The empirical exercises complete the knowledge chain by specifying how to move from a dataset or joint-distribution table to an economic conclusion. The original exercise datasets were not found in the inspected course files, so no empirical results are invented.
1. Joint-distribution workflow: age and earnings
SW E2.1 starts from a joint probability table for Age and AHE. First validate that cell probabilities are nonnegative and sum to one. Then:
- Sum each age row across earnings to obtain \(p_{A}(a)\).
- Divide the row'\(s\) joint cells by \(p_{A}(a)\) to obtain the conditional earnings PMF.
- Compute \(m(a)=\sum _{y}\)yP\((AHE=y|Age=a)\).
- Plot a against m(a), labeling units and population restrictions.
- Compute \(\mathbb{E}[AHE]=\sum _{a}m(a)p_{A}(a)\), and check against the marginal-earnings calculation.
- Compute \(\mathbb{E}[AHE^{2}]\), variance\(, \mathbb{E}[Age\cdot AHE]\), covariance, and correlation.
- Compare the correlation with the conditional-mean plot; curvature can make them tell different stories.
Self-contained substitute example. Let Age be 25 or 30 and earnings be 10 or 20. Joint probabilities are (.3,.2) for age 25 and (.1,.4) for age 30. Each age has mass .5; conditional means are 14 and \(18. \mathbb{E}[AHE]=16, \operatorname{Var}(AHE)=24, \mathbb{E}[Age]=27.5, \operatorname{Var}(Age)=6.25, \mathbb{E}[Age\cdot AHE]=445\), and covariance=5. Correlation\(=5/\sqrt{150}\approx .40825\). These are pedagogical numbers, not CPS data.
2. CPS earnings workflow
SW Appendix 3.1 describes the Current Population Survey underlying the book'\(s\) historical examples. Its actual sampling design is more complex than a literal i.i.d. draw. For real microdata analysis, definitions, weights, and design-based uncertainty must be checked. The textbook exercises use a simplified instructional approach.
In the supplied appendix, the earnings examples use the March survey'\(s\) information on earnings in the preceding year. Full-time workers are defined there as working more than 35 hours per week for at least 48 weeks in that preceding year. Average hourly earnings combine annual pretax wages, salaries, tips, and bonuses and divide by annual hours. The appendix describes geographic sampling followed by housing-unit sampling; these are the book'\(s\) historical data definitions, not a statement of current survey specifications.
- Read the data dictionary; identify year, education, earnings, hours, and the analysis sample.
- Document exclusions, missing values, full-time criteria, and whether weights are required.
- Compute mean, SD, and \(n\) by year in nominal units.
- Convert earnings to a common price level before comparing purchasing power.
- Compute intervals for each mean and for differences using the appropriate design.
- Compare education groups within each year, then compare the education gap across years.
- Report estimands, units, uncertainty, and limits on causal interpretation.
For independent year-by-group samples, a change in an education gap is \((\bar{Y} _{college,2}-\bar{Y} _{HS,2})-(\bar{Y} _{college,1}-\bar{Y} _{HS,1})\), with the four mean variances added. If observations overlap or survey clusters link the estimates, include the corresponding covariances instead.
A reporting sentence should identify the difference, the confidence interval, the price year, and the population. “The gap changed significantly” is incomplete without those definitions.
3. Sports-card experiment workflow
SW E3.2 uses the List (2003) experiment described in Module 13. Given the data, define \(Trade=1\) for trading and zero otherwise. Verify coding, assignment probabilities, and missing outcomes. Estimate the trading proportion and compare it with the justified no-endowment-effect benchmark.
Next estimate separate trading proportions for dealers and nondealers, and directly test their difference. Do not infer a significant group difference just because one group'\(s\) benchmark test rejects and the other'\(s\) does not. Distinguish randomized initial endowment from nonrandom market experience.
Data status. The required external dataset is a separate input. The learning text supplies the estimator, standard-error logic, benchmark derivation, and interpretation workflow; it does not report uncomputed trading fractions.
4. A Monte Carlo workflow that needs no external dataset
SW Exercise 2.16 approximates an event probability by simulation. If \(Y^{(b)}\) are independent draws from a specified distribution, then:
The indicator is Bernoulli, so Monte Carlo \(\operatorname{SE}\approx \sqrt{\hat{p} _{B}(1-\hat{p} _{B})/B}\). Here B controls simulation precision, while \(n\) controls the sampling precision of an estimator within each simulated dataset. These are separate sample sizes.
To study interval coverage, fix a known \(\mu\), draw \(B\) datasets of size \(n\), construct one interval per dataset, and average \(\mathbf 1\{\mu\text{ is covered}\}\). Compare coverage, average interval width, and rejection rates across \(n\) and population shapes. The sampling laboratory in Module 6 implements the repeated-sampling idea locally.
Sources: SW E2.1, p. 109; E3.1, pp. 149–150; E3.2, pp. 151–152; Appendix 3.1, p. 152; Exercise 2.16; Schellenberg Section 2, pp. 7–9. Synthetic table and reporting structure are original teaching additions.