Module riesz 

Linear functionals of the regression and their Riesz representers, for automatic debiased machine learning (AutoDml; Chernozhukov, Newey and Singh, 2022).

The target is θ = E[m(W; γ)], a mean of a linear functional m of the regression γ(w) = E[Y | W = w] with W = (D, X). Its Riesz representer α satisfies E[m(W; f)] = E[α(W) f(W)] for all f, and the debiased score is m(W; γ) + α(W)(Y − γ(W)) − θ. Instead of deriving α by hand (as the propensity weights of the ATE), it is learned from the functional alone by minimising the Riesz loss E[α(W)² − 2 m(W; α)].

A functional is written as evaluations of a function at transformed data points: m(W; f) = Σ_k c_k(W) · f(W_k) (Terms), e.g. for the ATE f(1, X) − f(0, X).

Structs

Ate

Average treatment effect of a binary treatment: m = f(1, X) − f(0, X).

AverageDerivative

Average derivative E[∂γ(D, X)/∂D] of a continuous treatment, by a central difference with step h (default 10⁻⁴ times the standard deviation of the treatment). Needs a regression that is smooth in the treatment (linear models, polynomials, neural networks); for trees use ShiftEffect.

CustomFunctional

A functional given by a closure returning its terms.

PolicyEffect

Average effect of a policy that sets each unit’s treatment to a given value: m = f(d_new, X) − f(D, X).

RieszLasso

The Lasso Riesz representer of Chernozhukov, Newey and Singh (2022): α(w) = b(w)'ρ with

ShiftEffect

Average effect of shifting the treatment by delta: m = f(D + δ, X) − f(D, X), divided by δ with per_unit.

Term

One evaluation in a functional: weight ⊙ f(w), with w the data with transformed columns (column 0 is the treatment).

Enums

Dictionary

Basis functions b(w) in which the Riesz representer is approximated.

RieszPenalty

How RieszLasso chooses its penalty.

Traits

LinearFunctional

A linear functional m(W; f) of a function of W = (D, X), given by the transformed data points at which f is evaluated (column 0 of w is the treatment).

RieszLearner

A learner for the Riesz representer α of a functional.

Functions

apply_terms

m(W; f) = Σ_k weight_k ⊙ f(w_k) for the terms of a functional.