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

TL;DR

This paper introduces a new inferential framework for matrix completion with non-random missing data, leveraging nuclear norm regularization and debiasing to achieve asymptotic normality even with weak signals, applicable to causal panel data analysis.

Contribution

It proposes a novel approach to matrix completion that handles non-random missingness without requiring strong signals, using group-based estimation and debiasing techniques.

Findings

Effective estimation of missing entries when missing is not at random.

Asymptotic normality achieved with weak signals through debiasing.

Application to SEC Tick Size Pilot reveals heterogeneity and dynamics in treatment effects.

Abstract

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.