Semiparametric Difference-in-Differences Estimation With Missing Not at Random Data: A Shadow Variable Approach

Abstract: This paper considers a semiparametric difference-in-differences (DID) framework for identifying and estimating treatment effects on the treated (ATT) when outcomes are missing not at random (MNAR), and a fully observed shadow variable is available. The shadow variable is assumed to be associated with the outcome evolution but independent of the missingness process, conditional on covariates and the possibly unobserved outcome evolution. We establish the identification conditions, derive the corresponding identification results and estimation algorithm, and evaluate the finite-sample performance of the proposed estimator through simulation studies and a real data application.

• The paper introduces a novel shadow variable approach to identify ATT under MNAR conditions in DID designs.
• It develops a semiparametric estimator using an odds ratio model that corrects bias compared to traditional MAR and naive methods.
• Monte Carlo simulations and an empirical study demonstrate its potential to yield unbiased ATT estimates in complex survey settings.

Semiparametric Difference-in-Differences Estimation with MNAR Data and Shadow Variables

Introduction and Motivation

This paper addresses the identification and estimation of the average treatment effect on the treated (ATT) in a difference-in-differences (DID) design with missing not at random (MNAR) post-treatment outcomes, utilizing a fully observed shadow variable. Traditional DID methods and recent semiparametric and doubly robust extensions require the missing at random (MAR) assumption, which often does not hold in settings such as household surveys, where nonresponse is linked to unobserved outcome values.

Existing solutions for MNAR DID typically rely on a "bespoke" instrumental variable (IV) that must influence missingness but not the outcome evolution, conditional on covariates. However, finding such an IV is challenging, especially in limited survey data. In contrast, the shadow variable approach leverages covariates associated with the outcome but—critically—independent of the missingness process when conditioning on other covariates and outcomes, providing a practical alternative for empirical applications.

Framework and Methodological Advances

The paper considers a standard DID setup with two periods (pre- and post-treatment) and two groups (treatment, control). Outcomes may be systematically missing in the post-treatment period, with missingness depending potentially on unobserved outcome evolution. Key assumptions include conditional parallel trends and appropriate overlap conditions. The primary identification target is: $\tau = \mathbb{E} [Y_{1}(1)-Y_{1}(0) \mid D=1],$ where outcomes may be unobserved for some units in the post-treatment period.

Under the MAR assumption, the ATT is identifiable using inverse probability weighting (IPW) based on observed data. However, when missingness is MNAR, additional structure is required. The main innovation of this work is to introduce a shadow variable $Z$, fully observed, that allows for nonparametric (or semiparametric, under certain model specifications) identification of the ATT even in the MNAR setting.

Shadow Variable Assumptions

$Z$ is valid if (i) independent of missingness given outcome evolution, treatment, and other covariates, and (ii) associated with the outcome evolution conditional on treatment and covariates. This exclusion restriction is weaker and more feasible in practice than that required for a valid IV.

Identification via Odds Ratio Model

Building on prior literature on shadow variables [e.g., Miao et al.], the paper employs an odds ratio model to encode the relationship between missingness and the outcome evolution: $$OR(\Delta Y, D, U, Z) = \frac{p(\Delta Y \mid D, R=0, U, Z) \cdot p(\Delta Y = 0 \mid D, R=1, U, Z)}{p(\Delta Y \mid D, R=1, U, Z) \cdot p(\Delta Y = 0 \mid D, R=0, U, Z)}.$$ This formulation enables recovery of the MNAR missingness mechanism under conditions including completeness of the conditional law $p(\Delta Y \mid D, R=1, U, Z)$ with respect to $Z$, which is satisfied in many common models (e.g., when $Z$ and $\Delta Y$ are jointly continuous).

ATT Estimation under MNAR

Given the odds ratio identification, an estimator for the ATT under MNAR is constructed through a four-stage procedure:

1. Specify working models for the treatment propensity score, baseline response mechanism, and odds ratio.
2. Use the odds ratio to recover the nonignorable missingness mechanism.
3. Solve GMM estimating equations for all model parameters.
4. Compute the ATT estimator via IPW using estimated quantities.

The resulting estimator is consistent and asymptotically normal if all working models are correct and retains double-robustness properties analogous to those in the MAR setting, provided appropriate linearity conditions.

Numerical Results and Empirical Illustration

Simulation Study

Extensive Monte Carlo simulations compare the naive DID, MAR DID, and proposed MNAR DID estimators under both MAR and MNAR missingness. Key findings:

• Under MAR, both MAR and MNAR methods are unbiased, as the latter nests the former theoretically.
• Under MNAR, only the proposed estimator is unbiased. Both the naive (complete-case) and MAR estimators show appreciable bias.
• The MNAR estimator not only reduces bias but often achieves greater efficiency due to sample reweighting.

Empirical Application: Household Debt and Policy Evaluation

Using Chinese Household Finance Survey (CHFS) data on the impact of the two-child policy on household debt, the method is applied with household registration (hukou) as the shadow variable. Findings:

• Estimated MNAR intensity parameter ($\hat{\gamma}$) is nonzero and economically meaningful, indicating nonignorably missing debt outcomes correlated with unobserved debt increases.
• The estimated ATT falls under the MNAR method ($0.234$) relative to both MAR ($Z$0) and matched DID ($Z$1), implying conventional analyses may overstate the policy effect by failing to account for selection.
• Bootstrap standard errors reflect the cost of weak shadow instruments and small treatment samples.

Implications and Future Research

This work substantially expands the identification frontier for ATT in DID designs with MNAR data. Practical ramifications include:

• Broader applicability of DID in empirical fields with nonignorable missingness (e.g., survey nonresponse, longitudinal studies).
• Viable estimation when supplemental instruments are lacking, given the presence of a valid shadow variable.
• The approach provides not only point identification and estimation of the ATT, but also circumstances for model checking and calibration vis-à-vis the missingness mechanism.

Theoretically, the framework bridges literatures on MNAR data, causal inference, and semiparametric efficiency. The methodology paves the way for:

• Development of efficient influence functions for the MNAR DID parameter (extending work, e.g., by Miao et al.).
• Multiply robust procedures that combine flexible machine learning models for nuisance estimation, given cross-fitting and higher-order correction schemes.
• Sensitivity analyses for potential shadow variable violations, and extensions to multi-period/multi-treatment DID and settings where both pre- and post-treatment outcomes may be missing.

Conclusion

The paper delivers a general, robust, and implementable semiparametric framework for difference-in-differences estimation in the presence of MNAR data, leveraging the shadow variable approach. Under standard regularity and identification conditions, ATT estimation remains feasible and demonstrably improves upon traditional procedures in both simulated and empirical domains. The introduced methodology thus substantially enhances the empirical relevance of DID estimation in contemporary panel and survey research.