High-dimensional inference: debiasedATT() when p >= NT

Gregory Faletto

2026-07-24

library(fetwfe)

This vignette covers high-dimensional inference in fetwfe: what changes when the design has as many or more columns than observations ($p \geq NT$), and how debiasedATT() restores a uniformly valid confidence interval for the overall ATT.

It is a companion to vignette("inference_vignette") (how the package’s standard errors are computed in the low-dimensional regime) and vignette("simultaneous_cis_vignette") (family-wise simultaneous bands).

1. High-dimensional inference ($p \geq NT$): debiasedATT()

The standard errors and simultaneous bands documented in the companion vignettes assume a low-dimensional design ($p < NT$), where the analytic Gram inverse exists and the fit’s standard errors are the asymptotically valid ones of vignette("inference_vignette"). When the design is high-dimensional — the number of design columns $p$ (cohort and time fixed effects, covariates, and all their treatment interactions) meets or exceeds the number of observations $NT$ — two things change:

1. The fused fit is a regularized (bridge) estimate, and its pointwise standard errors (and the simultaneous bands of vignette("simultaneous_cis_vignette")) are post-selection: valid conditional on the selected support, not uniformly valid. They can under-cover, because the bridge’s regularization biases the point estimate (it shrinks and prunes effects) and the post-selection standard error does not account for the selection.

2. debiasedATT() returns a uniformly valid confidence interval for the overall ATT by desparsifying the fit (the nodewise relaxed-inverse construction of the high-dimensional FETWFE theory in Faletto 2025), which removes the regularization bias. This is “one estimator, two regimes”: the call and the returned object are identical to the $p < NT$ case; only the internal nuisance changes.

The $p \geq NT$ regime arises with many covariates, short panels, or many cohorts. simulateData() can generate such panels.

A worked example

# d = 12 covariates, N = 40 units, T = 5 periods, G = 3 cohorts.
coefs_hd <- genCoefs(G = 3, T = 5, d = 12, density = 0.2, eff_size = 2, seed = 2)
sim_hd <- simulateData(
  coefs_hd, N = 40, sig_eps_sq = 1, sig_eps_c_sq = 0.5, seed = 2
)
c(p = sim_hd$p, NT = sim_hd$N * sim_hd$T) # p >= NT: high-dimensional
#>   p  NT 
#> 220 200

fit_hd <- fetwfeWithSimulatedData(sim_hd, q = 0.5)
# The fused (post-selection, pointwise) overall ATT:
round(c(att = fit_hd$att_hat, se = fit_hd$att_se), 3)
#>    att     se 
#> -2.416  0.186

That fused estimate is post-selection. For uniformly valid inference, pass the fit to debiasedATT(), which implements the high-dimensional debiased construction of Theorem debiased.highdim.thm (Faletto 2025). Its coverage is robust across the penalty values studied — a cross-validated selector below the theory scale and a fixed constant above it bracket the default 1.0 — so the default is fine; below we let lambda_c = "cv" select the constant per fit as an alternative.

db <- debiasedATT(fit_hd, lambda_c = "cv")
db
#> Debiased overall ATT (high-dimensional regime)
#>   Estimate:   -2.3684
#>   Std. Error: 0.2507
#>   95% CI: [-2.8597, -1.8771]
#> High-dimensional (p >= NT) desparsified interval
#>   [overall-ATT coverage validated near-nominally in simulation
#>   (Theorem debiased.highdim.thm, Faletto 2025); diagnostics below]
#>   nodewise penalty: lambda_c = 0.5 (CV-selected); lambda_node in [0.1836, 0.1836]
#>   nodewise directions: 1/1 converged, 1/1 KKT-feasible (worst feasibility/lambda_node = 1)

The print surfaces the high-dimensional diagnostics worth inspecting: the resolved nodewise penalty (lambda_c / lambda_node) and, per debiasing direction, whether it converged and met its KKT feasibility certificate. A direction that fails to converge or stay feasible flags a potentially unreliable standard error — raise riesz_max_iter and/or lambda_c and re-inspect.

