Statistical methods

Audience

This page assumes familiarity with applied econometrics. For a first result, start with Quickstart. For empirical size measurements and known finite-sample limits, use Inference calibration and limitations.

This page defines the cross-cutting statistical contracts shared by factrix metrics. Per-metric formulas, parameters, and return schemas remain on the Metrics API pages.

Periods, not calendar time

Every horizon, lag, stride, and window counts periods on the panel's own distinct-date grid. No method infers frequency or annualises from a date. See Period grid, not calendar.

1. HAC SE under overlapping returns

An h-period forward return makes adjacent observations share up to h - 1 periods. factrix exposes three series-mean approaches. None removes economic persistence that remains after the overlap horizon.

Method	Dependence treatment	Reference and main trade-off
NON_OVERLAPPING	Keep observations at least overlap_periods apart	Student-t reference on the strided series. Mechanical overlap is removed; the reference is exact only under iid normality. It spends roughly a factor of h in sample size.
NEWEY_WEST	Bartlett-kernel heteroskedasticity-and-autocorrelation-consistent (HAC) variance on the full series	Retains observations but uses an asymptotic HAC reference and a calibrated deterministic bandwidth.
STATIONARY_BOOTSTRAP	Stationary block resampling with an automatically selected mean block length	Avoids a parametric-normal reference for the series mean, but still requires stationarity or weak dependence, a suitable block length, and an adequate sample. It is a second read for long series with distributional doubt, not a rescue for a short series.

NON_OVERLAPPING is the default for the mainstream series-mean metrics. The caller can select the other admitted members explicitly; factrix never switches the requested method in response to a warning.

These metrics also accept alternative="two-sided", "greater", or "less". Keep the default while discovering a factor. Use a one-sided tail only when its direction was fixed independently and before inspecting the evaluated sample; factrix never infers a tail from the observed sign. Holm and BHY adjust the resulting p-values as supplied: they do not reinterpret tails, validate producer calibration, or repair a warning-coded inference regime.

Newey-West bandwidth families

The bandwidth depends on the rank of the tested restriction, not simply on whether a regression has multiple coefficients.

For a scalar series mean or rank-one regression contrast, factrix uses

\[ L = \min\!\left(\max\!\left(1.3\sqrt{T},\;3(h-1)\right),\;\lceil T/3\rceil\right), \]

scales the variance by \(T/(T-L-1)\), and reads the statistic against

\[ \nu = \max\!\left(\min\!\left(1.5T/L-1,\;T/h-1\right),\;1\right). \]

The \(1.3\sqrt{T}\) term follows LLSW (2018); the overlap floor extends the Hansen-Hodrick (1980) coverage requirement. The wider 3(h - 1) floor, finite-sample scale, and effective degrees of freedom are factrix calibration choices rather than textbook Newey-West.

Multi-restriction Wald tests instead use \(L=\max(\text{auto_bartlett}(T), h-1)\) and an F reference. A wide kernel can degrade the inversion of a multi-restriction HAC matrix, so the scalar recipe does not transfer to this family.

HAC path map

Path	Bandwidth and reference
ic, quantile_spread, quantile_spread_vw, k_spread with NEWEY_WEST; fm_beta stage 2	Scalar series-mean recipe and \(t_\nu\)
spanning_alpha, common_quantile_spread, common_asymmetry, single-restriction OLS contrasts	Scalar rank-one recipe and \(t_\nu\) or \(F_{1,\nu}\)
predictive_beta, h > 1	Scalar bandwidth and effective-df reference after the Amihud-Hurvich correction
pooled_beta	Driscoll-Kraay covariance over period-level score sums, using the scalar HAR bandwidth, variance scale, and \(t_\nu\) reference; it is not a series-mean covariance
Cross-sectional slice Wald tests	Narrow multi-restriction bandwidth and \(F\) reference
Date-disjoint slice_period_*_test	Per-slice NW variances for method="analytic"; independent stationary bootstrap draws for method="bootstrap"

See HAC family calibration for measured size bands and known oversized regimes.

Which metrics expose inference=

ic, quantile_spread, quantile_spread_vw, and k_spread expose the closed allowlist NON_OVERLAPPING, NEWEY_WEST, and STATIONARY_BOOTSTRAP. Each member was measured on the relevant IC or spread series before admission.

