In the potential outcomes framework, suppose that program...

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?

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?

Key Concepts

Difference in Means Estimator and Its Unbiasedness

The difference in means estimator assumes that the treatment and control groups are comparable in all aspects except for the treatment. When actual treatment receipt is influenced by factors related to the potential outcomes (due to noncompliance or self-selection), this condition fails, making the difference in means estimator biased for the true causal effect.

Endogeneity and Selection Bias

Endogeneity arises when an explanatory variable is correlated with the error term, often due to omitted variables that simultaneously affect treatment uptake and the outcome. In a context where individuals self-select into participation, their decision might be influenced by characteristics that also affect the outcome, leading to selection bias in estimating treatment effects.

Randomized Eligibility Versus Participation

Randomized eligibility implies that the assignment to being eligible for treatment is random and thus independent of potential outcomes. However, actual participation (or treatment take-up) might deviate from eligibility due to noncompliance, which introduces differences between those who are eligible and those who actually participate, potentially compromising the independence needed for unbiased estimation.

Potential Outcomes Framework

This framework conceptualizes causal effects by defining, for each individual, potential outcomes corresponding to each treatment status. It provides a basis for thinking about what the outcome would be if the individual were exposed to treatment versus if they were not, which is fundamental for causal inference and treatment effect estimation.

Noncompliance

Noncompliance occurs when individuals do not adhere to their assigned treatment status. In contexts where participation is voluntary, even random assignment to eligibility does not guarantee that the treatment is actually received, so the group that takes up treatment may differ in unobserved ways from the group that does not, leading to biased estimates.

Transcript

00:02 Okay, so this question we say we have two different statistics that we're comparing.

00:06 We're trying to get an good estimate about the parameter.

00:09 Part a says if everything else is the same about these two parameters.

00:15 So we have their variances are the same and things like that.

00:18 They want us to say, why would we choose the biased or the unbiased versus the unbiased? we would choose the unbiased versus the biased parameter.

00:34 And now, for an example of understanding this, an unbiased parameter means our population mean or the parameter mean is the same as our statistic.

00:52 So that would be something kind of like this.

00:55 Now a biased, so this is unbiased, i'm going to label it you.

00:59 Now, a biased mean would be something, and again, we're saying all else equal, so looks about the same, but a biased one has it centered off of the parameter mean.

01:14 And so this is our biased one.

01:18 And this is why we would choose our unbiased is because if we want to estimate it, we want most of our values to be centered around the mean.

01:27 The mean is kind of our most important part of the parameter.

01:32 And so that's why we would choose this unbiased one is because the means would be equivalent.

01:38 And that's just kind of by definition of unbiased.

01:47 Now, the second one says, so the second part b wants us to know, even if it's unbiased, does that ensure that it's a good, estimator.

02:06 So what we're saying here is even if the means are equivalent, so even if i have this, the means equivalent, is there any case, this is part b, is there any case that i wouldn't want to use this statistic? one reason might be that our distribution might look extremely flat.

02:42 And so we're not going to get very consistent values out of this.

02:49 Because to be able to estimate this, i might have to make an interval estimate if i want enough confidence in my estimate.

03:00 I might have to make an interval that is not meaningful.

03:04 For example, if we were trying to measure height, and let's say this was whatever estimate we were trying to use to measure height.

03:15 And we'd say, let's say the average height is like 5 '8 in a particular place.

03:27 But i had to say, well, to be confident enough, i have to say that i'm estimating that it's going to be between 3 .8 and...

03:41 7 .8, then that's not really meaningful to say, well, yeah, most people are between 3 feet 8 inches and 7 feet 8 inches.

03:56 Well, that's kind of just useless.

04:00 I mean, whether we're trying to do that estimate for pant lengths, i mean, there's a huge difference in those things...

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