Bayesian Propensity Score-Augmented Latent Factor Models for Causal Inference with Time-Series Cross-Sectional Data

1. Problem Statement

The paper addresses the challenge of estimating causal treatment effects in Time-Series Cross-Sectional (TSCS) data (panel data) where unobserved confounders exist.

• Limitations of Existing Methods:
  • Standard Propensity Score (PS) methods (e.g., matching, weighting) typically rely on the assumption of conditional ignorability (no unobserved confounders), which is often violated in TSCS settings.
  • Existing panel data methods (e.g., Interactive Fixed Effects, Difference-in-Differences) often model unobserved confounders via latent factors but fail to explicitly model the treatment assignment mechanism under latent ignorability. They treat latent factors as nuisance parameters in the outcome equation only.
  • Fully Bayesian approaches that attempt to model both stages simultaneously often suffer from the model feedback problem, where the estimation of latent factors in the outcome model biases the treatment assignment model (and vice versa), leading to inaccurate counterfactual predictions.

2. Methodology: The PS-LFM Framework

The author proposes a Bayesian Propensity Score-Augmented Latent Factor Model (PS-LFM) that jointly models treatment assignment and potential outcomes while addressing unobserved confounding and model feedback.

A. Model Setup

• Data: $N$ units over $T$ time periods with staggered treatment adoption.
• Assumptions:
  • Latent Ignorability: Potential outcomes are independent of treatment assignment conditional on observed covariates ($Z_i$) and unobserved, time-invariant latent confounders ($\gamma_i$).
  • Exclusion Restriction: No anticipation or carryover effects.
• The Joint Model (Equation 7):
  1. Treatment Assignment Stage: Modeled as a Probit model where the probability of treatment depends on observed covariates ($Z_i$) and latent factor loadings ($\gamma_i$).
     $$W_i = 1\{Z_i'\lambda_z + \gamma_i'\lambda_\gamma + \nu_i \geq 0\}$$
     This defines the propensity score: $ps(Z_i, \gamma_i) = \Phi(Z_i'\lambda_z + \gamma_i'\lambda_\gamma)$.
  2. Outcome Stage: The potential outcome under control ($Y_{it}(0)$) is modeled as a function of observed covariates, the propensity score (via stratification or continuous inclusion), and a residual latent factor structure.
     $$Y_{it}(0) = g(Z_i, ps(Z_i, \gamma_i))'\beta_t + \gamma_i'f_t + \epsilon_{it}$$
     Here, $g(\cdot)$ allows the propensity score to enter the model (e.g., via stratification), capturing heterogeneity across propensity strata. The term $\gamma_i'f_t$ captures residual unobserved variation.

B. Addressing Model Feedback

A fully Bayesian analysis (sampling from the joint posterior of all parameters) creates a feedback loop: the outcome model informs the latent factors, which inform the propensity score, which in turn biases the outcome model.

• Solution: The author adopts an Approximate Bayesian Analysis (inspired by Zigler et al., 2013).
  • Cutting the Loop: The algorithm updates the latent factor loadings ($\tilde{\gamma}_i$) and random effects using only the outcome model, ignoring the treatment assignment likelihood during these updates.
  • Post-Processing Rotation: To ensure identifiability of the latent factors (a common issue in factor models), the algorithm applies a rotation to the draws of factors and loadings to satisfy normalization constraints (e.g., Bai, 2009) after the outcome model updates.
  • Sequential Update: The rotated loadings are then inserted into the treatment assignment model to update the propensity score parameters, breaking the simultaneous feedback.

C. Estimation Details

• Shrinkage Priors: Hierarchical shrinkage priors (e.g., Horseshoe or similar) are used on factor loadings and time-varying coefficients to facilitate model selection (determining the number of latent factors and which covariates matter).
• Implementation: A custom MCMC sampler (Algorithm 1) is used, implemented in the R package bpCausal.

3. Key Contributions

1. Explicit Treatment Assignment Modeling: Unlike standard causal panel models, PS-LFM explicitly models the treatment assignment mechanism as a function of latent confounders, satisfying the requirements of latent ignorability.
2. Propensity Score Augmentation: It integrates the propensity score (derived from both observed and latent confounders) directly into the outcome model, allowing for flexible modeling of heterogeneity across propensity strata.
3. Approximate Bayesian Inference: It provides a robust solution to the model feedback problem in Bayesian causal inference with latent variables, ensuring stable counterfactual predictions.
4. Extension to TSCS: It extends Bayesian propensity score analysis (previously limited to cross-sectional data) to complex time-series cross-sectional settings with staggered adoption.

4. Results

A. Monte Carlo Simulations

• Setup: Simulated data with $N=200$, $T=50$, staggered adoption, and unobserved latent confounders.
• Comparisons: PS-LFM was compared against:
  • Oracle (known propensity scores and factors).
  • DM-LFM (Standard Bayesian Latent Factor Model without PS augmentation).
  • CS-DID (Callaway & Sant'Anna, 2021).
  • FEct (Fixed Effects Counterfactual).
• Findings:
  • PS-LFM achieved the lowest bias and RMSE among non-oracle methods and maintained coverage rates near the nominal 95% level.
  • DM-LFM and FEct exhibited higher bias and lower coverage because they ignored the propensity score stratification, failing to capture heterogeneity across strata.
  • CS-DID performed poorly because the data-generating process relied heavily on latent factors, which CS-DID (relying on observed covariates) could not capture.

B. Empirical Application: Political Connections and Firm Value

• Context: Re-analysis of the effect of political connections to Timothy Geithner on financial firm stock returns following his 2008 nomination as Treasury Secretary (based on Acemoglu et al., 2016).
• Findings:
  • Both PS-LFM and the benchmark DM-LFM produced similar pre-treatment trends.
  • Key Difference: On the announcement day and the 9th trading day post-announcement, the PS-LFM estimated abnormal returns with 95% credible intervals covering zero, whereas the DM-LFM intervals did not.
  • Interpretation: Incorporating propensity score stratification and latent factors changed the inference, suggesting the initial "significant" effect found by simpler models might be an artifact of unobserved heterogeneity.
  • Placebo Tests: Confirmed validity, showing effects near zero for pre-treatment periods.

5. Significance and Limitations

• Significance: The paper offers a rigorous framework for causal inference in observational panel data where unobserved confounding is likely. It bridges the gap between design-based propensity score methods and model-based latent factor approaches.
• Limitations:
  • Parametric Assumptions: The method relies on parametric forms for the treatment assignment (Probit) and outcome models. Mis-specification (e.g., incorrect stratification thresholds) can lead to bias.
  • Double Robustness: Due to the dependence of the outcome model on the estimated propensity score (which relies on the outcome model for latent factors), the method does not satisfy the classical double-robustness property (where consistency is guaranteed if either model is correct). It requires both models to be reasonably specified for consistent estimation under latent ignorability.

In summary, Liu's PS-LFM provides a sophisticated, flexible tool for researchers dealing with complex panel data, offering a way to explicitly account for unobserved confounders in the treatment assignment process while mitigating the statistical pitfalls of fully Bayesian feedback loops.