PO.ID.Exact.Variable­Intensity­IV

Treatment intensity level j, the lower endpoint of margin j → j+1.

Treatment intensity level j+1, the upper endpoint of margin j → j+1.

Numeric value of an ordered treatment level.

The margin increment f(j+1) - f(j) for j : Fin J.

A movement from a to b crosses margin j → j+1.

Real-valued indicator of Crossing a b j.

Generic normalized finite weight a_i / Σ_k a_k.

Instrument packaged as a POVar valued in 𝒵.

Treatment intensity packaged as a POVar valued in Fin (J+1).

Outcome packaged as a POVar valued in ℝ.

Potential treatment intensity D(z): the intensity that would be observed if the instrument were fixed to z. A genuine single-intervention counterfactual.

Treatment-indexed potential outcome Y(d): the outcome that would be observed if the treatment intensity were fixed to d. No instrument argument enters, so exclusion is structural.

Factual instrument Z.

Factual treatment intensity D.

Factual outcome Y.

Factual instrument cell {Z = z}.

Regimed variable for D(z) (treatment intensity under instrument set to z).

Regimed variable for Y(d) (outcome under treatment intensity set to d).

Homogeneous bundle of the J+1 treatment-indexed potential outcomes Y(0),…,Y(J).

Counterfactual bundle (D(z0), D(z1), Y(0),…,Y(J)) for the contrast (z0, z1). Index 0 is D(z0), index 1 is D(z1), index d+2 is Y(d).

Unit causal response on margin j → j+1: Δ_jY = Y(j+1) - Y(j).

Crossing event C_j(z0,z1) = {D(z1) ≥ j+1 > D(z0)}.

Crossing probability for a directed instrument contrast and margin.

Total crossing probability, equivalently the first-stage denominator under directed monotonicity.

Normalized Angrist-Imbens margin-crossing weight.

Indicator-weighted margin effect E[Δ_jY · 1_{C_j}], represented as a set integral over the crossing event.

Unnormalized sum of indicator-weighted crossing effects.

Indicator-weighted ACR ratio, avoiding partial conditional means in the core algebra.

The conditional margin response is the average unit-level causal response among units whose potential treatment crosses a selected treatment margin when the instrument changes along a selected ordered contrast.

Angrist-Imbens average causal response: Σ_j ω_j E[Δ_jY | C_j].

Potential outcome Y(D(z)): the outcome if the instrument were set to z. Equals Y(d) at d = D(z)(ω). No direct instrument effect enters because YofD fixes only the treatment intensity d, not z (structural exclusion).

Potential first-stage contrast E[D(z1) − D(z0)] using potential treatment intensities.

Potential reduced-form contrast E[Y(D(z1)) − Y(D(z0))].

E[D | Z = z] — observed first-stage conditional mean at instrument value z, defined via eventCondExp over the cell {Z = z}.

E[Y | Z = z] — observed reduced-form conditional mean at instrument value z, defined via eventCondExp over the cell {Z = z}.

Directed Wald estimand for two instrument cells.

Canonical single treatment margin transported from Fin 1 under J = 1.

First-stage fitted treatment D_S(ω) = γ_D^T S(Z(ω)); the linear projection of D onto the instrument score.

Population 2SLS estimand β_2SLS(S) = E[D_S Y] / E[D_S D]. This declaration is only the population ratio interface: the file does not derive the binary-instrument bridge beta2SLS T = wald z0 z1. Such a bridge would require centering algebra and the scalar FWL identity for a score of the form S(Z) = Z - E[Z].