Defining Average treatment effect (ATE) & Average treatment effect on the treated (ATT)

Question

I have some questions regarding the following theoretical model and finding the average treatment effect (ATE) and average treatment effect of the treated (ATT). I'm not sure if I'm defining them correctly.

$y_0 = \alpha_0+X_i\alpha_1+\epsilon_0$ where $y_0$ represents earnings for high school graduates if $S_i=0$.

$y_1 = \beta_0+X_i\beta_1+\epsilon_1$ where $y_1$ represents earnings for college graduates if $S_i=1$.

The error terms $\epsilon_0, \epsilon_1$are both Normal(0,1) and mutually independent.

Next the choice between high school and graduate studies is driven by the following utility function: $U(S_i)=y_0(1-S_i)+y_1S_i+(\epsilon_SS_i)$ where $\epsilon_S$ is also Normal(0,1) and is uncorrelated with both $\epsilon_0 , \epsilon_1$ and therefore uncorrelated with $(\epsilon_1-\epsilon_0)$.

My results:

Define ATE: $ATE = E(y_1) – E(y_0) = (\beta_0-α_0 )+X_i (β_1-\alpha_0 )+ϵ_1-ϵ_0$ I don't feel my answer is complete here. I'm not understanding how to define ATE. My understanding of the this measurement is that we take all the results and average the difference between the 2 subsets.

Define ATT: $ATT=E(y_1-y_0│X,S=1) =E(y_1│X,S=1)-E(y_0│X,S=1)$

Then the ATT would be: $ATT=pr(earnings|university)= (\beta_0-α_0 )+X_i(β_1-\alpha_1)+E(ϵ_1-ϵ_0>-(\beta_0-α_0 )-X_i(β_1-\alpha_1))$

Are my definitions complete or am I on the wrong track? What am I missing ?

Answer 1

You have a few errors.

$$ATE = E[y_1]-E[y_0] = \beta_0 - \alpha_0 + \beta_1 E[X] - \alpha_1 E[X] + E[\epsilon_1] - E[\epsilon_0]$$

We know $E[\epsilon_1] - E[\epsilon_0]=0$, so we can reduce this to

$$ATE = (\beta_0 - \alpha_0) + (\beta_1 - \alpha_1)E[X]$$

The ATT is $E[y_1|S=1]-E[y_0|S=1]$, so $$ATE = E[y_1|S=1]-E[y_0|S=1] = \beta_0 - \alpha_0 + \beta_1 E[X|S=1] - \alpha_1 E[X|S=1] + E[\epsilon_1|S=1] - E[\epsilon_0|S=1]$$ We can only assume $E[\epsilon_1|S=1] - E[\epsilon_0|S=1] = 0$ under selection on observables (i.e., $S \perp y_1|X$), which you don't claim here, but if that assumption was true, then we could rewrite the ATT as

$$ATT = (\beta_0 - \alpha_0) + (\beta_1 - \alpha_1)E[X|S=1]$$

The difference between the ATE and ATT is only in the group over which you are taking the expected value. For the ATE, it's the whole population; for the ATT, it's the population of those who receive treatment.