Due: At the start of class on Thursday, March 23. Please turn in a hard copy.
For this problem, we will be thinking about properties of the linear probability model: \(\textrm{P}(Y=1|X) = \textrm{E}[Y|X] = X'\beta\).
In the linear probability model, calculate the conditional variance of the CEF error \(\textrm{E}[e^2|X]\).
Now suppose that you are interested in estimating \(\beta\). We know that the usual OLS estimator of \(\beta\) is \(\hat{\beta} = \left(\frac{1}{n} \sum_{i=1}^n X_i X_i'\right)^{-1} \frac{1}{n} \sum_{i=1}^n X_i Y_i\). Instead, suppose that you estimate \(\beta\) using maximum likelihood; you can call this estimator \(\tilde{\beta}\). Derive an expression for \(\tilde{\beta}\).
Is \(\tilde{\beta}\) from part (b) different from \(\hat{\beta}\)? How is this related to the answer from part (a)?
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 average 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 \[\begin{align*} \sqrt{n}(\widehat{APE} - APE) &= \sqrt{n}\left( \frac{1}{n} \sum_{i=1}^n \phi(X_i'\hat{\beta}) \hat{\beta} - \textrm{E}[\phi(X'\beta) \beta] \right) \\ &= \sqrt{n}\left( \frac{1}{n} \sum_{i=1}^n \phi(X_i'\hat{\beta}) \hat{\beta} - \frac{1}{n} \sum_{i=1}^n \phi(X_i'\hat{\beta}) \beta \right) \\ & + \sqrt{n} \left(\frac{1}{n} \sum_{i=1}^n \phi(X_i'\hat{\beta}) \beta - \frac{1}{n} \sum_{i=1}^n \phi(X_i'\beta) \beta\right) \\ & + \sqrt{n} \left( \frac{1}{n} \sum_{i=1}^n \phi(X_i'\beta) \beta - \textrm{E}[\phi(X'\beta) \beta] \right) \end{align*}\]
which holds just by adding and subtracting some terms, and, just to be clear, the notation above is for the entire vector of average partial effects for all regressors. 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)