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Dependence-aware bootstrap references

Two optional interfaces supplement the existing spatial LP and GVAR diagnostics. They make different assumptions and do not replace the default asymptotic results. Neither is an exact finite-sample test or a claim that all documented size distortions have been eliminated.

Spatial LP: whole-cross-section time blocks

from puremacro.spatial import SpatialLPBootstrap, spatial_lp

result = spatial_lp(
    panel, "output", "shock", weights, horizons=range(6), n_lags=2,
    lagged_outcome="difference", alpha=0.05,
    bootstrap=SpatialLPBootstrap(
        n_boot=399, block_length=6, assume_stationary=True, seed=814,
    ),
)
inference = result.bootstrap_result
print(inference.summary())
print(inference.table)

result still contains the original horizon-specific fits. The bootstrap table contains a separate fit on the intersection of all horizon samples. Its point and se can differ from the original table. metadata['common_periods'] identifies the observations actually used. Every horizon must have a balanced, complete panel with the same entities, and the common sample must contain at least two block lengths.

The algorithm constructs the outcomes and controls before resampling, samples circular time blocks of complete cross-sections, and uses the same sampled periods at every horizon. Thus it preserves contemporaneous spatial dependence and the within-block cross-horizon dependence, without constructing fictitious leads or lags across block seams. Every draw re-estimates both fixed effects, coefficients, and the selected Driscoll–Kraay or Conley studentizing covariance. Entity-cluster studentization is rejected only for this optional bootstrap: with a fixed number of regions its scale has no supported fixed-N limit here. Legacy cluster fits remain available without the bootstrap.

table reports symmetric bootstrap-t pointwise intervals, simultaneous max-|t| intervals, centered two-sided p-values, and their conditional Monte Carlo standard errors. The simultaneous family contains all displayed nonzero contrasts at all requested horizons, including direct, indirect, and total responses. joint_p_value uses the maximum absolute t over the per-order indirect contrasts and horizons; metadata['joint_labels'] specifies that family. An intentionally zero-scaled indirect contrast is excluded. Without any nonzero indirect contrasts, the joint statistic and p-value are None. No chi-square Wald reference is used.

The stationarity assertion concerns the entire regression-row process, including controls. The dependent variable is still y[t+h] - y[t-1]. The new optional lagged_outcome='difference' uses lagged changes of y and, when requested, lagged changes of Wy; the default remains 'levels'. This permits an integrated outcome with stationary changes and controls. It does not make arbitrary integrated controls, structural breaks, endogenous shocks, or misspecified dynamics valid. The underlying argument is fixed N, increasing T, stationary weak dependence and an identified exogenous-shock regression. The total contrast remains relative to the period mean, as explained in the spatial guide.

Choose the block length before inspecting significance, report sensitivity to more than one defensible length, and remember that a long block leaves fewer effective blocks in a short panel. All requested draws are retained. A singular regression or invalid studentizing variance in any draw raises an error that names the draw; there is no selective retry or deletion. Empirical-quantile intervals and plus-one p-values may disagree exactly at a discrete Monte Carlo boundary by roughly one draw; neither is claimed to be an exact finite-sample inversion of the other. The stationary block-bootstrap asymptotic justification requires block length to grow with T while its ratio to T tends to zero; a fixed short block does not universally accommodate arbitrary serial dependence. The same condition applies to dependent-innovation blocks in the GVAR reference below; length one is its iid-innovation special case.

The circular moving-block construction follows the stationary resampling framework of Künsch (1989) and Politis and Romano (1992). This implementation is a panel regression-row application of that framework, not an implementation or proof of uniform unit-root LP inference. The latter requires the different assumptions and procedures studied by Montiel Olea and Plagborg-Møller (2021).

GVAR: an explicitly stronger independent-country null

# fitted_gvar was estimated with weak_exogeneity=True.
reference = fitted_gvar.independent_null_bootstrap(
    null="independent_countries", integration="i1",
    n_boot=399, block_length=1, null_lags=1, seed=815,
)
print(reference.summary())
print(reference.table[["country", "star_variable", "F",
                       "p_independent_null", "mc_se"]])

This reference assumes independent country innovation processes and no cross-country lag dynamics. It is stronger than general weak exogeneity. The reported p-values concern that specified reference distribution for the existing residual-augmented auxiliary statistic; they do not become the error-correction test of Dees et al. Country independence is not established by failing to reject, and the statistic can have low power against some departures from independence.

For each country, a VAR of order null_lags is estimated in levels under integration='stationary', or in first differences under integration='i1'. The latter simulates stationary increments and integrates them from the observed initial level. Initial VAR lags are conditioned on. No unit-root pretest silently chooses a mode. Mixed integration orders and cointegrated within-country blocks are outside the I(1) reference. A nonstationary fitted null VAR raises; roots are never clipped. Exogenous regressors and deterministic trends beyond 'n' or 'c' are not currently supported by the null simulator.

Centered innovation vectors are resampled in circular blocks independently across countries, retaining contemporaneous dependence among variables within each country. Block length one is the iid-innovation special case. Each synthetic global panel then re-estimates every original country VARX* and every auxiliary regression. The generated first-stage residuals are never frozen across draws. Original country and auxiliary lag orders are held fixed; prior information-criterion selection is not repeated. This does not simulate an unrestricted fitted GVAR under an alternative.

