Take-home final

EC 607

Author

Edward Rubin

1 Admin

1.1 Academic honesty

You are not allowed to work with anyone else. Working with anyone else will be considered cheating. You will receive a zero for both parts of the final exam and may fail the class.

You can use online materials, books, notes, solutions, etc. However, you still must put all of your answers in your own words. Copying other people’s (and chatbots’) words is also considered cheating.

Ngan and Ed will not help you debug your code. Please do not ask.

1.2 Instructions

Due Upload your answers to Canvas before 11:59 pm (Pacific) on Thursday, 15 June 2023.

Important You must submit your answers as an HTML or PDF file, built from an RMarkdown (.RMD) or Quarto (.qmd) file (you can also submit a link to an HTML page if you prefer that route).

2 Prompts

2.1 An analysis (100 points)

The data for this section come from the AEA’s data and code repository for a 2021 paper by Kelly Bedard, Maxine Lee, and Heather Royer titled Using Longitudinal Data to Explore the Gender Gap for Academic Economists.

Note While Bedard, Lee and Royer focus on a panel, we are going to focus on the last year that we observe each faculty member in their panel. I already created this cross-sectional dataset: aeapnp_bedard_lee_royer.csv. Do not use their panel dataset.

[1.01] (10 points) Let’s get to know the dataset a bit. Create two histograms: (1) the salaries (salary) of female (sex == 'Female') faculty members, and (2) the salaries of non-female faculty members.

Is the figure consistent with a pay gap between female and non-female economics faculty?

These figures (and all subsequent figures) should be well labeled and aesthetically pleasing.

Hint1 It will be helpful for you to create a binary indicator that equals 1 when the individual is female (sex == 'Female') and equals zero otherwise. We’re following the paper in the definition/approach. (There are five individuals in the data for who sex is not defined as Female or Male; they will fall into the non-female group with this indicator definition.)

Hint2 The facet_grid() function might be helpful for creating multiple figures.

[1.02] (10 points) Run two regressions to more directly test for a salary gap between female and non-female faculty:

  • Regress salary on your indicator for whether the individual is female.
  • Regress log(salary) on your indicator for whether the individual is female.

Which regression makes more sense in this setting? Explain your answer.

[1.03] (5 points) Using the potential outcomes framework, explain the treatment effect that we are attempting to estimate—and why we might or might not need controls.

[1.04] (10 points) Could omitting year from the regressions above in [1.02] lead to biased estimates of the effect of sex on salary?

Support your answer by regressing (a) log(salary) on year and (b) your indicator for female on year.

Based upon these regressions, is the bias likely to be positive or negative?

[1.05] (10 points) For now we’re going to stick with log(salary) as our outcome variable. The regressions will always have the female indicator. Try controlling for year

  • as a numeric variable
  • as a fixed effect

Does controlling for year “matter”? Is it a “good” control? Explain.

[1.06] (10 points) The authors show that years since PhD is an important factor to consider in this setting—as is tenure.

Continuing from the year fixed effects specification in [1.05]

  • control for years since PhD as a numeric variable
  • control for years since PhD as one of three groups (0–9, 10–19, 20+)
  • control for the three groups above and allow for a heterogeneous treatment effect with respect to the three groups

[1.07] (10 points) Keeping the year fixed effects and the three-group fixed effects for years-since-PhD, now (sequentially) add controls for

  • tenure (i_tenure)
  • whether the individual is at business school (business)
  • current institution (school_id)
  • PhD institution (phd_id)
  • economics subfields (micro + macromoney + ... + econhist + educ)

In other words: You should run five regressions for this question, where the last one has a lot of controls.

[1.08] (5 points) Are the controls we included in [1.07] going to help or hurt our issues with selection bias? Explain.

[1.09] (5 points) Up to this point, what type of standard errors have you been using? Do you think we should be using cluster-robust standard errors? Explain your answer.

[1.10] (15 points) Estimate whichever specification that you think is best—and use whichever inference approach you think is best. (It should still involve regressing log(salary) on an indicator for )

Explain the conditional independence assumption for your chosen model, and why you think this CIA is the most plausible compared to other models we estimated.

Also justify your inference approach.

[1.11] (5 points) Estimate the same specification but with salary as the outcome—rather than log(salary). If anything changes, does this affect your conclusions? Explain your answer.

[1.12] (5 points) Do you think a matching estimator would work in this setting? Explain your answer.

2.2 A simulation (50 points)

Important You may choose to skip this section of the test. However, if you skip this section, the highest grade you can earn in the entire course is a “B” (meaning no B+, A-, A, or A+). Completing this section does not guarantee you recieve higher than a “B”, but it does give you a chance.

DGP We are going to simulate a two-staged least squares setting.

You have two instruments

  • \(z_1 \sim \text{Uniform}(-5, 5)\)
  • \(z_2 \sim \text{Uniform}(-5, 5)\)

for your endogenous variable of interest \[x = 1 + z_1 + z_2 - 2 z_2 \times p + v + w\]

Also \(v \sim N(0, 1)\), and \(w \sim N(0, 2)\), and \(p \sim \text{Bernoulli}(0.2)\)

Hint: Use runif and rnorm to draw from the uniform and normal distributions. Similarly, you can use the rbinom() function to draw from the Bernoulli distribution.

Finally, your outcome of interest is \[y = 2 + 3 x + x \times p + 2 \times u + w\] with \(u \sim N(0, 1)\).

[2.01] (5 points) Using the defined DGP: Explain what the ATE represents in this setting and give me the ATE (I want a number). Feel free to justify your answer/show your work.

[2.02] (5 points) Using the defined DGP: Explain why a simple regression of \(y\) on \(x\) will fail to recover the ATE.

[2.03] (25 points) Run a simulation to show the distribution of the estimates for the effect of \(x\) on \(y\) for each of the following five estimation strategies:

  • Plain OLS: Regress \(y\) on \(x\)
  • IV2: Instrument \(x\) with \(z_1\)
  • IV2: Instrument \(x\) with \(z_2\)
  • 2SLS1: Instrument \(x\) with both \(z_1\) and \(z_2\)
  • 2SLS2: Instrument \(x\) with \(z_1\), \(z_2\), and \(z_2\times p\)

Include a plot of the results (densities of the distrubtions, like we did in class).

Requirements

  • Each iteration should have 1,000 observations.
  • You should run at least 100 iterations. A thousand would be better. Ten thousand would be even better.

[2.04] (5 points) Why are some of the IV and 2SLS approaches in [2.03] more biased than other approaches? Explain using the DGP.

[2.05] (5 points) What would be the best strategy for estimating the effect of \(x\) on \(y\)? Explain your answer (and what you’re defining as “best”).