Difference-in-differences (DiD) is a cornerstone of causal inference, yet extending it to functional outcomes is not a routine scalar generalization; rather, it entails three fundamental challenges in identification, inference, and observation. This paper develops a comprehensive semiparametric inference framework for functional DiD with discretely observed data. First, we define the functional average treatment effect under parallel trends and derive its efficient influence function (EIF), thereby establishing the semiparametric efficiency bound. Second, leveraging Neyman orthogonality and cross-fitting, we construct a debiased estimator that effectively mitigates regularization bias arising from nonparametric reconstruction. Third, we establish weak convergence of the estimator and propose an asymptotically valid uniform confidence band, enabling a rigorous transition from pointwise to curve-level inference. Finally, we demonstrate that reconstruction error under discrete sampling is asymptotically negligible for semiparametric inference, ensuring practical feasibility. Simulations and empirical applications confirm that the proposed method achieves superior coverage and testing power in finite samples, providing a theoretically grounded and computationally tractable foundation for causal evaluation with functional data.

翻译：双重差分法（DiD）是因果推断的基石，但将其扩展至功能结局并非简单的标量泛化；相反，它面临着识别、推断和观测方面的三个根本性挑战。本文针对离散观测数据下的功能型双重差分法，建立了一个综合的半参数推断框架。首先，我们在平行趋势假设下定义了功能平均处理效应，并推导出其有效影响函数（EIF），从而确立了半参数效率界。其次，利用内曼正交性和交叉拟合，我们构建了一个去偏估计量，有效缓解了由非参数重构导致的正则化偏差。第三，我们建立了估计量的弱收敛性质，并提出了渐近有效的统一置信带，实现了从点级推断到曲线级推断的严谨过渡。最后，我们证明离散采样下的重构误差对半参数推断是渐近可忽略的，从而确保了其实用可行性。模拟实验与实证应用均证实，所提方法在有限样本下实现了优越的覆盖率和检验功效，为功能数据的因果评估提供了兼具理论严谨性与计算可行性的基础。