Wild Cluster Bootstrap with Few Large Clusters
Suppose an experiment has only a few clusters, each containing many observations. More observations can improve the approximation to each cluster’s score, but do not create more independent clusters. The result of (Canay, Santos, and Shaikh 2021) uses the approximate sign symmetry of those few scores to construct a wild-bootstrap test.
1. The test and its guarantee
Consider \(Y_{ij}=Z_{ij}'\beta+W_{ij}'\gamma+\varepsilon_{ij}\) in clusters \(j=1,\ldots,q\), and test \(c'\beta=\lambda\). Fit the restricted regression, obtaining fitted values and residuals \(\widehat\varepsilon^r_{ij}\). For each \(g\in\{-1,1\}^q\), form
Refit and compare \(T_n=|\sqrt n(c'\widehat\beta-\lambda)|\) with the empirical \(1-\alpha\) quantile of \(|\sqrt n c'(\widehat\beta^*(g)-\widehat\beta^r)|\) over all signs.
Theorem 1. Under the paper’s fixed-\(q\), large-cluster conditions, the null rejection probability has limiting bounds
These conditions include independent Gaussian limits for cluster scores, positive limiting cluster proportions, proportional residualized-regressor covariance matrices across clusters, and compatibility of pooled and cluster-specific nuisance projections. The paper also establishes power tending to one as the local-alternative parameter becomes large, subject to the sign-grid resolution condition.
2. Where the approximation comes from
Partialling out \(W\) makes the leading estimation error a linear transformation of the cluster sums \(\sum_i\widetilde Z_{ij}\varepsilon_{ij}\). Each large cluster supplies a Gaussian limit. Independence across clusters makes the joint limit invariant under changing any subset of their signs.
Under the null, changing the signs of restricted residuals must reproduce the signed-score experiment after the regression is refitted. The design restrictions establish this connection. Sign symmetry of an abstract score vector alone would not justify the regression procedure.
The procedure then ranks the observed statistic among its sign transformations. There are only \(2^q\) transformations. Because the statistic is absolute, \(g\) and \(-g\) give the same value, leaving at most \(2^{q-1}\) distinct ranks. That is the source of the discreteness term in the size bound.
3. What this does and does not establish
The source of validity is symmetry of a finite limiting score vector. Large cluster sizes help establish that limit only under the within-cluster CLT and design assumptions. A different failure occurs when the number of clusters grows but extreme scores remain influential, as in the heavy-tailed cluster note.
For network-based groups, independence of cluster scores is an additional mathematical question. The conductance and network separation conditions address whether the proposed boundaries separate enough dependence for the sign argument.
4. The score experiment in a scalar example
A model with only an intercept makes the mechanism visible. Let \(Y_{ij}=\beta+\varepsilon_{ij}\), let \(n_j/n\to\xi_j>0\), and suppose
jointly, with independent limits. The pooled estimator satisfies
Under \(H_0:\beta=\beta_0\), restricted residuals are \(Y_{ij}-\beta_0\). Their cluster sign changes give
In this example the bootstrap and the desired signed-score experiment coincide exactly under the null. The Gaussian approximation is needed only to justify independent sign symmetry of the limits. Unequal \(\sigma_j^2\) do not spoil that symmetry: each centered Gaussian coordinate is symmetric separately.
Adding covariates changes the comparison. Estimated design matrices multiply the cluster scores, and restricted residuals include nuisance-estimation errors. The proportional-design and projection conditions make the resulting terms cancel in the tested direction. The intercept example has no such terms and therefore does not establish validity for arbitrary designs.
5. Why independent signs give a randomization test
For independent centered normal \(Z_j\) with positive variances, each sign is a fair coin conditional on the vector of magnitudes. Therefore, conditional on \((|Z_1|,\ldots,|Z_q|)\), every sign vector has probability \(2^{-q}\). For any statistic computed consistently across this orbit, its observed value has a uniform rank after appropriate treatment of ties.
For the absolute weighted sum,
Apart from these paired signs, additional ties have probability zero under nondegenerate continuous limits and nonzero weights. There are then \(M=2^{q-1}\) equally likely distinct ranks. Strict rejection at the empirical \((1-\alpha)\) quantile has probability
which lies between \(\alpha-1/M\) and \(\alpha\). For \(q=5\), \(M=16\), and a five-percent nonrandomized test cannot reject. For \(q=6\), \(M=32\), and its limiting rejection probability is \(1/32\). This is the source and practical meaning of the theorem’s discreteness term.
Randomizing at the boundary can fill the unused portion of a rank in an exact sign experiment. The stated theorem uses a nonrandomized test, whose rejection probability may remain below the nominal level because the sign grid is finite.
6. Why large signals need enough sign patterns
In the intercept example, take \(\beta=\beta_0+h/\sqrt n\). The original statistic has limit
whereas the restricted bootstrap statistic for signs \(g\) has limit
Because every \(\xi_j>0\) and their sum is one, any mixed sign pattern satisfies \(|\sum_jg_j\xi_j|<1\). As \(|h|\) becomes large, its bootstrap statistic grows more slowly than the observed one. Only the all-positive and all-negative patterns keep the full signal magnitude. If the upper rejection region contains at least the largest distinct rank, the test consequently detects sufficiently large local signals with probability approaching one. If the nominal level is below the sign-grid resolution, strict rejection remains impossible regardless of signal size.
Restricted residuals retain the deviation from the null in the bootstrap experiment. The displayed local-alternative calculation uses precisely that deviation. Another residual construction would change the comparison and require a new argument.
6.1. Restricted residuals retain a nuisance-estimation term
The intercept example avoids a complication that matters with covariates. To display it, let \(x_{ij}\) collect all regressors, let \(Q_j=n^{-1}\sum_i x_{ij}x_{ij}'\), and let \(Q=\sum_jQ_j\) be nonsingular. Write \(S_j=n^{-1/2}\sum_i x_{ij}\varepsilon_{ij}\). If \(\Delta^r=\sqrt n(\widehat\theta^r-\theta_0)\), then the signed restricted-residual score is exactly
The second term is order one; consistency of \(\widehat\theta^r\) does not make it disappear on the \(\sqrt n\) scale. Suppose, for this calculation, that \(Q_j=\xi_jQ\), and test a contrast \(c'\theta=c'\theta_0\). The restriction implies \(c'\Delta^r=0\), hence
After taking the tested contrast, the bootstrap therefore reduces to the signed sum \(c'Q^{-1}\sum_jg_jS_j\). With arbitrary \(Q_j\), the cancellation need not occur: the signed design matrices can turn a nuisance-estimation error into movement in the tested direction. In the regression with separately partialled-out controls, the paper’s projection conditions ensure the corresponding cancellation.
The fixed-cluster theorem connects two steps: sign symmetry of the limiting scores and its preservation by the fitted regression. The design conditions justify the second step while allowing the score variances to differ across clusters.
7. What cluster size must accomplish
The assumption that a large cluster score is approximately Gaussian is a statement about dependence within that cluster. If all its observations share the same unscaled shock, \(\sum_i\varepsilon_{ij}\) is of order \(n_j\), not \(\sqrt{n_j}\). Merely increasing \(n_j\) does not verify the displayed CLT. One must justify the normalization, the score limit, and independence across clusters from the sampling model.
These are distinct asymptotic regimes. Here a finite score vector has well-behaved limits. In the heavy-tail analysis, increasingly many clusters are observed but some extreme scores retain a substantial fraction of total variability. A count of observations by itself does not distinguish them.