Covariate-Adjusted Deep Causal Learning for Heterogeneous Panel Data Models

TL;DR

This work tackles unit-specific heterogeneous treatment effects in causal panel data with covariate influences and non-random missingness. It introduces Covariate-Adjusted Deep Causal Learning (CoDEAL), combining a nonlinear covariate adjustment via a deep neural network with a nonlinear latent-factor structure learned by a multi-output autoencoder to perform causal matrix completion under staggered adoption designs. The paper provides theoretical convergence guarantees for the estimated counterfactuals, and demonstrates through extensive simulations that CoDEAL consistently outperforms benchmark methods across linear and nonlinear factor and covariate settings. An empirical application to OxCGRT data on government interventions illustrates CoDEAL’s practical impact on counterfactual analysis in real-world policy contexts. The framework offers a scalable, unified approach to covariate-adjusted heterogeneous causal inference in panel data and can be extended to tensor data and multiple treatments in future work.

Abstract

This paper studies the task of estimating heterogeneous treatment effects in causal panel data models, in the presence of covariate effects. We propose a novel Covariate-Adjusted Deep Causal Learning (CoDEAL) for panel data models, that employs flexible model structures and powerful neural network architectures to cohesively deal with the underlying heterogeneity and nonlinearity of both panel units and covariate effects. The proposed CoDEAL integrates nonlinear covariate effect components (parameterized by a feed-forward neural network) with nonlinear factor structures (modeled by a multi-output autoencoder) to form a heterogeneous causal panel model. The nonlinear covariate component offers a flexible framework for capturing the complex influences of covariates on outcomes. The nonlinear factor analysis enables CoDEAL to effectively capture both cross-sectional and temporal dependencies inherent in the data panel. This latent structural information is subsequently integrated into a customized matrix completion algorithm, thereby facilitating more accurate imputation of missing counterfactual outcomes. Moreover, the use of a multi-output autoencoder explicitly accounts for heterogeneity across units and enhances the model interpretability of the latent factors. We establish theoretical guarantees on the convergence of the estimated counterfactuals, and demonstrate the compelling performance of the proposed method using extensive simulation studies and a real data application.

Figures (4)

• Figure 1: A graphical illustration of the proposed CoDEAL in a four-block design. Gray and orange refer to the untreated and treated blocks, respectively. Inputs are observed outcomes $\mathbf{Y}\in\mathbb{R}^{N\times T}$, covariates $\mathrm{\bf X}\in\mathbb{R}^{N\times P}$, indicator matrix $\boldsymbol{\Omega}\in\mathbb{R}^{N\times T}$. Output is the estimated unit-specific ATT $\widehat{\tau}_i$.
• Figure 2: Examples of constructing a four-block submatrix $\mathrm{\bf Y}^{(\xi_0,\eta_0)}$ to estimate $\mathrm{\bf Y}_{(\xi_0,\eta_0)}(0)$. Here, $r=5$. The question mark denotes the block of interest, with $(\xi_0,\eta_0)$ = (3,4) (Left) and (5,3) (Right). The associated four blocks are designed by green ($\mathrm{\bf Y}_{\mathcal{A}_{(\xi_0,\eta_0)}}$), red ($\mathrm{\bf Y}_{\mathcal{B}_{(\xi_0,\eta_0)}}$), blue ($\mathrm{\bf Y}_{\mathcal{C}_{(\xi_0,\eta_0)}}$), and orange ($\mathrm{\bf Y}_{\mathcal{D}_{(\xi_0,\eta_0)}}$) as in \ref{['eq:staggered_design']}.
• Figure 3: Indicator matrix of the implementation of mandatory vaccination policy. Blocks with darker color refer to policy executed.
• Figure 4: Comparison of the total confirmed cases (Left) and deaths (Right) across policy-in-effect states between observed results with the mandatory vaccination policy executed and estimated counterfactuals if no policy. Values are reported in ${\log_{10}}$ scale, and plots are visualized with 14-day moving averages.

Theorems & Definitions (2)

• Definition 1: Deep ReLU network class
• Theorem 2.1