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Unifying regression-based and design-based causal inference in time-series experiments

Zhexiao Lin, Peng Ding

TL;DR

This paper develops a design-based framework for regression-based causal inference in time-series experiments, enabling consistent estimation of lagged treatment effects by regressing a centered, scaled treatment history on future outcomes. It proves consistency and a central limit theorem for a diverging number of lags, and shows that HAC variance estimators provide conservative inference for the design-based variances, even under misspecification. The framework extends naturally to continuous treatments and includes Fisher randomization tests for sharp and weak nulls. Empirical studies—both simulations and a real trading dataset—demonstrate accurate finite-sample performance and reveal heterogeneous lagged effects across contexts. The approach offers a robust, regression-based pathway to dynamic causal inference in single-unit and potentially panel settings, with extensions to interactions and general exposure mappings discussed as future directions.

Abstract

Time-series experiments, also called switchback experiments or N-of-1 trials, play increasingly important roles in modern applications in medical and industrial areas. Under the potential outcomes framework, recent research has studied time-series experiments from the design-based perspective, relying solely on the randomness in the design to drive the statistical inference. Focusing on simpler statistical methods, we examine the design-based properties of regression-based methods for estimating treatment effects in time-series experiments. We demonstrate that the treatment effects of interest can be consistently estimated using ordinary least squares with an appropriately specified working model and transformed regressors. Our analysis allows for estimating a diverging number of treatment effects simultaneously, and establishes the consistency and asymptotic normality of the regression-based estimators. Additionally, we show that asymptotically, the heteroskedasticity and autocorrelation consistent variance estimators provide conservative estimates of the true, design-based variances. Importantly, although our approach relies on regression, our design-based framework allows for misspecification of the regression model.

Unifying regression-based and design-based causal inference in time-series experiments

TL;DR

This paper develops a design-based framework for regression-based causal inference in time-series experiments, enabling consistent estimation of lagged treatment effects by regressing a centered, scaled treatment history on future outcomes. It proves consistency and a central limit theorem for a diverging number of lags, and shows that HAC variance estimators provide conservative inference for the design-based variances, even under misspecification. The framework extends naturally to continuous treatments and includes Fisher randomization tests for sharp and weak nulls. Empirical studies—both simulations and a real trading dataset—demonstrate accurate finite-sample performance and reveal heterogeneous lagged effects across contexts. The approach offers a robust, regression-based pathway to dynamic causal inference in single-unit and potentially panel settings, with extensions to interactions and general exposure mappings discussed as future directions.

Abstract

Time-series experiments, also called switchback experiments or N-of-1 trials, play increasingly important roles in modern applications in medical and industrial areas. Under the potential outcomes framework, recent research has studied time-series experiments from the design-based perspective, relying solely on the randomness in the design to drive the statistical inference. Focusing on simpler statistical methods, we examine the design-based properties of regression-based methods for estimating treatment effects in time-series experiments. We demonstrate that the treatment effects of interest can be consistently estimated using ordinary least squares with an appropriately specified working model and transformed regressors. Our analysis allows for estimating a diverging number of treatment effects simultaneously, and establishes the consistency and asymptotic normality of the regression-based estimators. Additionally, we show that asymptotically, the heteroskedasticity and autocorrelation consistent variance estimators provide conservative estimates of the true, design-based variances. Importantly, although our approach relies on regression, our design-based framework allows for misspecification of the regression model.
Paper Structure (44 sections, 18 theorems, 198 equations, 3 figures, 2 tables)

This paper contains 44 sections, 18 theorems, 198 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction to time-series experiments
  2. Setup, estimand and estimator
  3. Setup
  4. Causal estimand
  5. Theory
  6. Consistency and asymptotic normality
  7. HAC variance estimation
  8. Fisher randomization test
  9. Relationship between full OLS, marginal OLS and weighted least squares
  10. Comparisons with the existing estimands
  11. Extension to continuous treatment
  12. Empirical studies
  13. Simulations
  14. Real data analysis
  15. Discussion
  16. ...and 29 more sections

Key Result

Theorem 3.1

Assume Assumption asp:no-anticipation, Assumption asp:design, Assumption asp:ps, Assumption asp:aniasp:ani:i with $K \sum_{k=1}^T \theta_{T,k}/T \to 0$ or Assumption asp:mdepasp:mdep:i with $K m/T \to 0$, Assumption asp:moment, and $K^2/T \to 0$. Then we have

Figures (3)

  • Figure 1: Histograms of $\widehat{\tau}_k-\tau_k$ for $k=0,\ldots,5$.
  • Figure 2: $\lVert \widehat{\tau} - \tau \rVert_2$ versus $T$. The center bar is the median.
  • Figure 3: Normalized Frobenius error $\lVert \widehat{\bm{V}}-\bm{V} \rVert_{\mathrm F}/\lVert \bm{V} \rVert_{\mathrm F}$ versus $T$ for different bandwidths $L$.

Theorems & Definitions (40)

  • Example 2.1
  • Example 2.2: Linear model
  • Example 2.3: Autoregressive model
  • Example 2.4: Moving-average model
  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.1: Consistency
  • Theorem 3.2: Asymptotic normality
  • ...and 30 more