Add SAVI e-values for linear regression - #9868

Summary

This PR adds support for safe anytime-valid inference (SAVI) for linear regression results, based on Lindon et al. (2026), "Anytime-Valid Inference in Linear Models with Applications to Regression-Adjusted Causal Inference," Journal of the American Statistical Association

The implementation reuses statsmodels' existing t and F statistics, including the active covariance estimator. This keeps the implementation thin while making e-values, SAVI p-values, and confidence sequences available from fitted regression results.

Motivation

This is useful from three related perspectives.

Anytime-valid inference

Classical regression p-values and confidence intervals are fixed-sample objects. They are not generally valid under continuous monitoring or optional stopping. SAVI replaces these with e-processes and confidence
sequences that remain valid over time, allowing users to monitor results as data accumulate while preserving time-uniform Type I error and coverage guarantees.

See Ramdas et al., "Game-theoretic statistics and safe anytime-valid inference".

Multiple testing

E-values are useful beyond sequential monitoring. They can be combined with multiple-testing procedures such as e-BH, which provides an e-value analogue of Benjamini-Hochberg and controls FDR under arbitrary dependence
between e-values. This makes e-values attractive for dependent regression screens, structured hypothesis tests, and exploratory workflows where many hypotheses are considered.

See Wang and Ramdas, "False discovery rate control with e-values".

Post-selection inference

Confidence intervals obtained by inverting e-values, or e-CIs, have strong post-selection properties. They can be adjusted after arbitrary data-dependent selection while controlling false coverage rate under arbitrary
dependence. This makes confidence sequences/e-CIs especially useful after model selection, variable screening, or exploratory regression analysis.

See Xu, Wang, and Ramdas, "Post-selection inference for e-value based confidence intervals".

API

This PR adds:

• RegressionResults.e_values(..., g=...) for parameter-wise t tests or joint linear restrictions via F tests.
• RegressionResults.p_values(savi=True, g=...) for reciprocal e-values as SAVI p-values.
• RegressionResults.conf_int(savi=True, g=...) for parameter-wise confidence-sequences/e-CIs.
• RegressionResults.summary(savi=True, g=...) to display e-values and confidence sequences in the regression summary table, including replacing the model-level Prob (F-statistic) with an F-test e-value.
• RegressionResults.sequential_p_values(..., g=...) and RegressionResults.confidence_sequences(..., g=...) as explicit aliases for users who prefer sequential-inference terminology.

Classical behavior is unchanged unless savi=True is requested.

Example

The example below uses the full statsmodels Longley dataset:

from statsmodels.datasets import longley
from statsmodels.regression.linear_model import OLS
from statsmodels.tools.tools import add_constant

data = longley.load_pandas()
y = data.endog
X = add_constant(data.exog, prepend=False)

res = OLS(y, X).fit()
print(res.summary(savi=True))

                            OLS Regression Results
==============================================================================
Dep. Variable:                 TOTEMP   R-squared:                       0.995
Model:                            OLS   Adj. R-squared:                  0.992
Method:                 Least Squares   F-statistic:                     330.3
Date:                Fri, 26 Jun 2026   e (F-statistic):              2.04e+05
Time:                        13:17:08   Log-Likelihood:                -109.62
No. Observations:                  16   AIC:                             233.2
Df Residuals:                       9   BIC:                             238.6
Df Model:                           6
Covariance Type:            nonrobust
==============================================================================
                 coef    std err          t          e      [0.025      0.975]
------------------------------------------------------------------------------
GNPDEFL       15.0619     84.915      0.177      0.247    -312.329     342.453
GNP           -0.0358      0.033     -1.070      0.425      -0.165       0.093
UNEMP         -2.0202      0.488     -4.136     29.336      -3.903      -0.137
ARMED         -1.0332      0.214     -4.822     70.649      -1.859      -0.207
POP           -0.0511      0.226     -0.226      0.249      -0.923       0.821
YEAR        1829.1515    455.478      4.016     24.928      73.044    3585.258
const      -3.482e+06    8.9e+05     -3.911     21.587   -6.92e+06   -4.92e+04
==============================================================================

Testing

Tests cover:

• e-values from t and F statistics,
• SAVI p-values as reciprocal e-values,
• confidence sequences containing classical confidence intervals,
• robust covariance behavior,
• summary(savi=True) replacing coefficient p-values, coefficient intervals, and the model-level F-test p-value,
• validation of g and alpha,
• convergence of the finite-sample e-process expression to the asymptotic Gaussian e-process for large n.
• numerical agreement with the avlm CRAN package on the same input data, up to floating-point precision.