Abstract

Bidirectional causal relationships arising from mutual interactions be

tween variables are commonly observed within biomedical, econometrical, and

social science contexts. When such relationships are further complicated by unobserved factors, identifying causal effects in both directions becomes especially

challenging.

For continuous variables, methods that utilize two instrumental

variables from both directions have been proposed to explore bidirectional causal

effects within linear models. However, the existing techniques are not immediately applicable when the key variables of interest are binary. To address these

issues, we propose a structural equation modeling approach that links observed

binary variables to latent continuous variables through a constrained mapping.

We firstly establish the identification results for bidirectional causal effects using

a pair of instrumental variables. Then, we develop an estimation method for

the corresponding causal parameters. We also conduct sensitivity analysis under

scenarios where certain identification conditions are violated. Finally, we apply

our approach to investigate the bidirectional causal relationship between heart

disease and diabetes, demonstrating its practical utility in biomedical researches.

Key words and phrases: Bidirectional causal inference; Binary outcomes; Instru- mental variables; Selection mechanism; Sensitivity analysis

Information

Preprint No.SS-2025-0374
Manuscript IDSS-2025-0374
Complete AuthorsYafang Deng, Kang Shuai, Shanshan Luo
Corresponding AuthorsShanshan Luo
Emailsshanshanluo@btbu.edu.cn

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Supplementary Materials

illustrates the regions of these configurations.

Sections C.2–C.4 of the Supplementary Materials provide the detailed

proofs of Corollaries 1–3, while Section C.5 presents the discussion of Corollary 3 under the alternative parameterization based on η/µxz and δ/µyw.

4.

Numerical Experiments

4.1

Simulation studies

In this section, we conduct numerical simulations to evaluate the finite sample performance of the proposed method in Section 2. We gener-

ate observed covariate Q ∼N(0, 1) and unobserved confounders U, V ∼

N(0, 0.752). Instrumental variables Z and W follow either N(0, 1) (Scenario 1) or Unif(−1, 1) (Scenario 2). Latent variables X◦and Y ◦are gen-

erated from Model (2.2) with parameters µx0 = µy0 = 0, βy→x = 0.45,

βx→y = −0.25, µxz = µyw = 0.65, µxq = µyq = 0.15, and then thresholded

to obtain binary X and Y .

For the two data-generating scenarios, we consider two estimation meth-

Supplementary materials are available for download.