Wild Cluster Bootstrap Inference

The wild cluster bootstrap tests a regression coefficient with few clusters by multiplying null-restricted residuals by random cluster-level weights and recomputing the cluster-robust t statistic many times.

군집이 적을 때 영가설을 부과한 잔차에 군집 단위 무작위 가중을 곱해 군집 강건 t 통계량을 반복 계산함으로써 회귀계수를 검정하는 부트스트랩입니다.

My data come from only 8 schools, and cluster-robust standard errors seem too small. How can I get a trustworthy p-value for the treatment coefficient?

학교 8곳에서만 모은 자료라 군집 강건 표준오차가 너무 작아 보여요. 처치 계수에 믿을 만한 p 값을 얻으려면 어떻게 하나요?

Use a wild cluster bootstrap-t: impose the null hypothesis, multiply each cluster's residuals by a common random weight (Rademacher, or Webb six-point weights when there are fewer than about 12 clusters), refit and recompute the cluster-robust t. The p-value is the share of bootstrap |t| values at least as large as the observed one, and inverting the test gives a confidence interval.

When to use Wild Cluster Bootstrap Inference

• There are few clusters, roughly fewer than 30–50, so cluster-robust standard errors over-reject.
• Clusters differ in size.
• You need a p-value and confidence interval for one coefficient in a linear model.

When to use something else

• There are many balanced clusters; ordinary cluster-robust standard errors are adequate.
• The outcome model is nonlinear (logit, count); use a score bootstrap or a model-specific method instead.

Assumptions and requirements

• Errors are independent across clusters but may be arbitrarily correlated within a cluster.
• The linear model is correctly specified.
• There are at least 4 clusters; with Rademacher weights and very few clusters, p-values move in steps of 1/2^G.

What the analysis reports

• Coefficient, CRV1 cluster-robust standard error and t
• Bootstrap p-value
• Confidence interval by test inversion (restricted) or percentile-t (unrestricted)
• Number of clusters, weight type and whether all sign patterns were enumerated
• Conventional cluster-robust p-value for comparison

How to report it

With [value] clusters, a wild cluster bootstrap-t test (restricted, [weights] weights, [value] replications) gave b = [value] for [predictor], t = [value], bootstrap p = [value], 95% CI [[value], [value]].

Related methods

• Linear Regression
• Generalized Estimating Equations
• Bootstrap Confidence Intervals (BCa)

Run Wild Cluster Bootstrap Inference on Quorant