This paper develops an inferential framework for matrix completion when missing is not at random and without the requirement of strong signals. Our development is based on the observation that if the number of missing entries is small enough compared to the panel size, then they can be estimated well even when missing is not at random. Taking advantage of this fact, we divide the missing entries into smaller groups and estimate each group via nuclear norm regularization. In addition, we show that with appropriate debiasing, our proposed estimate is asymptotically normal even for fairly weak signals. Our work is motivated by recent research on the Tick Size Pilot Program, an experiment conducted by the Security and Exchange Commission (SEC) to evaluate the impact of widening the tick size on the market quality of stocks from 2016 to 2018. While previous studies were based on traditional regression or difference-in-difference methods by assuming that the treatment effect is invariant with respect to time and unit, our analyses suggest significant heterogeneity across units and intriguing dynamics over time during the pilot program.

翻译：本文提出了一种在非随机缺失且无需强信号条件下的矩阵补全推断框架。我们的研究基于以下观察：当缺失条目数量相对于面板规模足够小时，即使缺失机制非随机，这些条目仍可被有效估计。基于这一事实，我们将缺失条目划分为更小的子集，并通过核范数正则化对每个子集进行估计。此外，我们证明经过适当的去偏处理后，即使在信号较弱的情况下，所提出的估计量仍具有渐近正态性。本研究的动力源于近期关于“最小报价单位试点计划”的研究——该实验由美国证券交易委员会（SEC）于2016年至2018年间实施，旨在评估扩大最小报价单位对股票市场质量的影响。与以往基于传统回归或双重差分方法（假设处理效应不随时间与个体变化）的研究不同，我们的分析揭示了试点计划期间处理效应存在显著的个体异质性及令人关注的动态时间演变特征。