Omitted-variable bound standard error

This study validates the standard error of the one-sided limits of the omitted-variable bound. Theorem 4 in Section 4 of Chernozhukov, Cinelli, Newey, Sharma and Syrgkanis (2026) gives those limits. For the ATT and the ATC, the curve of \(\nu^2\) carries the influence term of the estimated share of the conditioning arm, which RM22 added. The study reads each bound as an estimate of a known population value. It reads the bound’s standard error back off the reported limit.

No canonical implementation is compared. DoubleML omits the share term. The R package dml.sensemakr carries it. It estimates the share out of fold, and at its commit 58ac44d it divides the point estimate of \(\nu^2\) as well, so its bounds are a different estimate. The zero-row equivalence artifact records the absence of a comparator.

What was tested

setting	declaration
law	make_linear_ate(n=1000). Four standard normal covariates, the treatment mechanism \(\operatorname{expit}(0.3 W_1 - 0.2 W_2 + 0.1 W_3)\), standard normal outcome noise, and a constant effect of 1.5
fit	in-sample TMLE with a linear outcome regression and an unpenalized main-effects logistic treatment model. Both are correctly specified. The fit reports the ATE, the ATT and the ATC
estimator of \(\nu^2\)	the default, which resolves to the doubly robust estimator
strength	\(c_Y = 0.5\), \(c_D = 0.3\), \(\rho = 1\)
primary estimands	the lower and the upper bound of the ATT, the ATC and the ATE
primary standard error	(lower - ci_lower) / z_0.95 for a lower bound, and (ci_upper - upper) / z_0.95 for an upper bound
primary interval	the bound plus or minus 1.959964 standard errors
truth	closed form. \(\nu^2\) is \(4 e^{0.07}\) for the ATT and the ATC and \(2 + 2 e^{0.07}\) for the ATE, and \(\sigma^2 = 1\)
control	the same fits, with the standard error from the curve of \(\nu^2\) without the share term. That is the curve before RM22
Monte Carlo inference	99% intervals around every declared endpoint

The RM22 plan in the roadmap fixed the law, the budget, the seeds, the margins and the red-cell route before any run.

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
linear Gaussian-outcome law with a constant effect, make_linear_ate	atc_lower	lower omitted-variable bound on the ATC, at cf_y 0.5, cf_d 0.3, rho 1	cleverly omitted-variable bound	-0.0025 to 0.0011	0.9511	1.0026	pass
linear Gaussian-outcome law with a constant effect, make_linear_ate	atc_upper	upper omitted-variable bound on the ATC, at cf_y 0.5, cf_d 0.3, rho 1	cleverly omitted-variable bound	0.0015 to 0.0052	0.9482	0.9965	pass
linear Gaussian-outcome law with a constant effect, make_linear_ate	ate_lower	lower omitted-variable bound on the ATE, at cf_y 0.5, cf_d 0.3, rho 1	cleverly omitted-variable bound	-0.000286 to 0.0032	0.9507	1.0018	pass
linear Gaussian-outcome law with a constant effect, make_linear_ate	ate_upper	upper omitted-variable bound on the ATE, at cf_y 0.5, cf_d 0.3, rho 1	cleverly omitted-variable bound	-0.000631 to 0.0029	0.9487	0.9946	pass
linear Gaussian-outcome law with a constant effect, make_linear_ate	att_lower	lower omitted-variable bound on the ATT, at cf_y 0.5, cf_d 0.3, rho 1	cleverly omitted-variable bound	-0.0024 to 0.0012	0.9530	1.0042	pass
linear Gaussian-outcome law with a constant effect, make_linear_ate	att_upper	upper omitted-variable bound on the ATT, at cf_y 0.5, cf_d 0.3, rho 1	cleverly omitted-variable bound	0.0013 to 0.0050	0.9478	0.9916	pass

Theory properties

Each cell reads the same 10,000 draws, and one fit per draw gives every row. A positive cell must put its SE-ratio interval inside the calibration band and its coverage interval inside the coverage band. Each end of the ATT bound also has an inflated_se_control. The control reads the curve without the share term, which is too wide on this law, so its SE-ratio interval must rise above the band. The ATC has no control. The plan’s probe put its ratios without the term at 1.098 and 1.111, too close to 1.07 to predict a failure.

property	cell	role	what was tested	what must hold	measured	result
interval_calibration	atc_lower__correctly_specified	positive	lower omitted-variable bound on the ATC: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9462 to 0.9573, SE ratio 0.9909 to 1.0278	pass
interval_calibration	atc_upper__correctly_specified	positive	upper omitted-variable bound on the ATC: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9454 to 0.9566, SE ratio 0.9831 to 1.0198	pass
interval_calibration	ate_lower__correctly_specified	positive	lower omitted-variable bound on the ATE: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9463 to 0.9574, SE ratio 0.9883 to 1.0240	pass
interval_calibration	ate_upper__correctly_specified	positive	upper omitted-variable bound on the ATE: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9441 to 0.9554, SE ratio 0.9808 to 1.0173	pass
interval_calibration	att_lower__correctly_specified	positive	lower omitted-variable bound on the ATT: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9442 to 0.9555, SE ratio 0.9852 to 1.0217	pass
interval_calibration	att_lower__inflated_se_control	control	lower omitted-variable bound on the ATT: the standard errors come from the curve of nu^2 without the conditioning-share term, as before RM22	the SE-ratio interval must fall above the calibration band	coverage 0.9632 to 0.9724, SE ratio 1.0791 to 1.1189	pass
interval_calibration	att_upper__correctly_specified	positive	upper omitted-variable bound on the ATT: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9453 to 0.9565, SE ratio 0.9794 to 1.0157	pass
interval_calibration	att_upper__inflated_se_control	control	upper omitted-variable bound on the ATT: the standard errors come from the curve of nu^2 without the conditioning-share term, as before RM22	the SE-ratio interval must fall above the calibration band	coverage 0.9635 to 0.9726, SE ratio 1.0732 to 1.1129	pass

Result

Every primary test and every property cell passed. The controls show that the calibration band detects the curve without the share term on this law.

