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：双重差分法是因果推断的核心方法，但将其扩展至功能结局并非简单的标量推广，而是涉及识别、推断和观测三个基本挑战。本文针对离散观测数据，构建了功能型双重差分法的综合半参数推断框架。首先，我们定义平行趋势假设下的功能平均处理效应，并推导其有效影响函数，从而确立半参数效率界。其次，利用内曼正交性和交叉拟合，我们构造了一个去偏估计量，有效缓解非参数重构带来的正则化偏差。第三，我们建立了估计量的弱收敛性，并提出了渐近有效的统一置信带，实现从逐点推断到曲线级推断的严格过渡。最后，我们证明离散采样下的重构误差在半参数推断中具有渐近可忽略性，确保了实际可行性。模拟实验和实证应用证实，该方法在有限样本中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理论严谨且计算可行的基础。