Abstract

Proximal causal inference provides a framework for estimating the av

erage treatment effect (ATE) in the presence of unmeasured confounding by

leveraging outcome and treatment proxies. Identification in this framework relies

on the existence of a so-called bridge function. Standard approaches typically

postulate a parametric specification for the bridge function, which is estimated

in a first step and then plugged into an ATE estimator. However, this sequential

procedure suffers from two potential sources of efficiency loss: (i) the difficulty

of efficiently estimating a bridge function defined by an integral equation, and

(ii) the failure to account for the correlation between the estimation steps. To

overcome these limitations, we propose a novel approach that approximates the

integral equation with increasing moment restrictions and jointly estimates the

bridge function and the ATE. We show that, under suitable conditions, our estimator is efficient. Additionally, we provide a data-driven procedure for selecting

the tuning parameter (i.e., the number of moment restrictions). Simulation studies reveal that the proposed method performs well in finite samples, and an

application to the right heart catheterization dataset from the SUPPORT study

demonstrates its practical value.

Information

Preprint No.SS-2025-0104
Manuscript IDSS-2025-0104
Complete AuthorsChunrong Ai, Jiawei Shan
Corresponding AuthorsJiawei Shan
Emailsjiawei.shan@wisc.edu

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Acknowledgments

Chunrong Ai gratefully acknowledges funding from Project 72133005, supported by the NSFC.

A.

Regularity conditions

Smoothness classes of functions: Let p be the largest integer satisfying

p < p, and a = (a1, . . . , ad). A function ϕ(v) with domain V ⊂Rd is called

a p-smooth function if it is p times continuously differentiable on V and

∂pϕ(v)

∂vα1

1 · · · ∂vαd

d

−

∂pϕ(v′)

∂vα1

1 · · · ∂vαd

d

≤C ∥v −v′∥p−p ,

max

Pd

i=1 ai=p

for all v, v′ ∈V and some constant C > 0.

Regularity conditions: The following regularity conditions (A1)–(A4)

are common in the GMM literature (Newey and McFadden, 1994, p. 2132).

Condition (A5) imposes mild smoothness restrictions to ensure a good sieve

approximation.

(A1) Γ is a compact subset in Rp; γ0 lies in the interior of Γ and is the

unique solution to (3.4); T is a compact subset in R, and τ0 lies in

the interior of T ;

(A2) h(W, A, X; γ) is twice continuously differentiable in γ ∈Γ;

(A3) E[supγ∈Γ{Y −h(W, A, X; γ)}2] < ∞and E[supγ∈Γ ∥∇γh(W, A, X; γ)∥2] <

∞;

(A4) E[supγ∈Γ {h(W, 1, X; γ) −h(W, 0, X; γ)}2] < ∞and E[supγ∈Γ ∥∇γ{h(W, 1, X; γ)−

h(W, 0, X; γ)}∥2] < ∞.

(A5) E[∇γh(W, A, X; γ0) | z, a, x], E [{Y −h(W, A, X)}2 | z, a, x] and

E[{Y −h(W, A, X)}{h(W, 1, X) −h(W, 0, X) −τ0} | z, a, x] are psmooth functions for some p > 0.

B.

Notation

The following notations, adapted slightly from Donald et al. (2009), are

used in Section 5 for selecting K:

N

X

bΥK×K = 1

i=1

{Yi −h(Wi, Ai, Xi; ˇγ)}2uK(Zi, Ai, Xi)uK(Zi, Ai, Xi)

T,

N

N

X

b

BK×p = −1

i=1

uK(Zi, Ai, Xi)∇γh(Wi, Ai, Xi; ˇγ)

T,

bΩp×p = ( b

BK×p)

T bΥ−1

K×K b

BK×p,

N

edi = ( b

BK×p)

T

 1

N

X

j=1

uK(Zj, Aj, Xj)uK(Zj, Aj, Xj)

T

−1

uK(Zi, Ai, Xi),

N

eηi = −∇γh(Wi, Ai, Xi; ˇγ) −edi,

D∗

i = ( b

BK×p)

T bΥ−1

K×KuK(Zi, Ai, Xi),

bξij = uK(Zi, Ai, Xi)

T bΥ−1

K×KuK(Zj, Aj, Xj)/N.

For a fixed t ∈Rp,

N

X

i=1

bξii{Yi −h(Wi, Ai, Xi; ˇγ)}t

T bΩ−1

p×peηi,

bΠ(K; t) =

N

X

i=1

bξii

n

t

T bΩ−1

p×p

h

b

D∗

i {Yi −h(Wi, Ai, Xi; ˇγ)}2 + ∇γh(Zi, Ai, Xi; ˇγ)

io2

−t

T bΩ−1

p×pt.

bΦ(K; t) =

The loss function is

p

X

j=1

bΠ(K; ej)2/N + bΦ(K; ej),

SGMM(K) =

where ej is the unit vector with 1 in the j-th component and 0 in all others.

In addition, Tables A1–A3 summarize the random variables, notation,

and assumptions used in the paper for ease of reference.

Table A1: List of random variables.

A

Binary treatment assignment.

Y (a)

Potential outcomes under treatment A = a.

Y

Observed outcome.

X

Observed confounders.

U

Unobserved confounders.

(Z, W)

Treatment/outcome-inducing confounding proxies.

O

Observed variables O = (A, Y, W, X, Z).

Table A2: List of notation.

h(w, a, x)

Outcome-confounding bridge function; see Eq. (2.1).

q(z, a, x)

Treatment-confounding bridge function; see Eq. (4.8).

uK(z, a, x)

Vector of basis functions; see Eq. (3.4).

meff (z, a, x)

Efficient score for estimating h; see Eq. (2.3).

gK(O; γ, τ)

Joint score function for estimating h and τ0; see Section 3.

GK(γ, τ)

Sample average of gK(O; γ, τ); see Section 3.

Optimal weight in GMM estimation; see Section 3.

B(K+1)×(p+1)

Jacobian matrix; see Eq. (3.7).

{ψ1(O), ψ2(O)}

Influence functions of (bγ, bτ); see Theorem 1.

ψeff (O)

Efficient influence function of τ0; see Eq. (4.9).

{t(·), R(·), κ}

Terms in the influence function; see Theorem 1.

VK

Asymptotic variance of the GMM estimator; see Eq. (3.7).

(Vγ, Vτ)

Asymptotic variance of (bγ, bτ); see Theorem 1.

Vτ,eff

Semiparametric efficiency bound for τ0; see Eq. (4.9).

V plug-in

τ,eff

Variance of the optimal plug-in estimator; see Theorem 3.

Table A3: Summary of assumptions.

Assumption 1

Conditions for identifying treatment effects using proxies.

Assumption 2

Conditions for the use of sieve techniques.

Assumption 3

Conditions for developing semiparametric theory.

Conditions (A1)-(A5)

Conditions for the asymptotic analysis of the GMM estimator.

Supplementary Materials

The supplementary material includes intermediate lemmas, additional simulation studies, and all technical proofs.

Supplementary materials are available for download.