Matrix Completion When Missing Is Not at Random and Its Applications in Causal Panel Data Models

This article 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. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

本文针对非随机缺失且无需强信号假设的场景，构建了矩阵补全的统计推断框架。本研究的核心观察在于：若缺失条目数相较于面板规模足够小，则即便缺失机制为非随机，仍可对其实现精准估计。基于此观察，我们将缺失条目划分为若干小子集，并通过核范数正则化对每个子集进行估计。此外，本文证明：在施加适当去偏处理后，即便信号强度较弱，所提出的估计量仍满足渐近正态性。本研究的动机源自针对最小报价单位试点计划（Tick Size Pilot Program）的近期研究：该计划由美国证券交易委员会（Security and Exchange Commission, SEC）于2016至2018年间实施，旨在评估扩大股票报价最小单位对股票市场质量的影响。过往相关研究均基于传统回归或双重差分法，并假设处理效应不随时间与个体发生变化；但本文的分析结果显示，在该试点计划实施期间，不同个体间存在显著异质性，且随时间呈现出值得关注的动态变化特征。本文的补充材料可在线获取，其中包含可用于复现本研究的标准化材料说明。