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college distance fcollege income score tuition urban wage 1. Income → College: This arrow suggests that family income directly affects the likelihood of college enrollment, which is the primary causal effect we are interested in. 2. Fcollege → Income & College: Having a father who attended college (fcollege) likely influences both family income and the child’s likelihood of attending college, due to factors like higher earning potential and valuing education. 3. Income → Score: This suggests that higher family income can lead to better academic scores, possibly due to better educational resources. 4. Score → College: High scool academic scores directly impact college admission chances due to the college cut-off percentages. 5. Distance → Income & College: Geographic distance from college might affect family income (e.g., rural vs. urban income disparities) and the practicality of attending college. 6. Urban → Distance, Income & College: Urban living affects the distance to college, family income levels, and access to college education. 7. Wage → Income: Indicates that state minimum wages are a component of or influence on family income. 8. Tuition → College: The cost of tuition directly impacts the feasibility of college enrollment. By the backdoor criterion, all the backdoor paths from the ‘income’ (treatment) to ‘college’ (outcome) must be blocked in order for the effect of family income on college enrollment to be identifiable. A path is blocked when either it contains a non-collider that we condition on, or when it contains a collider that we do not condition on. In our case, the variables ‘fcollege’, ‘distance’, and ‘urban’ are non-colliders or confounders, i.e. variables that influences both the treatment and the outcome. Therefore, we must condition on these variables to understand the causal effect. 3

set.seed ( 123 ) nBoot <- 2000 # Number of bootstrap iterations boot_results <- rep ( NA , nBoot) # Prepare a vector to store the bootstrap results for (iter in 1 : nBoot) { # Resample the matched data with replacement bootstrap_sample <- matched_set[ sample ( nrow (matched_set), replace = TRUE ), ] # Fit a robust linear model to the treatment group treatment_model_boot <- lm (college ~ fcollege + urban + distance_college, data = subset (bootstrap_sample, income == 1 )) # Fit a robust linear model to the control group control_model_boot <- lm (college ~ fcollege + urban + distance_college, data = subset (bootstrap_sample, income == 0 )) # Predict the potential outcome under treatment for all units bootstrap_sample $ predicted_treatment <- predict (treatment_model_boot, newdata = bootstrap_sample) # Predict the potential outcome under control for all units bootstrap_sample $ predicted_control <- predict (control_model_boot, newdata = bootstrap_sample) # Calculate and store the ATT for this bootstrap iteration boot_results[iter] <- mean (bootstrap_sample $ predicted_treatment[bootstrap_sample $ income == 1 ]) - mean (bootstrap_sample $ predicted_control[bootstrap_sample $ income == 1 ]) } # Calculate the standard error of the bootstrapped ATT estimates se_hat <- sd (boot_results) # Calculate the 95% confidence interval for the ATT conf_interval <- quantile (boot_results, probs = c ( 0.025 , 0.975 )) # Output the estimated ATT, its standard error, and the confidence interval list ( ATT = mean (boot_results), SE = se_hat, ConfInt = conf_interval) ## $ATT ## [1] 0.1101 ## ## $SE ## [1] 0.01741 ## ## $ConfInt ## 2.5% 97.5% ## 0.07502 0.14371 The analysis indicates a notable positive impact of the treatment on the outcome. With an ATT of 0.1101, it suggests that the treatment (family income is above USD 25,000 per year) is associated with an 11.01% higher likelihood of the outcome (college enrollment). The precision of this estimate is high, as indicated by the standard error of 0.01741. The 95% confidence interval, ranging from 0.07502 to 0.14371, not encompassing 6