Simulating a Confounded DGP (Research Design)

In this exercise we are going to practice simulating a confounded DGP and diagnosing bias.

Set the random seed to 24601 and simulate 30,000 observations.
Generate an unobserved confounder called ability that is distributed Normal(0, 1)
Generate a variable called p_train using a probability rule (invlogit) that increases with ability: invlogit(-0.2 + 1.0*ability)
Then assign training as a Bernoulli draw train = (runiform() < p_train)
Create a variable eps ~ rnormal(0, 3) so there is meaningful noise
Generate wages so that the true causal effect of training be +2 and ability raises wages by +4 while wages without training (train = 0) or ability (ability = 0) is 10
Add noise to the wage equation so that wage_lat = 10 + 2*train + 4*ability + eps
Enforce wage >= 0 by truncating at 0 using max(wage_lat, 0)
Add variable labels to all variables and value labels for train (0 = no, 1 = yes)

1. What is the naive difference in means by training status and is it larger or smaller than the true causal effect?

2. Show that selection is happening by showing that ability differs by training status.