y = β0 + β1X +
1. [10pts] Suppose we obtain a sample of size n from this process. State the assumptions under which the OLS estimator βˆ1 is BLUE (Best Linear Unbiased Estimator) for β1.
2. [10pts] For each of the assumptions, discuss what will go wrong when the assumption is violated.
3. [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. A simple Google search will lead you to a way to simulate from this distribution (Don’t get scared away from the new distribution and the fancy Greek letters!). Simulate a dataset of size n = 3,000 from this process in which all of the assumptions you’ve discussed above hold.
4. [20pts] For each of the assumptions, create a plot which illustrates how the violation of the assumption affects the regression results (all other assumptions should be satisfied). This can be a scatter plot, a plot of the regression line, the sampling distribution of the OLS estimator (comparing what the regression table suggest with actual simulations), or anything else. The important point is to show how the violation leads us to false decisions if we assume the assumption is true, not just showing that the assumption is violated. When simulating data, you don’t have to use the parameters set in the previous problem. Also, you can skip the “no autocorrelated errors” assumption, which goes beyond the course materials.
Further bigger hint: For the violation of exogeneity assumption, the easiest way is to create two variables, x1 and x2, with non-zero covariances between the two. y is a function of x1 and x2, while when you specify the OLS regression, you omit either x1 or x2. In this case, the error term, by construction, will be correlated with the independent variable. Then discuss the difference between the true model, where exogeneity assumption is satisfied.
Part 2 Simulate and Plot Correlation
[15pts] Replicate the following figure which demonstrates the different levels of Pearson’s correlation.
Hint: Look up mvrnorm() from the MASS package.

Part 3 Simulate Interactions
1) [15pts] Simulate a data generation process that will provide you data to roughly replicate the two-panel figure below. You should show your code and replicate the two-panel example figure.
Note: Your output does not need to replicate the exact layout of the example figures. You will get credit as long as you generate a similar plot that illustrates the relationship clearly. Remember to use set.seed() for any random process.

2) [15pts] Simulate a scatter plot that demonstrates the Simpson’s Paradox:

3) [10pts] Choose one of the above data simulations you just generated, write down the regression equation you estimated using OLS, and interpret the coefficients in your own words.
Hint: You will need to specify group-specific slopes and intercepts.
Part 4: The Replication Project
Read through the article and determine which variables you will need to replicate the project. Go to the IPUMS webpage and submit a screenshot of the page where you can see the samples and variables that you have selected (you can obtain this from your “Data Cart”).