Homework 3

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

Textbook Questions 7.14, 7.17, 7.28 (parts (b)-(e) only)

Extra Question 1

Repeat part (a) of question 7.28 except compute standard errors using the bootstrap. Provide your code and compare the standard errors you get from the bootstrap to the standard errors you got for part (a) in the previous homework by direct calculation.

Extra Question 2 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,

  1. Compare the estimates that you get to those coming from the glm function

  2. Calculate and report average marginal contrasts for each regressor.

  3. Derive the asymptotic variance for the average marginal contrasts.

    Hint: It is helpful to notice that \[\begin{align*} \sqrt{n}(\widehat{AMC} - AMC) &= \sqrt{n}\left( \frac{1}{n} \sum_{i=1}^n \phi(X_i'\hat{\beta}) \hat{\beta} - \E[\phi(X'\beta) \beta] \right) \\ &= \sqrt{n}\left( \frac{1}{n} \sum_{i=1}^n \phi(X_i'\hat{\beta}) \hat{\beta} - \frac{1}{n} \sum_{i=1}^n \phi(X_i'\hat{\beta}) \beta \right) \\ & + \sqrt{n} \left(\frac{1}{n} \sum_{i=1}^n \phi(X_i'\hat{\beta}) \beta - \frac{1}{n} \sum_{i=1}^n \phi(X_i'\beta) \beta\right) \\ & + \sqrt{n} \left( \frac{1}{n} \sum_{i=1}^n \phi(X_i'\beta) \beta - \E[\phi(X'\beta) \beta] \right) \end{align*}\] which holds just by adding and subtracting some terms, and, just to be clear, the notation above is for the entire vector of average marginal contrasts for all regressors. To derive the asymptotic distribution, you should be looking for a way to apply the CLT And CMT arguments that we have used before to the terms above. Handling 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).

  4. Based on your result in part (c), compute the standard error of each average marginal contrast (again, use your own code to compute these).

  5. Use the bootstrap to calculate standard errors for the average marginal contrasts that you calculated in part (b). Compare these standard errors to the analytical standard errors that you calculated in part (d).