Bidirectional causal relationships arising from mutual interactions between 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 in linear models. However, the existing techniques are not 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 continuous latent variables through a constrained mapping. We further establish identification results for bidirectional causal effects using a pair of instrumental variables. Additionally, 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 research.

翻译：在生物医学、计量经济学和社会科学领域中，由变量间相互交互作用产生的双向因果关系普遍存在。当此类关系进一步受到未观测因素的干扰时，双向因果效应的识别变得尤为困难。对于连续变量，已有研究提出利用来自两个方向的工具变量来探索线性模型中的双向因果效应。然而，当关键关注变量为二元变量时，现有方法并不适用。为解决这一问题，我们提出一种结构方程建模方法，通过约束映射将观测的二元变量与连续潜变量相关联。我们进一步利用一对工具变量建立了双向因果效应的识别结果。此外，我们开发了相应因果参数的估计方法，并对某些识别条件不满足的情形进行了敏感性分析。最后，我们将所提方法应用于研究心脏病与糖尿病之间的双向因果关系，展示了其在生物医学研究中的实际应用价值。