Inference and Variance Estimation

Inference depends on whether the compressed table preserves the score or residual information needed for the target covariance estimator.

Linear HC1

For DBRegression.fit_vcov(), the covariance has sandwich form:

\[ \hat{V} = \frac{N}{N-k} (X'WX)^{-1} \left(\sum_g RSS_gx_gx_g'\right) (X'WX)^{-1}. \]

The grouped residual sum of squares is exact from compressed sufficient statistics:

\[ RSS_g = n_g\hat{y}_g^2 -2\hat{y}_g\sum_{i\in g}y_i +\sum_{i\in g}y_i^2. \]

This is the fastest inference path when iid or HC1-style robust standard errors are enough.

Cluster Bootstrap

Cluster bootstrap resamples clusters, applies multiplicities to grouped cluster-by-cell aggregates, collapses back to design cells, and refits. The DBRegression path does this without backend-specific array parameter expansion.

Conceptually:

grouped = group_by(covariate_cell, cluster).aggregate(
    count=count(),
    sum_y=sum(Y),
    sum_y_sq=sum(Y * Y),
)

for bootstrap_draw in range(B):
    multiplicities = sampled_clusters.value_counts()
    boot = grouped.join(multiplicities).collapse_to(covariate_cell)
    beta_b = wls(boot_X, boot_y, boot_n)

Panel event-study bootstrap uses the same idea with the generated event-study design.

GLM Covariance

For canonical GLMs, fit_vcov() uses Fisher information:

\[ \hat{V}_{model} = I(\hat{\beta})^{-1}. \]

For binary and Poisson models, fit_vcov(robust=True) computes a grouped sandwich:

\[ \hat{V}_{robust} = I(\hat{\beta})^{-1} \left(\sum_g U_g(\hat{\beta})U_g(\hat{\beta})'\right) I(\hat{\beta})^{-1}. \]

The grouped score contribution \(U_g\) is available because the compressed table stores covariates, outcome sums, and cell counts.

DML Bootstrap

DBDML bootstraps over compressed control groups. The point estimator is built from group-level leave-one-out cross-products, so resampling compressed groups is a natural low-cost uncertainty calculation.

Current Matrix

Estimator	Inference path
`DBRegression`	HC1 via `fit_vcov()`; iid or cluster bootstrap through `fit()` when `n_bootstraps > 0`.
`DBDML`	Bootstrap over compressed groups.
`DBMundlak`	Cluster bootstrap when `cluster_col` is provided.
`DBDoubleDemeaning`	Point estimate currently; bootstrap is not implemented in the `DB*` path.
`DBMundlakEventStudy`	Cluster bootstrap by generated event-study design.
`DBLogisticRegression`	Fisher covariance, or grouped robust sandwich with `fit_vcov(robust=True)`.
`DBPoissonRegression`	Fisher covariance, or grouped robust sandwich with `fit_vcov(robust=True)`.
`DBMultinomialLogisticRegression`	Inverse multinomial Fisher information.
`DBPoissonMultinomialRegression`	Point estimates only.
`DuckRidge`	Point estimates, paths, and cross-validation; no standard errors.

Keep n_bootstraps=0 for GLMs unless a bootstrap implementation is added.