Problems - Ch5

1. 1. Derive the maximum likelihood estimator using the standard normal density function. Compare that to the least squares estimator discussed in Chapter 1.

   2. Write up a maximum likelihood estimator and use it to estimate the data created in 1 of Chapter 1’s problem. Compare results to your answer in 2 of Chapter 1’s problems. How close is your estimate of the variance of the observed term?

   3. Create the following simulated data.

      1. x1 ~ U[0,1], x2 ~ U[0,1], e ~ N(0,1)

      2. y = 1 if a + b*x1 + c*x2 + e > 0, a=-2,b=2, c=5

0 otherwise

1. 3. 3. N = 1000 (1000 data points)

      4. Run ordinary least squares on the relationship between y and x’s (y as a function of x). Present estimates for a, b and c. Discuss why they are or are not close to the true values.

      5. Repeat (d) but use a logit.

      6. Repeat (d) but use a probit.

   4. Use the “Fishing” data in R (library Ecdat)

      1. estimate demand for charter boat fishing relative to pier fishing. In your analysis account for prices, catch rates and income.

      2. Explain your results from (a).

      3. What concerns should we have with this analysis?

   5. Download the data for Mortgage Lending in Boston. Available here: HMDA data (Google Sheets)

      1. Create variables for mortgage denials, race, income, credit worthiness, housing characteristics, demographics.

      2. Using ordinary least squares, logits and probits can you show that there is evidence that banks are illegally denying mortgages based on race? Or that there is evidence that banks are behaving legally? Explain your method. Explain the concerns with your analysis.

   6. Duke labor economist, Peter Arcidiacono, has been hired by a group of people suing Harvard University for racial discrimination in admissions. Arcidiacono’s report is here: http://samv91khoyt2i553a2t1s05i-wpengine.netdna-ssl.com/wp-content/uploads/2018/06/Doc-415-1-Arcidiacono-Expert-Report.pdf You have been hired by Berkeley labor economist, David Card, to help Harvard University rebut the claims made in Arcidiacono’s report.

      2. 1. Table 2.1 presents a summary of the raw admissions. Explain why this does or does not show discrimination.

         2. Summarize the results of Arcidiacono’s analysis, particularly his logit estimates of admissions presented in Table B.7.1 (p. 135). Section 3.7 discusses his analysis.

         3. Write a brief critique of the logit analysis presented in Table B.7.1. Explain why it may not show evidence of discrimination.

   7. Download the National Household Transportation Use survey data (National Household Travel Survey).

      1. Create multinominal login estimator with bootstrap standard errors. Hint: Estimate the model once, then use those values as starting values for the bootstrap.

      2. Create bivariate probit estimator removing the restriction that the variance is 1, and with bootstrap standard errors. Hint: you should reduce the number of reps for doing the integration.