Treatement Parameters

What are Treatment Parameters?

• it is generally accepted that calculating the treatement effect (TE) for an individual \[\Delta_i = TE \]

• Instead of trying to find the treatment effect for any one indiviudal, we want to know the average treatment for the population and the treatment distribution of Te

• the proportion of people who would benefit from the program:

\[Pr(Y_{1i} > Y_{0i}) = Pr( \Delta_{i} > 0)\]

• the porportion of people taking the program who benefit from it:

\[Pr(Y_{1i} > Y_{0i}|D =1) = Pr( \Delta_{i}|D=1 > 0)\]

• the increase in porportion of outcomes above some threshold due to the policy \[\hat{y} = threshold\] \[Pr(Y_{1i} > \hat{y}|D=1) - Pr( Y_{0i} > \hat{y} | D = 1)\]

library(tidyverse)

#the accountant makes on average $65,000 a year with a standard deviation of $5,000
accountant <- rnorm(100000,
                    65000,
                    5000)
#the economist makes on average $60,000 a year with a standard deviation of $10000
economist <- rnorm(100000,
                   60000,
                   10000)

#using a simple economy for this simulation - where people will choose the careeer with a #higher  
model <- data.frame(accountant = accountant, 
                    economist = economist,
                    Decision = ifelse(economist > accountant, 1, 0)
                    )
head(model)

mean_accoutnant <- mean(accountant)
mean_accoutnant

[1] 64995.84

mean_economist <-  mean(economist)
mean_economist

[1] 59962.94

accountant_df <- filter(model, Decision == 0)
#mean accountant salary if they become accountants:
accountant_fact <- mean(accountant_df$accountant)%>%print()

[1] 66206.02

#mean acccountant salary if they became economists:
accountant_counterfact <- mean(accountant_df$economist)%>%print()

[1] 55146.89

economist_df <- filter(model, Decision == 1)
#mean economist salary when they are an economst
economist_fact <- mean(economist_df$economist)%>%print()

[1] 69818.77

#mean economist salary when theywould have been an accounant
economist_counterfact <- mean(economist_df$accountant) %>% print()

[1] 62519.25

#number of observations of accountant:
num_accountants <- nrow(accountant_df)
print(num_accountants)

[1] 67175

#number of observations for economists:
num_economists <- sum(economist_df$Decision)
print(num_economists)

[1] 32825

#total number of observations:
total_obs <- num_accountants + num_economists
print(total_obs)

[1] 1e+05

Naive Estimator

What would the niave estimator suggest is the impact

remember that the niave estimator is simple subtracting the average treatment effect for the treated minus the average treatment effect for the untreated

\[\Delta_i =\bar{Y}_{1i} - \bar{Y}_{0i}\]

#Here, the treatement is being an economist, untreated are those that are accountants.
#We subtract the average of both salaries to get the average treatment effect
avg_treatment_effect <- economist_fact - accountant_fact
print(avg_treatment_effect)

[1] 3612.752

This suggests that it is better for people to choose to be economists.

However!

this is niave (i.e. incorrect/inaccurate) because it assumes that

\[E(Y_{1}) = E(Y_{i} | D = 1) \]

AND

\[E(Y_{0}) = E(Y_{0} | D=0)\]

Average Treatment Effect

Since we get to observe all potential outcomes in this excercise, we can calculate that Average Treatment Effect (ATE)

\[\Delta^{ATE} = \bar Y_{1} - \bar Y_{0} \]

average_treatment_effect <- mean_economist - mean_accoutnant
print(average_treatment_effect)

[1] -5032.901

This is the opposite sign of the naive estimator

It tells us what the impact would be if we made everyone become economists. We know that this must be true since it reflects the distributions from which we simulated the data

Average Treatment Effect on the Treated:

we can calculate the real gain from being an economist for the people who actually became economist:

\[\Delta^{ATET} = \bar y_{1, D=1} - \bar y_{0, D=1}\]

average_treatment_effect_treated <- economist_fact - economist_counterfact
print(average_treatment_effect_treated)

[1] 7299.521

Average Treatment Effect on the Not Treated:

estimate the impact of being an economist for those who actually become accountats:

\[ \Delta^{ATEN} = \bar y_{1,D=0} - \bar y_{0,D=0} \]

average_treatment_effect_not_treated <- accountant_counterfact - accountant_fact
print(average_treatment_effect_not_treated)

[1] -11059.13

It looks like those who became accountants made the right decision

They would have been worse off if they had become economists!

These demonstrate why the niave estimator is a bad metric to track program effectiveness