The original weak_exogeneity.table is unchanged and remains diagnostic-only. The new table uses the explicit names p_independent_null, p_holm_independent_null, and reject_independent_null. Holm adjustment covers all table rows. The p-value is (1 + number of bootstrap F >= observed F)/(B+1); mc_se describes simulation noise conditional on the fitted null, not model uncertainty. The attainable resolution is 1/(B+1): for example, 99 draws cannot produce a p-value below 0.01 and therefore cannot reject a six-test Holm family at 5% at its first step. All draws and fixed reference settings are retained; any failed draw makes the inference fail explicitly. Restricted and unrestricted auxiliary designs must both have full column rank in the observed data and every draw. For example, extra lags in a levels auxiliary regression can span the generated residual exactly, making its tested coefficient non-estimable; the bootstrap rejects that specification even though the legacy diagnostic-only table remains available.

For the distinction between VARX* and cointegrating VECX* weak-exogeneity tests, see Chudik and Pesaran, Theory and Practice of GVAR Modelling and the GVAR guide. This restricted simulator does not add a VECX* layer.

Reproducible size, coverage, and power evidence

tools/reference_validation/validate_dependence_bootstrap.py runs bounded, seeded experiments and writes JSON after every design. Its spatial designs use N=20/40, T=50/100, six horizons, spatially correlated AR(0.4) errors, and both zero and nonzero spillovers. They use differenced outcome controls: they are not a relabeling of the older level-control experiment. GVAR designs include independent I(1) countries at T=200/300, stationary AR(0.5), block-length sensitivity, and two innovation-leakage alternatives, including the existing persistent-foreign-country power design.

PYTHONPATH=. OPENBLAS_NUM_THREADS=1 python tools/reference_validation/validate_dependence_bootstrap.py \
  --domain spatial --replications 200 --draws 199 --output /tmp/spatial-mc.json
PYTHONPATH=. OPENBLAS_NUM_THREADS=1 python tools/reference_validation/validate_dependence_bootstrap.py \
  --domain gvar --replications 200 --draws 399 --output /tmp/gvar-mc.json

The output reports rejection, coverage and power with replication-level Monte Carlo uncertainty, plus failed replications. Correlated GVAR tests within a replication are averaged before computing its Monte Carlo standard error. Binary rates use Wilson intervals, so zero observed failures do not imply zero uncertainty. Inference failures are recorded separately; successful-replication rates must be read alongside their denominator and failures. These are design-specific experiments, not an automatic calibration gate or a universal nominal-size claim.

Observed limitations in the preset matrix

The October 2026 run used 200 independent DGP replications per design, seed 812000, 199 spatial bootstrap draws and 399 GVAR draws. All 2,200 DGP replications completed without inference failures. Settings were fixed before the results; none were tuned to obtain nominal rejection rates. Percentages below include 95% Monte Carlo intervals in brackets, using the uncertainty conventions above.

Spatial N / T / block / true spillover Pointwise coverage range Simultaneous coverage (target 95%) Joint rejection at 5%
20 / 50 / 4 / 0 94.5–98.0 93.5 [89.2, 96.2] 3.0 [1.4, 6.4]
20 / 50 / 8 / 0 92.0–95.5 90.5 [85.6, 93.8] 2.5 [1.1, 5.7]
40 / 50 / 4 / 0 90.0–94.5 88.5 [83.3, 92.2] 9.0 [5.8, 13.8]
20 / 100 / 6 / 0 92.5–97.5 91.0 [86.2, 94.2] 5.5 [3.1, 9.6]
20 / 50 / 4 / 0.25 92.5–97.5 92.0 [87.4, 95.0] 100.0 [98.1, 100.0] (power)

Spatial pointwise coverage here concerns the indirect response over six horizons; the simultaneous family also contains direct and total responses. The bootstrap does not solve the short-panel coverage problem: N=40 has 88.5% simultaneous coverage and 9% null rejection, and even T=100 under-covers. A longer block did not uniformly improve results. Treat this as an experimental, design-dependent reference and report the sensitivity, not as a calibrated replacement for all default bands. These runs use Driscoll–Kraay studentization; they do not establish corresponding finite-sample performance for Conley.

GVAR reference/design Legacy nominal rejection Independent-null bootstrap rejection
Independent I(1), T=200, block 1 10.25 3.25 [2.20, 4.30]
Independent I(1), T=300, block 1 10.92 4.08 [2.96, 5.21]
Independent AR(0.5), T=200, block 1 0.08 4.58 [3.40, 5.76]
Independent I(1), T=200, block 4 10.25 3.33 [2.30, 4.37]
I(1) innovation leakage 1, T=200 (power) 9.00 3.00 [1.35, 4.65]
Stationary leakage 2, T=250 (power) 100.00 100.00 [98.12, 100.00]

Null rows average six auxiliary tests per replication; power rows average the two tests for the leaking country. The last design has own persistence 0.5 and foreign persistence 0.9. The restricted reference reduces the documented null overrejection, sometimes conservatively, but its 3% power against the I(1) leakage alternative is a material blind spot. It is not an omnibus independence test, and nonrejection cannot validate the original GVAR weak-exogeneity assumption.