Measured values

Names beginning margin: are thresholds declared before the run. Everything else is measured from the committed results and checked at the precision printed.

quantity	value	source
replicates	10000	primary replications
n	1000	observations per primary replication
independent_tests_total	6	bound-end tests against truth
independent_tests_passed	6	of those, passing
paired_tests_total	0	external comparisons declared
paired_tests_passed	0	external comparisons passing
property_cells_total	8	repeated-sampling property cells
property_cells_passed	8	cells whose own and family verdicts pass
max_standardized_bias	0.0475	largest absolute primary bias in empirical standard deviations
min_coverage	0.9478	lowest measured primary-study coverage
min_coverage_ci_lower	0.9418	lowest exact 99% primary coverage endpoint
min_se_ratio_ci_lower	0.9743	lowest bootstrap primary SE-ratio endpoint
max_se_ratio_ci_upper	1.0228	highest bootstrap primary SE-ratio endpoint
margin:confidence_level	0.9900	confidence level of every Monte Carlo interval
margin:alpha	0.0500	nominal size of the reported intervals
margin:nominal_coverage	0.9500	nominal coverage those intervals claim
margin:bootstrap_replicates	10000	resamples behind every bootstrap interval
margin:standardized_bias	0.2500	bias equivalence margin, in empirical standard deviations
margin:coverage_floor	0.9000	validity floor the coverage lower endpoint must clear
margin:over_coverage_ceiling	0.9900	above this, coverage is conservative rather than invalid
margin:se_ratio_sanity_lower	0.8000	SE-ratio screen, lower limit
margin:se_ratio_sanity_upper	1.2000	SE-ratio screen, upper limit
margin:calibration_se_ratio_lower	0.9300	calibration-cell SE-ratio band, lower limit
margin:calibration_se_ratio_upper	1.0700	calibration-cell SE-ratio band, upper limit. A control must rise above it
margin:calibration_coverage_lower	0.9200	calibration-cell coverage band, lower limit
margin:calibration_coverage_upper	0.9800	calibration-cell coverage band, upper limit
margin:type_i_ceiling	0.1000	largest size a one-sided type-I bound may establish
margin:paired_difference	0.1500	paired similarity margin, in pooled empirical standard deviations
margin:rmse_noninferiority	1.1000	largest external-comparison RMSE ratio bound
margin:coverage_noninferiority	-0.0250	smallest external-comparison coverage difference bound
margin:calibration_noninferiority	0.0500	largest external-comparison calibration excess bound
margin:minimum_power	0.8000	rejection lower bound a power control must clear
margin:root_n_slope	-0.5000	contraction rate root-n asymptotics predict
margin:root_n_slope_lower	-0.6250	accepted root-n slope band, lower limit
margin:root_n_slope_upper	-0.3750	accepted root-n slope band, upper limit
margin:excluded_slope	-0.2500	slower rate a root-n interval must exclude

Limitations

limitation	what it means for use
The row publishes under the reporting policy, not gated	Every declared cell is green. The reporting policy does not assert that, so the fast tier recomputes each verdict and does not fail on a red one
There is no cross-implementation evidence	No maintained implementation computes the same bounds with the same curve
The fit is in sample, with correctly specified GLMs	The row does not cover cross-fitting, a flexible learner, or a misspecified nuisance
One law, one sample size and one strength	The row covers make_linear_ate at \(n = 1000\) and \(c_Y = 0.5\), \(c_D = 0.3\), \(\rho = 1\). At the default strength of 0.03 the share term moves no ratio by more than 0.001
Two arms, no weights and no clusters	The exact-law witness covers three arms, weights and clusters. This row does not
Only the doubly robust estimator	The plug-in estimator reports no limits, so the row says nothing about it
Positive residual variance	The law has \(\sigma^2=1\). Theorem 4 needs a positive maximal-bias scale, so the row does not test the zero-variance refusal
The ATC has no control	The ATC’s positive cells pass, and no control shows that the band would detect its omission. The plan left it out because its 500-fit probe put the ATC ratios without the term at 1.098 and 1.111, too close to 1.07 to predict a failure. The ATT controls then measured about the same inflation, 1.0987 and 1.0922. No ATC control was added after the run
The att_upper control passed narrowly	Its lower edge is 1.0732 against the band edge of 1.07. The Monte Carlo standard deviation of its ratio is about 0.0077, the interval width over \(2 z_{0.995}\). A new run fails when its ratio falls at or below 1.0890, which puts the lower edge at or below 1.07. That happens about 34% of the time if the true ratio is this cell’s 1.0922, and about 20% if it is 1.0954, the mean of the two controls. The plan sized the budget on the 500-fit ratio of 1.118, whose Monte Carlo error did not cover the measured inflation of about 1.0954. A future design would pair each control with its positive cell and read the ratio of their standard errors, which cancels most of that error
The law is symmetric	On make_linear_ate the ATT and the ATC are equal, and so are their \(\nu^2\) and their bounds. The study cannot see an implementation that swaps the conditioning arm. The exact-law witness covers that case: mutations M5 and M7 in tests/unit/test_omitted_variable_standard_error.py
Point treatment only	The bound refuses longitudinal fits, so the row covers none

Reproduction

The fixture README gives the regeneration command. The manifest records the seeds, margins, exact estimator configuration, source hashes, and result hashes. The replications and property results carry every published row.