Influence-Function Procedure

This is the default procedure and the one the whole package is built around. It is the Farrell, Liang, Misra correction. This page explains the one object it produces per observation, how that object becomes a point estimate and a standard error, and the two knobs that change the variance.

The influence function, in words

For each observation i the procedure builds a single number ψ_i, the influence function value. Read it as “the target, plus the correction for the bias that regularization put into observation i”. Concretely,

\[\psi_i = \underbrace{H_i}_{\text{plug-in target}} \;-\; \underbrace{H_\theta \,\Lambda(x_i)^{-1}\, \ell_\theta(i)}_{\text{bias correction}}\]

where H_i is the target evaluated at the fitted parameters (the naive, biased piece), ℓ_θ(i) is the score (gradient of the loss in θ), Λ(x_i) is the expected Hessian from Estimation, and H_θ is the Jacobian of the target in θ. You do not assemble this by hand; the engine does it. The point is that the second term is exactly the first-order bias, so subtracting it removes the bias to first order.

From ψ to an estimate and an interval

The debiased point estimate is just the average of the influence values, and the standard error is the sample standard deviation of those values divided by √n. Both entry points (structural_dml() and inference()) use the same code for this step, compute_se_ci.

\[\hat\mu = \frac{1}{n}\sum_{i=1}^n \psi_i, \qquad \widehat{SE} = \sqrt{\hat\Psi / n}, \qquad \text{CI}_{95\%} = \hat\mu \pm 1.96\,\widehat{SE}\]

The only thing that varies is how the variance Ψ̂ is computed.

Variance estimator (the variance= argument)

variance=

Formula

Notes

'pooled' (default)

\(\hat\Psi = \frac{1}{n-1}\sum_i (\psi_i - \hat\mu)^2\)

Bessel-corrected sample variance centered at the global mean. This is the variance the paper prescribes (FLM2025 §3.2; FLM2021 §4).

'within_fold' (legacy)

\(\hat\Psi = \frac{1}{K}\sum_k \operatorname{mean}_{i \in \text{fold } k}(\psi_i - \hat\mu_k)^2\)

Mean of per-fold variances, each centered at its own fold mean. Drops the between-fold component, so it is always smaller than pooled and gives narrower intervals. Use 'pooled'.

Repeated cross-fitting (the n_repeats= argument)

Cross-fitting splits the data randomly, so a single split is a little noisy. Setting n_repeats > 1 repeats the whole procedure on fresh splits and combines them with the median rule of Chernozhukov et al. (2018), median_dml_aggregate:

\[\hat\mu = \operatorname{median}_r(\hat\mu_r), \qquad \widehat{SE}^2 = \operatorname{median}_r\!\big(\widehat{SE}_r^2 + (\hat\mu_r - \hat\mu)^2\big)\]

The (μ̂_r − μ̂)² term folds the variation across splits into the standard error, so the SE can only widen, never average away. This is the lever that fixes residual under-coverage. The package recommendation for linear ATE coverage is n_folds >= 50 with n_repeats >= 5.

What you get back

The result object (DMLResult or InferenceResult) exposes:

Attribute

Meaning

mu_hat

Debiased point estimate (mean of ψ)

se

Standard error

ci_lower, ci_upper

95% confidence interval bounds

psi_values

The (n,) influence values themselves

theta_hat

The (n, d_θ) fitted heterogeneous parameters

diagnostics

Regime, n_folds, lambda_method, min Hessian eigenvalue, correction ratio, and more

Call result.summary() for a statsmodels-style printout.

See also