Definitions

# Average treatment effects

Average treatment effects (ATE) is an econometric measure of treatments used in from policy evaluation to medicine.

## General definition

The expression "treatment effect" refers to the causal effect of a given treatment or policy (for example, the administering of a drug) on an outcome variable of interest (for example, the health of the patient). Originating from medicine, the term "treatment" is now applied, more generally, to other fields of science such as, for example, the evaluation of the impact of public policies.

Since the effect of a given treatment can vary across individuals, and since it is usually not possible to estimate the treatment effect for each individual, researchers have to rely on aggregate measures of treatment effect for a given group. Among these measures, the Average treatment effects (ATE) is the average of the individual treatment effects across the whole population of interest. For example, if the variable of interest is the impact a job search monitoring policy on the length of an unemployment spell, the ATE is the average expected reduction in the duration of the unemployment spell among all those who experience unemployment.

Other aggregate measures widely used are the Average Treatment Effect on the Treated (ATET) and the Local Average Treatment Effect (LATE).

## Formal definition

In order to define formally the ATE, we define two potential outcomes : $y_\left\{0i\right\}$ is the value of the outcome variable for individual $i$ if she is not treated, $y_\left\{1i\right\}$ is the value of the outcome variable for individual $i$ if she is treated. For example, $y_\left\{0i\right\}$ is the health status of the individual if she is not administered the drug under study and $y_\left\{1i\right\}$ is the health status if she is administered the drug.

The treatment effect for individual \$i\$ is given by $y_\left\{1i\right\}-y_\left\{0i\right\}=beta_\left\{i\right\}$. In the general case, there is no reason to expect this effect to be constant across individuals.

Let $E\left[.\right]$ 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: $E\left[y_\left\{1i\right\}-y_\left\{0i\right\}\right]$.

If we could observe, for each individual, $y_\left\{1i\right\}$ and $y_\left\{0i\right\}$ among a large representative sample of the population, we could estimate the ATE simply by taking the average value of $y_\left\{1i\right\}-y_\left\{0i\right\}$ for the sample: $frac\left\{1\right\}\left\{N\right\} cdot sum_\left\{i=1\right\}^N \left(y_\left\{1i\right\}-y_\left\{0i\right\}\right)$ (where $N$ is the size of the sample).

The problem is that we can not observe both $y_\left\{1i\right\}$ and $y_\left\{0i\right\}$ for each individual. For example, in the drug example, we can only observe $y_\left\{1i\right\}$ for individuals who have received the drug and $y_\left\{0i\right\}$ for those who did not receive it; we do not observe $y_\left\{0i\right\}$ for treated individuals and $y_\left\{1i\right\}$ 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.

## Estimation

Depending on the data and its underlying circumstances, many methods can be used to estimate the ATE. The most common ones are

Once a policy change occurs on a population, a regression can be run controlling for the treatment. The resulting equation would be $y = Beta_\left\{0\right\} + delta_\left\{0\right\}d2 + Beta_\left\{1\right\}dT + delta_\left\{1\right\}d2 cdot dT$ where y is the response variable and $delta_\left\{1\right\}$ measures the effects of the policy change on the population.

The difference in differences equation would be $hat delta_\left\{1\right\} = \left(bar y_\left\{2,T\right\} - bar y_\left\{2,C\right\}\right) - \left(bar y_\left\{1,T\right\} - bar y_\left\{1,C\right\}\right)$ where T is the treatment group and C is the control group. In this case the $delta_\left\{1\right\}$ measures the effects of the treatment on the average outcome and is the average treatment effect.

From the diffs-in-diffs example we can see the main problems of estimating treatment effects. As we can not observe the same individual as treated and non-treated at the same time, we have to come up with a measure of counterfactuals to estimate the average treatment effect.