Other metrics keep estimator-specific parameters instead:

Metric family	Surface
fm_beta, predictive_beta, common_quantile_spread, common_asymmetry	newey_west_lags; None selects the documented rule, while an integer is an uncalibrated research override
pooled_beta	driscoll_kraay_lags; None selects the calibrated rule, while an integer replaces the automatic base but cannot undercut the overlap floor or estimability cap
spanning_alpha and slice Wald tests	Bandwidth determined from overlap_periods; no caller lag knob

2. Multiple-testing under dependence

The literature names two step-up procedures three ways:

• BH is the Benjamini-Hochberg (1995) procedure. Its FDR guarantee assumes independence or positive regression dependence on a subset (PRDS).
• BY is the Benjamini-Yekutieli (2001) arbitrary-dependence procedure. It divides the BH threshold by \(c(m)=\sum_{i=1}^{m}1/i\).
• BHY is common quant shorthand for the BY procedure. factrix follows that convention in the function name bhy().

Factor-test dependence is usually difficult to prove to be PRDS. factrix therefore uses the BY/BHY threshold when a family must be valid under arbitrary dependence. Bonferroni also remains valid under arbitrary dependence, but it controls the family-wise error rate (FWER), a stricter and usually more conservative target than FDR; its conservatism does not come from assuming independence.

The declared hypothesis family is part of the analysis. Placeholder results are excluded rather than treated as p-values, and separate economic questions should not be pooled merely to increase m. See BHY screening for family construction and hierarchical procedures.

Resampling knobs

Entry points that turn resamples into a p-value or interval use n_resamples and rng consistently.

For STATIONARY_BOOTSTRAP, result metadata keeps the requested draw count in n_resamples so seed can reproduce the run. n_resamples_used reports the finite studentized roots that actually enter the empirical p-value; invalid roots are excluded from both its denominator and p_value_mc_se. If none are usable, the p-value is unavailable rather than 1.0.

Entry point	Default n_resamples	Enforces the 199 floor	Reports resolved seed
STATIONARY_BOOTSTRAP on admitted metrics	999	Yes	In metric metadata
monotonicity	999	Yes	In metric metadata
bootstrap_mean_ci	999	Yes	No; returns an interval
slice_period_pairwise_test / slice_period_joint_test with method="bootstrap"	999	Yes	In the result column
stationary_bootstrap_resamples	999	No; returns draws, not inference	No

A smoothed empirical p-value lies on the 1/(B + 1) grid. Values such as 199, 399, and 999 align common significance levels with that grid; 999 gives roughly 0.001 resolution. The Monte Carlo standard error \(\sqrt{p(1-p)/B}\) describes resampling noise, not uncertainty in the factor estimate. Automatic block-length selection is a separate operation; it does not determine the number of resamples.

rng accepts an integer, None, or numpy.random.Generator. An integer is reproducible and reported unchanged. None draws and reports an integer seed. A generator is advanced in place and reports no seed because the caller owns the stream.

3. Robust scale and trend handling

For each period, mad_winsorize clips a factor to

\[ \text{centre} \pm n_{\text{MAD}}\,b_n\,1.4826\,\operatorname{MAD}, \]

where the centre can be the median (default) or mean. The MAD itself is always computed about the median. 1.4826 gives asymptotic Gaussian consistency; the Croux-Rousseeuw \(b_n\) factor corrects finite-cross-section bias. This keeps small cross-sections from receiving a materially narrower band solely because raw MAD is biased downward.

Sn and Qn have higher Gaussian efficiency than MAD, but require more pairwise order-statistic machinery. factrix keeps the simpler MAD contract and applies the explicit finite-sample correction instead.

ic_trend pairs two rank-based quantities:

• Theil-Sen supplies the trend magnitude and descriptive confidence interval.
• Hamed-Rao Mann-Kendall supplies stat and p_value after non-overlapping sampling.

The p-value is not backed out of the Theil-Sen interval. A constant series returns the zero slope while withholding stat and p_value with a degenerate-variance warning.

