Dynamic covariate balancing: estimating treatment effects over time with potential local projections

1. Problem Statement

The paper addresses the challenge of estimating Average Treatment Effects (ATE) for specific treatment histories in panel data settings where treatments change dynamically over time.

• Context: Researchers observe $n$ independent units over $T$ periods with time-varying covariates ($X$), outcomes ($Y$), and binary treatments ($D$).
• The Core Difficulty: Unlike static cross-sectional studies or standard Difference-in-Differences (DiD) designs, treatment assignment at time $t$ can depend on the entire history of past outcomes, covariates, and treatments. This creates dynamic selection.
• Limitations of Existing Methods:
  • Inverse Probability Weighting (IPW/AIPW): These methods require estimating the propensity score (the probability of a specific treatment history). In dynamic settings, the propensity score is the joint probability of the entire history, which can be near zero for long horizons, leading to extreme variance and instability. Furthermore, in high-dimensional settings, correctly specifying the propensity score is difficult and often requires "doubly robust" conditions (consistent estimation of both the outcome and propensity models) that are hard to satisfy.
  • Standard Local Projections (LP) & DiD: Standard LPs model observed outcomes, which imposes restrictions on the distribution of treatment assignments. If treatments are dynamically selected based on past outcomes, standard LPs yield biased estimates. DiD relies on parallel trends, which is violated when units dynamically select treatments based on their own history.
  • High-Dimensionality: When the number of covariates ($p$) exceeds or is comparable to the sample size ($n$), modeling the complex non-linear relationships between all past variables and current outcomes leads to massive estimation errors.

2. Methodology: Dynamic Covariate Balancing (DCB)

The authors propose a novel estimator called Dynamic Covariate Balancing (DCB). The method combines potential local projections with sequential covariate balancing to avoid modeling the propensity score directly.

A. Theoretical Framework: Potential Local Projections

Instead of modeling observed outcomes, the authors model the conditional expectation of potential outcomes as a linear function of past history.

• Assumption: The expected potential outcome $E[Y_{i,T}(d_{1:T}) | \text{History}]$ is approximately linear in the high-dimensional history vector $H_{i,t}$.
• Recursive Identification: Using the logic of Marginal Structural Models (MSM) and Sequential Ignorability, the target parameter $\mu_T(d_{1:T})$ is identified by recursively projecting potential outcomes onto past histories.
  • $E[Y_{i,T}(d_{1:T}) | H_{i,T}] = H_{i,T} \beta^{(T)}$
  • $E[H_{i,T} \beta^{(T)} | H_{i,T-1}] = H_{i,T-1} \beta^{(T-1)}$
  • ...and so on, back to baseline covariates.

B. The Estimation Algorithm

The estimator proceeds in two main stages:

1. Recursive Coefficient Estimation (Lasso):

   • The authors estimate the coefficients ($\beta$) of the linear potential outcome models recursively using Lasso (or other high-dimensional penalized regression).
   • Starting from the final period $T$, they regress outcomes on the history $H_T$. They then use the fitted values from this step as the "outcome" for the regression at period $T-1$, and so on, back to period 1.
   • This allows for high-dimensional covariates without needing to model the evolution of every covariate over time.

2. Sequential Dynamic Balancing (Reweighting):

   • To correct for the bias introduced by high-dimensional estimation errors and dynamic selection, the method constructs balancing weights ($\gamma$) sequentially.
   • Objective: Minimize the variance of the weights (minimize $\sum \gamma_i^2$) subject to dynamic balancing constraints.
   • Constraints:
     • At period $t=1$: Balance the distribution of baseline covariates $X_1$ between treated and control units.
     • At period $t>1$: Balance the distribution of the entire history $H_t$ between units with the target treatment path and the rest, reweighted by the weights obtained in the previous period.
   • This is formulated as a sequence of Quadratic Programs (QP).
   • Key Insight: By balancing the history $H_t$ weighted by previous weights, the method ensures that the product of the estimation error ($\|\hat{\beta} - \beta\|_1$) and the covariate imbalance is negligible, achieving a "strongly doubly robust" property without needing a correctly specified propensity score.

