\[ \newcommand{\E}{\mathbb{E}} \]

Due: At the start of class on Tuesday, April 19. Please turn in a hard copy.

  1. Hansen 25.15, for this question, you need to code the probit estimator yourself (compute both estimates of the parameters and their standard errors); i.e., you cannot use built-in R functions such as glm. Hint: you can use R’s optimization routines such as optim. In addition to what is asked in 25.15,

    • compare the estimates that you get to those coming from the glm function

    • report average partial effects (which the book refers to as average marginal effects) for each regressor

    • compute the standard error of each partial effect (again, use your own code to compute these). For computing standard errors of the partial effects, you have two options:

      Option 1: Analytical Standard Errors, hint: Obtaining the standard errors for this part is challenging. One thing to notice is that \[\begin{align*} \sqrt{n}(\widehat{APE} - APE) &= \sqrt{n}\left( \frac{1}{n} \sum_{i=1}^n \phi(X_i'\hat{\beta}) \hat{\beta} - \E[\phi(X'\beta) \beta] \right) \\ &= \sqrt{n}\left( \frac{1}{n} \sum_{i=1}^n \phi(X_i'\hat{\beta}) \hat{\beta} - \frac{1}{n} \sum_{i=1}^n \phi(X_i'\hat{\beta}) \beta \right) \\ & + \sqrt{n} \left(\frac{1}{n} \sum_{i=1}^n \phi(X_i'\hat{\beta}) \beta - \frac{1}{n} \sum_{i=1}^n \phi(X_i'\beta) \beta\right) \\ & + \sqrt{n} \left( \frac{1}{n} \sum_{i=1}^n \phi(X_i'\beta) \beta - \E[\phi(X'\beta) \beta] \right) \end{align*}\]

      which holds just by adding and subtracting some terms. Figuring out the asymptotic distributions for the first and last lines is not too hard, but the middle expression requires using some kind of mean value theorem type of argument (as we have done before in the context of the delta method).

      Option 2: This is also a place where it might make sense to use the bootstrap. You can use the bootstrap to compute standard errors for the average partial effect.

      You can compute standard errors for the average partial effects using either Option 1 or Option 2 for full credit (or both if you are feeling ambitious)

  2. For this problem, we will be interested in computing an estimate of the \(ATT\) of a job training program. You can download the data here and download a description here. For this problem, the outcome of interest is re78, the treatment is train, and suppose that unconfoundedness holds after conditioning on age, educ, black, hisp, married, re75, and unem75.

    • Given our discussion in class, we know that if we believe unconfoundedness and that untreated potential outcomes are linear in covariates, then we have that

      \[\begin{align*} ATT = \E[Y|D=1] - \E[X'|D=1]\beta \end{align*}\]

      where \(\beta\) can be estimated from the regression of \(Y\) on \(X\) using untreated observations. Estimate \(ATT\) based on the above expression for it and report your result.

    • Show that \(\sqrt{n}(\widehat{ATT} - ATT) \xrightarrow{d} N(0,V)\) and provide an expression for \(V\). Based on this result, provide standard errors for your estimate of \(ATT\).

    • Use the bootstrap to compute standard errors for your estimate of \(ATT\). How do these compare to the standard errors that you reported previously?

    • Run a regression of \(Y\) on \(D\) and \(X\) (where \(X\) includes the same additional variables as above). Compare the coefficient on \(D\) to the estimate of \(ATT\). How similar are they? What about their standard errors? Do you have any comment on these results?