Abstract: To address the problems of increased causal-effect estimation error and unstable small-sample results caused by the coexistence of weak and invalid instrumental variables (IVs), as well as the rapidly increasing complexity of combinatorial searches as the number of candidate variables grows, an E-CAT algorithm applicable to additive nonlinear constant-effects models is proposed. The algorithm adopts a three-stage framework of “screening–joint scoring–estimation.” First, the linear association coefficients between the candidate IVs and the treatment, together with their standard errors, are estimated. An adaptive hard-thresholding rule is then used to remove weak IVs and reduce the candidate set. Second, candidate subsets of a prespecified size are enumerated from the screened set. A distance-correlation score and a residual-orthogonality score are constructed using distance correlation coefficients and sample moments, respectively. The two scores are then normalized using the Min–Max method and combined through a weighted sum to form a joint score, and the subset with the minimum joint score is selected. Finally, moment conditions are constructed using the selected IVs, and the causal effect is estimated by the generalized method of moments. Theoretical analysis shows that both scores are statistically consistent. Under linear detectability and other regularity conditions, the probabilities that the prescreening stage retains a sufficient number of truly valid IVs and that E-CAT selects a truly valid subset both converge to one. Synthetic experiments consider valid-IV-dominant and invalid-IV-dominant scenarios, comprising 16 settings with 6–20 candidate IVs. Sample sizes are set to 200, 500, and 750, and each setting is repeated 100 times. Sensitivity analysis shows that a joint-score weight of 0.5 provides relatively stable overall performance. Compared with OLS, MR-Egger, TSHT, CIV, and CAT, E-CAT achieves a mean squared error below 0.2 in 14 settings at both sample sizes of 500 and 750. In the two complex settings with 20 candidate IVs and a sample size of 750, the average runtime of E-CAT is 4–5 s, representing a reduction of more than 96% compared with the 145–150 s required by CAT. In the ablation experiment, the full E-CAT achieves a lower mean squared error than all five ablated variants at all three sample sizes. Real-data experiments were further conducted on the maternal labor supply dataset and the colonial origins dataset. In the maternal labor supply dataset, which contains 12 candidate instrumental variables, E-CAT selected “two boys” and “two girls” and estimated the causal effect of having more than two children on mothers’ weeks worked to be −5.824. The colonial origins dataset contains only two candidate instrumental variables, namely settler mortality and the proportion of Europeans in the population in 1900. The P-values of the two cross-independence tests were 0.1966 and 0.2590, respectively, and that of the joint residual orthogonality test was 0.7961; thus, none of the corresponding null hypotheses was rejected. These results demonstrate that E-CAT improves the accuracy and stability of causal-effect estimation in complex nonlinear settings while substantially reducing the computational cost relative to CAT.

摘要： 针对候选工具变量中弱工具变量与无效工具变量并存导致因果效应估计误差增大、小样本结果不稳定，以及候选变量增加后组合搜索复杂度迅速上升的问题，提出适用于加性非线性常效应模型的E-CAT算法。该算法采用“筛选—联合评分—估计”三阶段框架：首先估计候选工具变量与处理变量的线性关联系数及标准误，利用自适应硬阈值剔除弱工具变量，压缩候选集合；其次在筛选后的集合中枚举给定规模的候选子集，利用距离相关系数和样本矩分别构造距离相关评分与残差正交评分，采用Min-Max方法对两类评分进行归一化，再通过加权求和形成联合评分，并选择联合评分最小的子集；最后根据所选工具变量构造矩条件，采用广义矩方法估计因果效应。理论分析表明，两类评分具有统计相合性；在线性可检测性等条件下，预筛选保留足够真实有效工具变量及E-CAT选出真实有效子集的概率均趋近于1。合成数据实验包含有效工具变量占优和无效工具变量占优两类场景，共16种情形，候选工具变量总数为6～20个，样本量为200、500和750，每种设置重复100次。敏感性分析表明，联合评分权重取0.5时整体表现较为稳定。与OLS、MR-Egger、TSHT、CIV和CAT相比，当样本量为500和750时，E-CAT均有14种情形的均方误差低于0.2；在候选工具变量总数为20、样本量为750的两种复杂情形下，其平均运行时间为4～5 s，较CAT的145～150 s缩短96%以上。消融实验中，完整E-CAT在三种样本量下的均方误差均低于五种消融版本。进一步在母亲劳动力供给数据和殖民起源数据上开展真实数据实验。在包含12个候选工具变量的母亲劳动力供给数据中，E-CAT筛选出“两名男孩”和“两名女孩”，估计生育超过两个孩子对母亲工作周数的因果效应为−5.824；在殖民起源数据中，只包含殖民者死亡率和1900年欧洲人口比例两个候选工具变量，两个交叉独立性检验的P值分别为0.1966和0.2590，残差联合正交性检验的P值为0.7961，均未拒绝相应原假设。结果表明，E-CAT能够提高复杂非线性场景下因果效应估计的准确性与稳定性，并显著降低相较于CAT的计算开销。