Conditioning on the dependent variable changes the L in LATE

sample_selection_bias

# Load packages
library(dplyr)

# Create data
set.seed(1)

x <- rnorm(1000000, mean = 0, sd = 1)
y <- x*x

full <- data.frame(x, y)

# Full data OLS
lm(y ~ x, data = full) %>% summary()

# Condition sample on tails of observed distribution
partial <- full %>% dplyr::filter(y > quantile(y, 0.8) | y < quantile(y, 0.2))

# Partial data OLS
lm(y ~ x, data = partial) %>% summary()