4. Persistence diagnostics under near-unit-root predictors

Persistent predictive regressors can carry finite-sample coefficient bias (Stambaugh 1999). HAC changes the variance estimate; it does not remove that coefficient bias. predictive_beta therefore applies the Amihud-Hurvich correction unconditionally and reports the raw OLS slope in metadata. The persistence warning describes the remaining inference regime; it is not the switch that decides whether the correction runs.

At overlap_periods > 1, the corrected predictive-slope test remains oversized on the measured null grid. overlapping_predictive_inference makes that limit explicit on every such result. Use the estimate as an effect size, raise the evidentiary hurdle for its p-value, and compare a one-period or genuinely non-overlapping design. factrix does not silently substitute IVX, Bonferroni-Q, or another model-dependent estimator.

For ic_trend, non-overlapping sampling happens before Theil-Sen, Mann-Kendall, the augmented Dickey-Fuller (ADF) screen, and residual autocorrelation diagnostics. The ADF test therefore reads the strided tested series, not the raw MA(h - 1) overlap. Its lag order is selected by AIC up to the Schwert ceiling. The default adf_threshold=0.10 is a conventional practitioner cutoff, not a direct prescription from Stock-Watson (1988).

A series that remains autocorrelated after the stride can still make HAC, bootstrap, or Mann-Kendall p-values optimistic. factrix raises serial_correlation_detected; changing inference member does not make that regime calibrated. See Persistence beyond the overlap horizon.

For stage-1 per-asset regressions inside common-factor metrics, factrix keeps plain OLS standard errors. Adding HAC there would not correct Stambaugh bias and would imply robustness the coefficient estimator does not provide.

5. Event-study inference

Event metrics differ in their inference unit. Compare n_obs only after checking n_obs_axis and the metric metadata.

Metric	Inference unit	Test and dependence treatment	n_obs
caar	Event periods	Mean signed abnormal return on a non-overlapping event-period series; Student-t reference	Sampled event periods
bmp_z	Valid event rows	Standardised abnormal returns; per-asset event overlap removal plus optional within-period design-effect deflation	Valid events
corrado_rank	Event periods	Signed within-asset ranks collapsed by event period; z reference on the event-period series	Event periods
event_hit_rate / event_ic	Event rows with period clustering adjustment	Metric-specific test with the same-period design effect where applicable	Metric-specific event count; inspect n_event_periods separately

CAAR event-period t

caar forms a per-event-period cross-sectional mean of signed abnormal returns, samples event periods at the declared overlap stride, and tests the sampled mean. It is not a raw-event-count t-test. Event-induced variance can invalidate the simple reference, which motivates reading bmp_z alongside it.

BMP-style standardised abnormal returns

bmp_z standardises each valid event by its estimation-window volatility. The default is a documented BMP-style simplification; set include_prediction_error_variance=True for the constant textbook prediction-error factor. Same-period dependence is estimated with an ANOVA ICC(1) and a design-effect deflator. The result reports both valid events and distinct event periods because the latter controls the residual small-sample limit under clustering.

The published Kolari-Pynnönen statistic also includes a (1 - r) numerator for variance estimated within one event date. factrix pools standardised abnormal returns across event periods, so that between-period variance is already present; applying the numerator again would deflate the statistic twice.

Corrado rank

corrado_rank replaces abnormal returns with signed within-asset ranks and collapses them by event period before inference. It is the distribution-robust read when standardised abnormal returns are sensitive to skew or extreme returns.

Event skewness is descriptive

event_skewness reports Fisher skewness without stat or p_value. A pooled D'Agostino test was removed because non-normal signed returns and same-period shocks break it in different directions; a single clustering deflator did not calibrate both. Use caar, bmp_z, corrado_rank, event_hit_rate, or event_ic for a declared directional hypothesis.

6. Calibration and known limitations

The full size tables, known oversized regimes, and test provenance now live on Inference calibration and limitations. The split keeps method contracts scannable without hiding the evidence behind them.

Useful direct links:

• HAC family calibration

• Single-restriction Wald contrasts

• Joint period tests on short slices

• Persistence beyond the overlap horizon

For null versus NaN handling, use the canonical Data schema and Preparing data guidance.