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Matrix Completion Methods for Causal Panel Data Models (1710.10251v5)

Published 27 Oct 2017 in math.ST, econ.EM, and stat.TH

Abstract: In this paper we study methods for estimating causal effects in settings with panel data, where some units are exposed to a treatment during some periods and the goal is estimating counterfactual (untreated) outcomes for the treated unit/period combinations. We propose a class of matrix completion estimators that uses the observed elements of the matrix of control outcomes corresponding to untreated unit/periods to impute the "missing" elements of the control outcome matrix, corresponding to treated units/periods. This leads to a matrix that well-approximates the original (incomplete) matrix, but has lower complexity according to the nuclear norm for matrices. We generalize results from the matrix completion literature by allowing the patterns of missing data to have a time series dependency structure that is common in social science applications. We present novel insights concerning the connections between the matrix completion literature, the literature on interactive fixed effects models and the literatures on program evaluation under unconfoundedness and synthetic control methods. We show that all these estimators can be viewed as focusing on the same objective function. They differ solely in the way they deal with identification, in some cases solely through regularization (our proposed nuclear norm matrix completion estimator) and in other cases primarily through imposing hard restrictions (the unconfoundedness and synthetic control approaches). The proposed method outperforms unconfoundedness-based or synthetic control estimators in simulations based on real data.

Citations (386)

Summary

  • The paper introduces a nuclear norm matrix completion estimator that imputes missing control outcomes to accurately assess causal effects using panel data.
  • It extends traditional methods by incorporating time-series dependency and robust regularization for improved causal inference in varied data configurations.
  • Simulations demonstrate that the proposed estimator consistently outperforms unconfoundedness and synthetic control approaches across diverse panel settings.

An Overview of Matrix Completion Methods for Causal Panel Data Models

The paper presents a novel approach to estimating causal effects using panel data through matrix completion techniques. The authors focus on settings where some units experience a treatment during specific periods, necessitating the estimation of counterfactual outcomes for treated unit/period combinations. The proposed method employs matrix completion estimators using observed control outcomes from untreated unit/periods to impute missing control outcomes for treated units/periods. This imputation results in a matrix that closely approximates the original incomplete matrix while maintaining a low rank based on the nuclear norm.

Theoretical Developments and Innovations

The authors extend the matrix completion literature by incorporating time series dependency structures, often occurring in social science applications, into the matrix of missing data. This innovation allows for a more realistic depiction of the data's dependency structure. The paper also elucidates the connections between matrix completion methods, interactive fixed effects models, and program evaluation approaches such as unconfoundedness and synthetic control methods. All these estimators share a common objective function, differing primarily in their approach to regularization and the identification of parameters.

The proposed method stands out by outperforming unconfoundedness-based and synthetic control estimators in simulations involving real data. Unlike traditional methods, the nuclear norm matrix completion estimator is adaptable to various matrices’ configurations, delivering consistently strong performance.

Estimator Details and Implementation

The matrix completion with nuclear norm minimization (MC-NNM) estimator is central to the paper. It aims to estimate a matrix representing the complete outcomes, partitioned into an observed set and an imputed set. The estimator relies on minimizing a penalized objective function where the penalty is imposed through the matrix's nuclear norm. The algorithm iteratively applies a singular value decomposition (SVD) shrinkage operator to achieve convergence, allowing for efficient estimation even in large data settings.

Simulation Studies and Practical Implications

The paper includes simulations based on real-world data such as the California Smoking Data and daily returns for a comprehensive set of stocks. These studies demonstrate the proposed method's robustness and superiority across various configurations and adoption patterns, including simultaneous and staggered adoption. The MC-NNM shows adaptability, excelling in both "thin" matrices with more units than time periods and "fat" matrices with more time periods than units. This characteristic enhances its practical applicability across diverse research settings involving panel data.

Theoretical Contributions and Future Directions

The authors present detailed consistency results for the MC-NNM estimator, including theorems and proofs underpinning the estimation error bounds. These theoretical advancements affirm the estimator's validity and scalability, emphasizing its strength when the matrix is low-rank or can be economically approximated by one.

Future research directions include extending the method to account for covariates in a more integrated manner, handling dependent error structures, and exploring different weighting schemes to enhance imputation accuracy. These extensions promise to increase the estimator's versatility and efficacy, especially in more complex, real-world applications that require causal inference from panel data.

Conclusion

The paper advances the methodological landscape for causal inference in panel data settings. The MC-NNM estimator offers a robust, flexible solution, seamlessly integrating into existing frameworks while providing superior imputation capabilities. Through its theoretical rigor and empirical demonstrations, the paper sets the stage for subsequent developments in causal inference methodologies leveraging the potential of matrix completion techniques.

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