Average Treatment Effect - Formal Definition

Formal Definition

In order to define formally the ATE, we define two potential outcomes : is the value of the outcome variable for individual if he is not treated, is the value of the outcome variable for individual if he is treated. For example, is the health status of the individual if he is not administered the drug under study and is the health status if he is administered the drug.

The treatment effect for individual is given by . In the general case, there is no reason to expect this effect to be constant across individuals.

Let denote the expectation operator for any given variable (that is, the average value of the variable across the whole population of interest). The Average treatment effects is given by: .

If we could observe, for each individual, and among a large representative sample of the population, we could estimate the ATE simply by taking the average value of for the sample: (where is the size of the sample).

The problem is that we can not observe both and for each individual. For example, in the drug example, we can only observe for individuals who have received the drug and for those who did not receive it; we do not observe for treated individuals and for untreated ones. This fact is the main problem faced by scientists in the evaluation of treatment effects and has triggered a large body of estimation techniques.