In this assignment, we will create a simulated dataset to explore: A) how sample size influences coefficient precision, B) how collinearity influences point estimates, and C) how variable functional form influences model fit.

Create a simulated dataset with the following characteristics:

• N = 100,000
• X1 = random_uniform
• X2 = random_uniform
• X3 = 0.2*X1 + random_normal(mean=1, sd=.2)
• Y = 0.2*X1 + 0.2*log(X2) + 0.1*X3 + random_normal(mean=1, sd=.2)

Perform and report on the following analysis:

1. Draw random samples of varying sizes (N=50,100,200,1000, etc.) and run regression models
   1. What happens to estimate precision across models? Why?
2. Using the population run and report the following regression specifications:
   1. Y ~ X1
   2. Y ~ X1 + X2
   3. Y ~ X1 + X2 + X3
   4. Comparing models: Why does the coefficient on X1 change? Why does the coefficient on X2 not change?
3. Using the population and report the following regression specifications
   1. Y ~ X1 + X2 + X3
   2. Y ~ X1 + log(X2) + X3
   3. Comparing models: why does model fit (R2) change?