[Paper Review] Simple Difference-in-Differences Estimation in Fixed-T Panels

Nicholas M. Brown, Kyle Butts+1 more|arXiv (Cornell University)|Jan 26, 2023

Global trade and economicsEconomics, Econometrics and Finance3 citations

TL;DR

This paper proposes a new difference-in-differences estimator, C²ED², for fixed-T panels that remains valid under interactive fixed effects and allows for treatment effects on covariates. By leveraging the common correlated effects (CCE) approach, it enables consistent estimation of both direct and indirect treatment effects even when T is small and the parallel trends assumption fails due to unobserved heterogeneity.

ABSTRACT

The present paper proposes a new treatment effects estimator that is valid when the number of time periods is small, and the parallel trends condition holds conditional on covariates and unobserved heterogeneity in the form of interactive fixed effects. The estimator also allow the control variables to be affected by treatment and it enables estimation of the resulting indirect effect on the outcome variable. The asymptotic properties of the estimator are established and their accuracy in small samples is investigated using Monte Carlo simulations. The empirical usefulness of the estimator is illustrated using as an example the effect of increased trade competition on firm markups in China.

Motivation & Objective

• Address the limitation of standard difference-in-differences estimators when the number of time periods T is small and parallel trends fail due to unobserved interactive fixed effects.
• Develop a computationally simple estimator that does not require large T or known factor dimension, overcoming constraints of existing GMM or PCA-based methods.
• Allow for treatment effects on control variables, enabling separation of direct and indirect treatment effects on the outcome.
• Establish asymptotic validity under minimal assumptions, relying only on large N and the CCE framework.
• Provide a practical, implementable estimator that supports standard inference and is robust to unobserved common trends and factor loadings.

Proposed method

• Estimate common factors using cross-sectional averages of the outcome and covariates from the never-treated group.
• Use the estimated factors to jointly estimate slope coefficients of control variables and heterogeneous factor loadings via OLS.
• Impute untreated values of covariates in post-treatment periods using the estimated model parameters.
• Construct counterfactual outcomes for treated units by plugging imputed covariates and estimated coefficients into the structural model.
• Compute the average treatment effect on the treated (ATT) as the difference between observed and estimated counterfactual outcomes.
• Leverage the CCE approach to avoid non-convex optimization and ensure closed-form, computationally efficient estimation.

Experimental results

Research questions

• RQ1 Can a simple, computationally efficient difference-in-differences estimator be developed that remains valid when T is fixed and the parallel trends assumption fails due to interactive fixed effects?
• RQ2 To what extent can treatment effects on covariates be accounted for in DD estimation, and can direct and indirect effects be separated?
• RQ3 How does the performance of the proposed C²ED² estimator compare to existing methods in small-sample settings with limited time periods?
• RQ4 Does the inclusion of treatment-affected covariates lead to biased estimates in standard DD models, and can the new estimator correct for this?
• RQ5 What is the empirical relevance of distinguishing direct and indirect treatment effects in real-world applications, such as trade liberalization impacts?

Key findings

• The C²ED² estimator is consistent and asymptotically normal under standard regularity conditions, with validity relying only on large N and the CCE framework.
• Monte Carlo simulations confirm good finite-sample performance, with accurate size and power even when T is small.
• In the empirical application to China’s WTO accession, the estimated average ATT on markup dispersion is approximately -0.1 and statistically significant.
• The indirect effect—via changes in TFP dispersion—accounts for nearly half of the total ATT, indicating substantial transmission through covariates.
• Pre-treatment ATT estimates are close to zero, supporting the validity of the identification strategy and the absence of pre-trends.
• Conditioning on treatment-affected covariates in standard models leads to a 50% underestimation of the total ATT, highlighting the importance of separating direct and indirect effects.