where b(i) is the individual response coefficient to the policy or treatment x.
An extreme version of equation (2) assumes a common response effect across all individuals, i.e., a “homogeneous effects” model. An intermediate specification might allow the response parameters to vary according to observed covariates (the z variables defined above). In general, however, the “heterogeneous effects” model of equation (2) has become the standard reference model for evaluating policy interventions.
To fully understand the impact of a policy intervention or treatment ‘x' on the outcome measure ‘y', the best-case situation would be to know the full distribution of the response parameters b(i). For example, although the mean or median response may be positive, the lower quartile of the response distribution could still show a negative impact. However, we do not see the same individual with and without the treatment at the same time and in the same country. Typically, therefore, we must settle for the average effect.
A properly designed experiment measures the expected impact of the treatment on individuals drawn at random from the population. Again, this can usually be broken down into the average response for subgroups according to observed covariates ‘z'.
For nonexperimental data, a popular alternative parameter of interest is the average impact of the intervention on those who are included in the program, that is, the average treatment effect on the treated. Suppose we divide a particular group according to the observed variables z; for example, we might choose women who are between 50 and 60 years of age who live in a high-unemployment area. Among these women, let some subsample be subject to the treatment, and the average response for this subsample is the impact of the treatment on the treated.
When the treated and comparison groups are chosen randomly as in an experiment, the average treatment of the treated measures the average treatment effect. But when the treatment group occurs by self-selection or by some other nonrandom mechanism, we are simply measuring the average treatment effect among the treated. This is a much less interesting parameter but one that is used regularly in the ex-post evaluation of policy interventions.
One simple measure of the average response parameter is to take the difference in the outcomes between the treated group and the comparison