The core chapters explain inference through sampling distributions. Bootstrap methods preserve that question while changing how the distribution is approximated. This extension connects the course to three papers without making them prerequisites.
1. A local model of the sampling process
Let F be the unknown population distribution and \(\hat{F} _{n}(y)=n^{-1}\sum 1{Y_{i}\le y}\) the empirical CDF. The nonparametric bootstrap replaces draws from F by draws from \(\hat{F} _{n}\), conditional on the observed data.
- Draw \(n\) observations with replacement from the original \(n\) observations.
- Compute the estimator \(\hat{\theta } *\) in that bootstrap sample.
- Repeat B times to approximate its conditional distribution.
- Use an appropriate centered or studentized statistic to approximate sampling error.
The star indicates probability or expectation conditional on the observed sample. The empirical distribution has variance \(n^{-1}\sum (Y_{i}-\bar{Y} )^{2}\), and averaging \(n\) independent draws from it divides that variance by \(n\). Thus the bootstrap reproduces the familiar mean SE up to the same finite-sample divisor issue discussed earlier.
For a studentized bootstrap, compute \(T*=(\hat{\theta } *-\hat{\theta } )/\operatorname{SE}*\). If its bootstrap quantiles are \(q*_{\alpha /2}\) and \(q*_{1-\alpha /2}\), inversion gives \([\hat{\theta } -q*_{1-\alpha /2}\operatorname{SE}, \hat{\theta } -q*_{\alpha /2}\operatorname{SE}]\). This differs from simply taking percentiles of \(\hat{\theta } *\) or using 1.96 times a bootstrap SE.
2. Why resampling is not a universal cure
Consider i.i.d. Uniform\([0,\theta ]\) data and the sample maximum \(M_{n}\). In an n-out-of-n empirical bootstrap, the probability of redrawing the original maximum at least once is \(1-(1-1/n)^{n}\to 1-e^{-1}\approx .632\). Therefore \(n(M*_{n}-M_{n})\) has a mass at zero of about .632. The target \(n(M_{n}-\theta )\) has a continuous nonpositive limiting distribution with no mass at zero. The bootstrap fails to reproduce the required distribution.
This is the sample-maximum example in Princeton'\(s\) bootstrap lecture, p. 12. Boundary problems, some nonsmooth estimators, and dependent sampling can require different resampling procedures. More bootstrap repetitions reduce simulation error, not the underlying approximation error.
3. Direct textbook paper connection: List (2003)
John A. List (2003), “Does Market Experience Eliminate Market Anomalies?” Quarterly Journal of Economics 118(1), 41–71. SW Empirical Exercise E3.2 explicitly cites this paper and describes a sports-collectible trading experiment. Publication details were checked against the publisher record.
Connection. Binary trading decisions become Bernoulli outcomes; a trading fraction becomes a sample mean; testing a benchmark becomes proportion inference. The exercise then compares experienced and inexperienced traders. This note uses the experiment as described in SW, and does not claim to reproduce the paper'\(s\) full analysis.
Derivation. Suppose a share q prefers item A, everyone has stable preferences without an endowment effect, and assignment of A or B is equally likely and independent of preferences. An A-preferring subject trades if assigned B; a B-preferring subject trades if assigned A. The trading probability is \(q/2+(1-q)/2=.5\).
4. Method connection: Efron (1979)
Bradley Efron (1979), “Bootstrap Methods: Another Look at the Jackknife.” Annals of Statistics 7(1), 1–26. DOI / primary publication.
Connection. The empirical-bootstrap idea substitutes an estimated population distribution into the sampling experiment. The construction above is supported by Princeton'\(s\) lecture §1.2, pp. 3–5, which attributes it to Efron. The original article was not fully read for these notes. Its role here is a methodological origin and a reading lead, not evidence that all bootstrap procedures work.
5. Research frontier: Cattaneo–Jansson–Nagasawa (2020)
“Bootstrap-Based Inference for Cube Root Asymptotics.” Econometrica 88(5), 2203–2219. Primary publication DOI. The local paper'\(s\) abstract and introduction, pp. 2203–2204, were read.
Question. How can inference work for estimators with cube-root rates and non-Gaussian Chernoff-type limits, where the usual empirical bootstrap is inconsistent? The paper proposes modifying the shape of the criterion function used for bootstrap estimation to recover a valid approximation under its conditions.
Connection to the core. Consistency alone does not determine a convergence rate; a rate does not determine a normal limit; normal critical values cannot be used without the appropriate limiting distribution. Princeton'\(s\) lecture p. 7 explicitly points to this paper as a bootstrap modification. Full theorem assumptions and proofs are beyond this chapter-level note.
6. Optional reading path
Start with the empirical-CDF construction and the mean-variance calculation above. Then read Princeton pp. 3–5 for roots and intervals, p. 12 for failure examples, and only then the local Cattaneo–Jansson–Nagasawa introduction. Graham'\(s\) Econ 240a syllabus p. 5 places conditional expectation and Bayesian bootstrap in a broader graduate sequence; it is a roadmap rather than proof of any result here.
Connection exercise
Question. If a bootstrap standard error is stable when B increases from 1,000 to 10,000, does that establish correct confidence-interval coverage?
Solution. No. It indicates reduced Monte Carlo error in the chosen bootstrap procedure. The conditional bootstrap law may still approximate the wrong sampling distribution.
Sources: SW E3.2, pp. 151–152; Princeton bootstrap lecture pp. 3–5, 7, 12–13; local Cattaneo–Jansson–Nagasawa (2020), pp. 2203–2204; Graham syllabus p. 5. External links identify primary publications.