Question

In the potential outcomes framework, suppose that program eligibility is randomly assigned but participation cannot be enforced. To formally describe this situation, for each person \(i, z_{i}\) is the eligibility indicator and \(x_{i}\) is the participation indicator. Randomized eligibility means \(z_{i}\) is independent of \(\left[y_{i}(0), y_{i}(1)\right]\) but \(x_{i}\) might not satisfy the independence assumption. i. Explain why the difference in means estimator is generally no longer unbiased. ii. In the context of a job training program, what kind of individual behavior would cause bias?

Short answer

Non-compliance with eligibility leads to bias because the treated group isn't purely randomized. In job training, motivated individuals self-select, skewing results.

Step-by-step solution

Understanding the Difference in Means Estimator

The difference in means estimator calculates the average causal effect by comparing the average outcomes between those eligible and ineligible for a program: \[ \text{Difference} = \text{mean}(y_i|z_i=1) - \text{mean}(y_i|z_i=0) \]This estimator assumes that eligibility (\(z_i\)) ensures participation (\(x_i\)), such that no differences exist, other than treatment, between groups.

Introducing Participation Issues

In the scenario where participation cannot be enforced, eligibility does not always lead to participation (i.e., \(z_i=1 \Rightarrow x_i\) is not always true). This means some who are eligible may not participate, and some who are ineligible might not have participated even if given the opportunity. This deviation introduces bias because treated and control groups may intrinsically differ in unobservable ways.

Effect of Non-Compliance On Estimator

Non-compliance (eligible individuals who do not participate) causes the average treatment effect calculated using the difference in means estimator to only reflect the effect among compliers, the subset of participants who comply with the treatment assigned, potentially biasing results if compliers differ systematically from non-compliers.

Behavioral Causes of Bias in Job Training Context

Bias occurs if individuals who choose to participate differ significantly from those who do not. In a job training example, motivated individuals might self-select into the program regardless of assignment, having a naturally higher likelihood of success. This self-selection implies any difference in outcomes might reflect individual motivation rather than program effectiveness.

Key concepts

Randomized Eligibility

In the potential outcomes framework, randomized eligibility refers to a situation where individuals are randomly assigned as eligible or ineligible for a program, such as a job training program. This process ensures that the eligibility indicator, denoted as \( z_i \), is distributed independently of the potential outcomes, \( y_i(0) \) and \( y_i(1) \). In simpler terms, whether someone is eligible or not is decided by chance and not influenced by their inherent characteristics or potential behaviors.

This random assignment of eligibility is intended to help create equivalent groups, much like flipping a coin to decide who gets access to an opportunity. However, the catch here is that while eligibility is random, actual participation (denoted by \( x_i \)) might not be random. This means that although someone is selected as eligible, they may not actually participate, introducing complexities in analyzing the program's impact.

Difference in Means Estimator

The difference in means estimator is a simple yet powerful tool used to estimate causal effects in experiments. It works by calculating the average difference in outcomes between eligible and ineligible groups:\[ \text{Difference} = \text{mean}(y_i|z_i=1) - \text{mean}(y_i|z_i=0) \]This method assumes that all eligible individuals (\( z_i=1 \)) participate and those who are ineligible (\( z_i=0 \)) do not.

However, when participation is not guaranteed, this assumption fails. Not all eligible individuals may participate, and this discrepancy, known as non-compliance, leads to biased results. The estimator then tends to reflect outcome differences not purely from the effect of the program but also from other differing characteristics between the compliant participants and non-participants.

Non-Compliance Bias

Non-compliance bias occurs when participants' behavior does not align neatly with their assigned eligibility for a program. For example, consider a job training program where eligible individuals (\( z_i=1 \)) are supposed to participate, but only some actually do so (\( x_i \) might not equal 1 for all \( z_i=1 \)).

The problem arises because the difference in outcomes (the difference in means estimator) now only considers those who complied with the treatment, i.e., those who both were eligible and chose to participate. This subset of *compliers* might have different characteristics than non-compliers, creating a bias. Thus, the effect observed is not an accurate representation of the program's influence across the whole eligible population; instead, it pertains to this selective group that may naturally have different outcomes.

Individual Behavior Bias

Individual behavior bias; this is a concept that arises when the personal choices and motivations of individuals skew the results of a study or program evaluation. In the context of randomized program eligibility, such as a job training program, certain individuals may self-select to participate based on their inherent characteristics.

For instance, in this scenario, individuals who are more motivated or have a greater need for the program might participate regardless of their eligibility status. These individuals may have better outcomes simply because of their initial motivation or circumstances, not just because of the program's impact, creating a bias.

This bias challenges our understanding of how effective the program is because the outcomes can no longer be solely attributed to the intervention itself. Instead, the difference might partly reflect the varying levels of drive and need among participants.