Homework 1

library(tidyverse)

Part 1: Introduction to for Loops and Simulation

In this section you are going to write a for loop to help with simulation. I have written an example below that you can use to help you.

Assume the following data generating process. \(X_{n,1}\) is a normal random variable that can be written as:

\[ X_{n,1} = \mu + \epsilon_{n,1} \]

where \(\epsilon_{n,1} \sim \mathcal{N}(0,\sigma^2)\). \(X_{n,2}\) is a normal random variable given by:

\[ X_{n,2} = \mu + \rho\times\epsilon_{n,1} + \epsilon_{n,2} \]

where \(\epsilon_{n,2} \sim \mathcal{N}(0,\sigma^2)\). Each random variable \(\epsilon_{n,j}\) is independent of \(\epsilon_{m,k}\) for all \(m\), \(n\), \(j\), and \(k\).

Here is a function to simulate a sample of \((X_{n,1},X_{n,2})\) of size \(N\) given parameter values \(\mu\), \(\sigma\), and \(\rho\).

draw_data <- function(mu,sig,rho,N) {
  X = matrix(0,N,2) #<- this is an N x 2 array that will store X_{n,1} in the n-th row and 1st column, and X_{n,2} in the second column of the n-th row
  eps_1 = rnorm(N,0,sig) #<- this command simulates N random realizations from a normal with mean zero and standard deviation sig. The n-th entry of this vector will be taken as the valuate of epsilon_{n,2}
  eps_2 = rnorm(N,0,sig) #<- another vector of normal random variables, this time the n-th entry is epsilon_{n,2}
  X[,1] = mu + eps_1 #<- This command sets of the value of the first column of X (i.e. each X_{n,1} for n=1,...N)
  X[,2] = mu + rho*eps_1 + eps_2 #<- And setting the value for the second column (each X_{n,2} for n=1,...,N)
  X
}

Question 1:

Suppose we were to treat each observation of \(X_{n,j}\) as taken from an iid sample of size 2\(N\). In this case the best estimator of \(\mu\) would be the sample mean:

\[\hat{\mu} = \frac{\sum_{n=1}^{N}\sum_{j=1,2}X_{n,j}}{2N}\]

What value of \(\rho\) would be necessary for the iid assumption to be true? When \(\rho=0\), write a formula for the variance (i.e. \(E[(\hat{\mu}-\mu)^2]\)) of \(\hat{\mu}\) in terms of \(\sigma\) and \(N\).

Question 2:

Below is code that “tests” the estimator by repeatedly drawing a sample of size \(N\) and calculating the estimator on that sample. Effectively, we are simulating draws from the sampling distribution of \(\hat{\mu}\). The argument \(B\) is the number of times to repeat the test.

montecarlo <- function(mu,sig,rho,N,B) {
  mu_hat = matrix(0,B) #<- this will store an estimate from each trial
  for (b in 1:B) {
    xb = draw_data(mu,sig,rho,N) #<- simulate a sample of size N
    mu_hat[b] = sum(xb) / (2*N) #<- store the estimator for this trial
  }
  mu_hat
}

Assuming that \(\mu=2\), \(\sigma=1\) and \(N=100\) (and \(\rho=0\)), use this function to calculate the variance of \(\hat{\mu}\) for \(B=10,50,100,1000\) repetitions of the test. Show that the sample variance of \(\hat{\mu}\) converges to what your formula in Question (1) predicts.

Question 3:

Repeat the above exercise using \(\rho=0.5\) instead. What do you notice about the sample variance of \(\hat{\mu}\) relative to your formula? Explain what is happening here.

Question 4:

Update the simulation function above to instead calculate a confidence interval for \(\mu\). Using a value of \(B=1000\), show that the confidence interval works as it is intended when \(\rho=0\). Repeat the exercise for \(\rho=0.5\). Does the confidence interval still work as intended? Why or why not?

Part 2: An introduction to reading and analyzing data

You are going to document some patterns using the Current Population Survey which you have been introduced to in recitation and in class.

Question 1:

Following the example from recitation, load the csv file cps-econ-4261.csv and replicate the steps for cleaning the data and creating the Wage variable, with one key difference: keep only observations between 2014 and 2018.

Question 2:

Write code to calculate mean wages separately by age (variable AGE), sex, and fertility status (hint: in recitation you did this by year, sex, and fertility status).

Question 3:

Write code to plot mean wages by age, sex, and fertility status. You can present the plot in whichever way makes the patterns most clear. Offer a brief interpretation of what the evolution of wage gaps by age and fertility status tell us about one potential source of gender gaps in wages.