Matrix Completion for Causal Inference

Matrix completion for causal inference estimates treatment effects in panel data by treating untreated potential outcomes of treated unit-periods as missing and imputing them with a low-rank matrix estimated by nuclear norm regularization.

인과추론을 위한 행렬완성법은 처치된 단위-시점의 비처치 잠재결과를 결측으로 보고 핵노름 정규화로 추정한 저계수(low-rank) 행렬로 대체하여 패널 자료의 처치효과를 추정하는 방법이다.

How do I estimate a treatment effect in panel data when few units are treated and timing varies?

처치 단위가 적고 처치 시점이 다른 패널 자료에서 처치효과를 어떻게 추정하나요?

The matrix completion estimator fills in the counterfactual untreated outcomes for treated unit-periods using a low-rank approximation of the outcome matrix plus unit and time fixed effects, then compares observed with imputed outcomes. It fits panels with staggered or irregular treatment timing and unobserved factors that affect units differently over time, bridging synthetic control and fixed effects approaches.

When to use Matrix Completion for Causal Inference

• Panel data with many units and periods where some units become treated
• Staggered or irregular treatment adoption
• Concern that unobserved time-varying factors affect units differently

When to use something else

• There is one treated unit with a long pre-period: use the synthetic control method
• Parallel trends are credible and timing is simultaneous: use difference in differences

Assumptions and requirements

• The untreated outcome matrix is approximately low rank
• Treatment assignment depends on observed outcomes and latent factors, not on unobserved shocks
• Enough untreated unit-periods, including pre-treatment periods for treated units
• Regularization parameter chosen by cross-validation

What the analysis reports

• Average treatment effect on the treated with bootstrap or placebo-based uncertainty
• Imputed counterfactual outcomes for treated units
• Effects by period relative to treatment
• Placebo or pre-treatment fit diagnostics

How to report it

The matrix completion estimator yielded an average treatment effect on the treated of [value] (95% CI [[value], [value]]), with pre-treatment placebo effects close to zero.

Related methods

• Synthetic Control Method
• Difference in Differences (DiD)
• Interactive Fixed Effects Model

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