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Forecasted Treatment Effects

Irene Botosaru, Raffaella Giacomini, Martin Weidner

Abstract

We consider estimation and inference of the effects of a policy in the absence of an untreated or control group. We obtain unbiased estimators of individual (heterogeneous) treatment effects and a consistent and asymptotically normal estimator of the average treatment effect. Our estimator averages, across individuals, the difference between observed post-treatment outcomes and unbiased forecasts of their counterfactuals, based on a (short) time series of pre-treatment data. The paper emphasizes the importance of focusing on forecast unbiasedness rather than accuracy when the end goal is estimation of average treatment effects. We show that simple basis function regressions ensure forecast unbiasedness for a broad class of data generating processes for the counterfactuals. In contrast, forecasting based on a specific parametric model requires stronger assumptions and is prone to misspecification and estimation bias. We show that our method can replicate the findings of some previous empirical studies but it does so without using an untreated or control group.

Forecasted Treatment Effects

Abstract

We consider estimation and inference of the effects of a policy in the absence of an untreated or control group. We obtain unbiased estimators of individual (heterogeneous) treatment effects and a consistent and asymptotically normal estimator of the average treatment effect. Our estimator averages, across individuals, the difference between observed post-treatment outcomes and unbiased forecasts of their counterfactuals, based on a (short) time series of pre-treatment data. The paper emphasizes the importance of focusing on forecast unbiasedness rather than accuracy when the end goal is estimation of average treatment effects. We show that simple basis function regressions ensure forecast unbiasedness for a broad class of data generating processes for the counterfactuals. In contrast, forecasting based on a specific parametric model requires stronger assumptions and is prone to misspecification and estimation bias. We show that our method can replicate the findings of some previous empirical studies but it does so without using an untreated or control group.
Paper Structure (30 sections, 5 theorems, 57 equations, 7 figures, 8 tables)

This paper contains 30 sections, 5 theorems, 57 equations, 7 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Baseline case: Unobserved-Components FAT
  3. Parameter of interest and estimator
  4. Unbiased forecasts of counterfactuals
  5. Stationary or stochastic trends DGPs
  6. Deterministic trends DGPs
  7. Choice of basis functions and tuning parameters
  8. Individual treatment timing and limited anticipation
  9. Balanced panels and pooled estimation
  10. Parametric-Model FAT
  11. Homogeneous parameters
  12. Heterogeneous parameters
  13. Simulation studies
  14. Performance across forecast horizons
  15. Estimators.
  16. ...and 15 more sections

Key Result

Lemma 1

For each $i=1,\ldots,n$, let the forecast $\widehat{y}_{i\tau+h}(0)$, $h\geq 1$, be a function of $\{y_{it}\}_{t\leq\tau}$. Let Assumptions Unbiasedness and ass:Sampling hold. ThenEstimation of the variance $\bar{\sigma}_n^2$ is discussed in Appendix app:VarEstimation.

Figures (7)

  • Figure 1: Average suicide rates across states as a function of time-to-adoption.
  • Figure 2: FAT estimates for different forecast horizons (horizontal axis) and polynomial orders (different colors). To the left of the dashed vertical line, the figure shows placebo FAT (for $h=1$) as a function of the lag (horizontal axis) and polynomial order (different lines). The error bars in both panels represent $95\%$ confidence intervals.
  • Figure 3: Overdose mortality rate as a function of time to adoption using not-yet-treated-states as control units.
  • Figure 4: Overdose mortality rate as a function of time to adoption via generalized SC.
  • Figure 5: Log mortality rate averaged across states as a function of time-to-adoption.
  • ...and 2 more figures

Theorems & Definitions (17)

  • Lemma 1: Consistency and asymptotic normality
  • Remark 1
  • Remark 2
  • Theorem 1: Unbiasedness for stationary or stochastic trends DGP
  • Remark 3
  • Definition 1: Forecasts via basis function regressions
  • Lemma 2
  • Theorem 2: Unbiasedness for deterministic trend DGPs
  • Remark 4
  • Remark 5
  • ...and 7 more