Why debiased, and how well it covers

In any single draw the fused and debiased point estimates can be close, so the value of debiasedATT() is not visible from one fit — it is a repeated-sampling guarantee. A coverage study at a $p > NT$ anchor (reported in Faletto 2025) found:

Overall-ATT interval	Coverage (nominal 95%)
Debiased (debiasedATT() with lambda_c = "cv")	≈ 94%
Fused plug-in (pointwise)	≈ 77%

The fused interval under-covers because of the post-selection non-uniformity; the debiased interval is near-nominal — the direct simulation confirmation of Theorem debiased.highdim.thm, robust across the penalty values studied (a CV selector below the theory scale and a fixed constant above it, bracketing the default 1.0). The per-direction converged / feasibility diagnostics above remain worth a glance to confirm the nodewise directions are well-behaved on your own fit.

An asymptotic alternative: the wild bootstrap

method = "bootstrap" offers an alternative, asymptotically-valid reference distribution for the overall-ATT interval: a studentized score / influence-function wild cluster bootstrap that re-signs the per-unit influence summands (no refit per replicate), leaving the point estimate and the reported se unchanged and changing only the critical value. It is floored at the Gaussian quantile, so the interval is never narrower than the analytic Wald interval.

It is not a few-clusters remedy. The analytic sandwich SE is downward-biased with few clusters, but this bootstrap does not fix that: as an unrestricted, no-refit score bootstrap it does not reproduce the tail inflation the restricted wild-cluster bootstrap-t uses for few-cluster coverage, and under heterogeneous cluster influence its critical value (before the floor) actually falls below the Gaussian — so the floor reduces it to the analytic interval in exactly the small-N regime. Its only effect is to widen the interval when cluster influence is near-homogeneous. For genuine few-clusters inference a restricted bootstrap-t or a CR2-type analytic correction is required (future work).

# Works on both single-sample and two-sample (`indep_counts`) fits -- e.g. the
# fetwfeWithSimulatedData() fit from the worked example above (a two-sample fit).
debiasedATT(fit, method = "bootstrap", multiplier = "webb", seed = 1)

The multiplier ("webb", the default; "rademacher"; or "mammen") only affects the near-homogeneous case where the bootstrap widens above the floor — in the small-N / heterogeneous regime the interval is the analytic one regardless of the weight. The bootstrap supports both single-sample fits (real observational panels are single-sample) and indep_counts two-sample fits (the cohort-weight channel is perturbed on its own independent multiplier stream over the reconstructed count-sample units).

Simultaneous bands at $p \geq NT$

simultaneousCIs() also switches to the desparsified construction in the high-dimensional regime (via method = "bootstrap"; the analytic Gram-inverse band does not exist when $p \geq NT$). Pass method = "bootstrap" for the valid band: the analytic default — and eventStudy(ci_type = "simultaneous") — instead return the post-selection band on the selected support, which under-covers. On a gls = FALSE fit only the bootstrap band is available.

sc_hd <- simultaneousCIs(
  fit_hd,
  family = "cohort",
  method = "bootstrap",
  B = 500,
  seed = 1
)
sc_hd$regime
#> [1] "high-dimensional"

Both regimes are now simulation-validated: Theorem debiased.highdim.thm covers the scalar overall-ATT (debiasedATT(), ≈0.94), and Theorem debiased.highdim.joint.thm covers the family-wise band (simultaneousCIs(method = "bootstrap")) — family-wise coverage ≈0.92 over the event-study family ($K = 7$), tighter than Bonferroni — both at the $p > NT$ anchor of Faletto (2025). Like the scalar, the band is validated at a single high-dimensional anchor and rests on the sparsity / restricted-eigenvalue assumptions (assumed, not proved); the cohort and all-post-treatment families are covered by the same theorem but were not separately simulated.

Controlled high-dimensional truth

The default density placement spreads signal across all coordinates, which when $p$ greatly exceeds the number of treatment-effect parameters rarely lands signal on the treatment-effect block. For a controlled, sparse, non-degenerate high-dimensional data-generating process, genCoefs() offers a targeted-sparsity mode (n_signal_cohorts or treat_base_levels) that places a heterogeneous treatment-effect signal directly on the per-cohort base levels — the DGP the #88 coverage study uses. See ?genCoefs.

References

Faletto, G. (2025). “Fused Extended Two-Way Fixed Effects for Difference-in-Differences with Staggered Adoptions.” arXiv preprint arXiv:2312.05985.