3. Key Contributions

1. Dynamic Balancing Conditions: The paper derives the first set of balancing conditions specifically for dynamic treatment settings. Unlike static balancing (which balances $X$), dynamic balancing requires balancing the history $H_t$ sequentially, reweighted by previous period weights.
2. Robustness to Propensity Score Misspecification: Unlike AIPW, DCB does not require consistent estimation of the propensity score. It only requires that the outcome model coefficients are estimated at a rate of $O_p(n^{-1/4})$ (achievable via Lasso under sparsity). This bypasses the "product of rates" condition required by standard doubly robust estimators in high dimensions.
3. Stability in High Dimensions: The method minimizes the $L_2$ norm of the weights, ensuring they do not explode (a common issue with IPW in dynamic settings where joint probabilities are tiny).
4. Theoretical Guarantees:
   • Existence: Proves that feasible balancing weights exist even when $p \gg n$, provided an overlap condition holds.
   • Convergence: Establishes a parametric convergence rate ($O_p(n^{-1/2})$) for the treatment effect estimator, even with high-dimensional covariates growing with $n$.
   • Inference: Provides a valid asymptotic normal distribution for inference, allowing for confidence intervals.

4. Results

Numerical Studies (Simulations)

• Performance: DCB significantly outperforms competitors (AIPW, IPW, Double Lasso, DiD, Standard LP) in terms of Mean Squared Error (MSE), particularly in:
  • High-dimensional settings ($p > n$).
  • Poor overlap scenarios (where propensity scores are close to 0 or 1).
  • Longer time horizons ($T > 2$), where IPW variance explodes.
• Misspecification: DCB remains robust even when the true outcome model is non-linear (misspecified), whereas methods relying on correct propensity score specification fail.
• Comparison: In simulations, DCB often outperforms even the "Oracle" AIPW (which knows the true propensity score) because the variance reduction from balancing weights outweighs the bias reduction from knowing the true propensity score in finite samples.

Empirical Application

• Study: Re-evaluating the effect of Democracy on Economic Growth using data from Acemoglu et al. (2019).
• Findings:
  • Standard Local Projections and the linear regression used by Acemoglu et al. (2019) yield significantly smaller point estimates.
  • DCB estimates show a statistically significant positive impact of democracy on GDP growth after 3 years, with a magnitude roughly twice as large as the original study's estimates.
  • Diagnosis: The original study and standard IPW methods suffer from imbalance in lagged GDP (a key predictor of treatment), leading to bias. DCB successfully balances these covariates, reducing bias and increasing precision.
  • Weight Stability: Figure 4 and Table 2 demonstrate that IPW weights become highly unstable (high variance) for longer treatment histories, whereas DCB weights remain stable.

5. Significance

This paper provides a critical advancement in causal inference for dynamic panel data.

• Practical Utility: It offers a feasible, computationally efficient (solving a sequence of QPs) method for researchers dealing with time-varying treatments and high-dimensional controls, a common scenario in economics, political science, and biostatistics.
• Theoretical Breakthrough: It resolves the tension between high-dimensional inference and dynamic selection by showing that balancing is superior to propensity score weighting in these contexts. It shifts the burden of consistency from the propensity score (hard to estimate in dynamics) to the outcome model (easier to approximate linearly).
• Policy Relevance: By providing more accurate estimates of long-run treatment effects (e.g., the long-term impact of policy changes), it helps avoid the underestimation of effects that plagues standard dynamic models.

In summary, Viviano and Bradic introduce a robust, high-dimensional framework that replaces unstable inverse probability weighting with sequential covariate balancing, enabling reliable inference on dynamic treatment effects where previous methods fail.