## 6.11 Extra Questions

What is the difference between treatment effect homogeneity and treatment effect heterogeneity?

Why do most researchers give up on trying to estimate the individual-level effect of participating in a treatment?

Explain what unconfoundedness means.

What is the key condition underlying a difference-in-differences approach to learn about the causal effect of some treatment on some outcome?

What are two key conditions for a valid instrument?

Suppose you are interested in the causal effect of participating in a union on a person’s income. Consider the following approaches.

Suppose you run the following regression

\[\begin{align*} Earnings_i = \beta_0 + \alpha Union_i + \beta_1 Education_i + U_i \end{align*}\]

Would it be reasonable to interpret \(\hat{\alpha}\) in this regression as an estimate of the causal effect of participating in a union on earnings? Explain.

Suppose you have access to panel data and run the following fixed effects regression \[\begin{align*} Earnings_{it} = \beta_{0,t} + \alpha Union_{it} + \beta_1 Education_{it} + \eta_i + U_{it} \end{align*}\]

where \(\eta_i\) is an individual fixed effect. Would it be reasonable to interpert \(\hat{\alpha}\) in this regression as an estimate of the causal effect of participating in a union on earnings? Explain. Can you think of any other advantages or disadvantages of this approach?

Going back to the case with cross-sectional data, consider the regression \[\begin{align*} Earnings_i = \beta_0 + \alpha Union_i + U_i \end{align*}\] but using the variable \(Z_i = 1\) if birthday is between Jan. 1 and Jun. 30 while \(Z_i=0\) otherwise. Would it be reasonable to interpert \(\hat{\alpha}\) in this regression as an estimate of the causal effect of participating in a union on earnings? Explain. Can you think of any other advantages or disadvantages of this approach?

Suppose that you are interested in the effect of lower college costs on the probability of graduating from college. You have access to student-level data from Georgia where students are eligible for the Hope Scholarship if they can keep their GPA above 3.0.

What strategy can use to exploit this institional setting to learn about the causal effect of lower college costs on the probability of going to college?

What sort of data would you need in order to implement this strategy?

Can you think of any ways that the approach that you suggested could go wrong?

Another researcher reads the results from the approach you have implemented and complains that your results are only specific to students who have grades right around the 3.0 cutoff. Is this a fair criticism?

Suppose you are willing to believe versions of unconfoundedness, a linear model for untreated potential outcomes, and treatment effect homogeneity so that you could write \[\begin{align*} Y_i = \beta_0 + \alpha D_i + \beta_1 X_i + \beta_2 W_i + U_i \end{align*}\] with \(\mathbb{E}[U|D,X,W] = 0\) so that you were willing to interpret \(\alpha\) in this regression as the causal effect of \(D\) on \(Y\). However, suppose that \(W\) is not observed so that you cannot operationalize the above regression.

Since you do not observe \(W\), you are considering just running a regression of \(Y\) on \(D\) and \(X\) and interpreting the estimated coefficient on \(D\) as the causal effect of \(D\) on \(Y\). Does this seem like a good idea?

In part (a), we can write a version of the model that you are thinking about estimating as \[\begin{align*} Y_i = \delta_0 + \delta_1 D_i + \delta_2 X_i + \epsilon_i \end{align*}\] Suppose that \(\mathbb{E}[\epsilon | D, X] = 0\) and suppose also that \[\begin{align*} W_i = \gamma_0 + \gamma_1 D_i + \gamma_2 X_i + V_i \end{align*}\] with \(\mathbb{E}[V|D,X]=0\). Provide an expression for \(\delta_1\) in terms of \(\alpha\), \(\gamma\)’s and \(\beta\)’s. Explain what this expression means.

Suppose you have access to an experiment where some participants were randomly assigned to participate in a job training program and others were randomly assigned not to participate. However, some individuals that were assigned to participate in the treatment decided not to actually participate. Let’s use the following notation: \(D=1\) for individuals who actually participated and \(D=0\) for individuals who did not participate. \(Z=1\) for individuals who were assigned to the treatment and \(Z=0\) for individuals assigned not to participate (here, \(D\) and \(Z\) are not exactly the same because some individuals who were assigned to the treatment did not actually participate).

You are considering several different approaches to dealing with this issue. Discuss which of the following are good or bad ideas:

Estimating \(ATT\) by \(\bar{Y}_{D=1} - \bar{Y}_{D=0}\).

Run the regression \(Y_i = \beta_0 + \alpha D_i + U_i\) using \(Z_i\) as an instrument.

Suppose you and a friend have conducted an experiment (things went well so that everyone complied with the treatment that they were assigned to, etc.). You interpret the difference \(\bar{Y}_{D=1} - \bar{Y}_{D=0}\) as an estimate of the \(ATT\), but your friend says that you should interpret it as an estimate of the \(ATE\). In fact, according to your friend, random treatment assignment implies that \(\mathbb{E}[Y(1)] = \mathbb{E}[Y(1)|D=1] = \mathbb{E}[Y|D=1]\) and \(\mathbb{E}[Y(0)] = \mathbb{E}[Y(0)|D=0] = \mathbb{E}[Y|D=0]\) which implies that \(ATE = \mathbb{E}[Y|D=1] - \mathbb{E}[Y|D=0]\). Who is right?