Continuous difference-in-differences with double/debiased machine learning

Aug 4, 2026

Uploaded by Liam

This paper extends difference-in-differences to settings with continuous treatments by identifying the average treatment effect on the treated (ATT) at any treatment intensity under a conditional parallel trends assumption. Estimation requires nuisance functions, especially the conditional density of the continuous treatment, which can generate substantial bias. The work proposes causal estimators within the double/debiased machine learning framework, proving asymptotic normality and consistent variance estimation. Methods are applied to the 1983 Medicare PPS reform study to obtain nonparametric treatment effects with richer detail.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages67

Reading Time2 h 49 min

AccessFree to download

Introduction

Continuous difference-in-differences and treatment intensity

Identification, covariates, and generalized propensity score

Double/debiased machine learning for nuisance estimation

Empirical illustration using Medicare PPS reform

arXiv :2408 . 10509v1 [ econ .EM] 20 Aug 2024 Continuous difference-in-differences with double/debiased machine learning Lucas Z. Zhang This Version: August 21, 2024 Abstract This paper extends difference-in-differences to settings involving continuous treatments. Specifically, the average treatment effect on the treated (ATT) at any level of continuous treatment intensity is identified using a conditional parallel trends assumption. In this framework, estimating the ATTs requires first estimating infinite-dimensional nuisance parameters, especially the conditional density of the continuous treatment, which can introduce significant biases. To address this challenge, estimators for the causal parameters are proposed under the double/debiased machine learning framework. We show that these estimators are asymptotically normal and provide consistent variance estimators. To illustrate the effectiveness of our methods, we re-examine the study by Acemoglu and Finkelstein (2008), which assessed the effects of the 1983 Medicare Prospective Payment System (PPS) reform. By reinterpreting their research design using a difference-in-differences approach with continuous treatment, we nonparametrically estimate the treatment effects of the 1983 PPS reform, thereby providing a more detailed understanding of its impact. Keywords: Difference-in-differences, causal inference, continuous treatment, machine learning 1 Introduction Difference-in-differences (DiD) is one of the most popular research designs in empirical work. While the more common DiD settings focus on binary or discrete multi-valued treatments, there has been increasing interest in DiD with continuous treatments. The main idea of continuous DiD is simple: the treatment group rarely receives the treatment at the same level, and the treatment effect can vary with the “dose/intensity” of the treatment. Therefore, instead of comparing the outcomes of the treated and the controls before and after the treatment at the group level, one can further examine the treated group and compare the outcomes at different treatment intensities. In fact, continuous treatment is prevalent in many empirical settings. For instance, each affected individual can have varied exposure to policy interventions, marketing campaigns, or environmental pollutants, all of which can be modeled as continuous treatments. Several recent studies in various fields have considered DiD with continuous treatments, including the study by Zeng et al.(2022) on the impact of shutting down online advertising sites, Cook et al. (2023)’s work on racial discrimination in public accommodations, and Ananat et al. (2022)’s study on the effects of the expanded child tax credit. While continuous DiD is popular among empirical studies, its theoretical foundation is still limited. A few recent studies have started to fill this gap, notably Callaway et al. (2024), D’Haultfoeuilleet al. (2023), and de Chaisemartin et al. (2022) . For instance, Callaway et al. (2024) examine continuous DiD in the context of the commonly used two-way fixed effect (TWFE) regression setting. Concurrently, D’Haultfoeuille et al. (2023) generalize the change-in-changes model studied in Athey and Imbens (2006) to continuous treatment. On the other hand, de Chaisemartin et al. (2022) study the average slope of stayers in the continuous DiD setting. In contrast to these studies, our results build upon the semiparametric framework proposed in Abadie (2005), broadening its applicability to settings involving continuous treatments. The main advantage of our approach is that it explicitly accounts for the presence of covariatesand focuses directly on causal parameters: the average treatment effect on the treated (ATT) at any given continuous treatment intensity. As noted in Abadie (2005), the (unconditional) parallel trends assumption can be restrictive if there are covariates that affect outcome dynamics and their distributions differ between c

FAQs

How does the paper extend difference-in-differences to continuous treatments?

It identifies the ATT at any continuous treatment intensity using a conditional parallel trends assumption, rather than relying on binary or discrete treatment changes.

Why does estimating nuisance parameters create bias in continuous DiD?

The causal parameter depends on an infinite-dimensional nuisance object, particularly the conditional density of the continuous treatment, and naive machine-learning nuisance estimates can introduce substantial bias and overfitting effects.

What role does double/debiased machine learning play in the proposed estimation?

The framework uses orthogonalization and cross-fitting to reduce the impact of nuisance estimation error, yielding asymptotically normal causal estimators with consistent variance estimates.

Aug 4, 2026

Uploaded by Liam