Evaluation of continuous treatment

Authors

  • Elena Kotyrlo HSE University (Moscow, Russian Federation) Автор

DOI:

https://doi.org/10.22394/1993-7601-2025-80-93-116

Keywords:

обобщенный метод соответствия, функция отклика на непрерывное воздействие, метод разности разностей, оценка политики, несмещенный алгоритм машинного обучения, условный средний эффект воздействия

Abstract

Continuous treatment effect evaluation is complicated by confounders. The dose is commonly related to observed characteristics of the treated unit. This paper provides an overview of approaches for evaluating continuous treatment effects. They are 1) the dose-response function; 2) application to panel data based on difference-in-differences; and 3) conditional treatment effect evaluation with debiased machine learning. Empirical examples of the teleworkability effect on unemployment indicators during the COVID‑19 period illustrate the approaches.
Received: 7 June 2025 · Accepted: 21 August 2025

Agencies

Данная статья является результатом исследовательского проекта, реализованного в рамках Программы фундаментальных исследований Национального исследовательского университета «Высшая школа экономики» (НИУ ВШЭ).

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References

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Bia M., Mattei A. (2008). A Stata package for the estimation of the dose-response function through adjustment for the generalized propensity score. The Stata Journal, 8 (3), 354–373. DOI: 10.1177/1536867X1401400307.

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Kennedy E. H., Ma Z., McHugh M. D., Small D. S. (2017). Non-parametric methods for doubly robust estimation of continuous treatment effects. Journal of the Royal Statistical Society Series B: Statistical Methodology, 79 (4), 1229–1245. DOI: 10.1111/rssb.12212.

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Mattei A., Bia M. (2009). DoseResponse: Stata module to estimate dose-response function through adjustment for the generalized propensity score. Statistical Software Components S457096. Boston College Department of Economics.

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Semenova V., Chernozhukov V. (2021). Debiased machine learning of conditional average treatment effects and other causal functions. The Econometrics Journal, 24 (2), 264–289. DOI: 10.1093/ectj/utaa027.

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Sostero M., Milasi S., Hurley J., Fernandez-Macías E., Bisello M. (2020). Teleworkability and the COVID‑19 crisis: A new digital divide? JRC Working Papers Series on Labour, Education and Technology, 2020/05.

Wasserman L. (2006) All of nonparametric statistics. New York: Springer.

Wooldridge J. M. (2023). Simple approaches to nonlinear difference-in-differences with panel data. The Econometrics Journal, 26 (3), C31–C66.

Zhao S., van Dyk D. A., Imai K. (2020). Propensity score-based methods for causal inference in observational studies with non-binary treatments. Statistical Methods in Medical Research, 29 (3), 709–727. DOI: 10.1177/0962280219888745.

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Abadie A., Drukker D., Herr J. L., Imbens G. W. (2004). Implementing matching estimators for average treatment effects in Stata. The Stata Journal, 4 (3), 290–311. DOI: 10.1177/1536867X0400400307.

Belloni A., Chernozhukov V., Wei Y. (2016). Post-selection inference for generalized linear models with many controls. Journal of Business & Economic Statistics, 34, 606–619. DOI: 10.1080/07350015.2016.1166116.

Bia M., Flores C., Flores-Lagunes A., Mattei A. (2014). A Stata package for the application of semiparametric estimators of dose-response functions. The Stata Journal, 14 (3), 580–604. DOI: 10.1177/1536867X1401400307.

Bia M., Mattei A. (2008). A Stata package for the estimation of the dose-response function through adjustment for the generalized propensity score. The Stata Journal, 8 (3), 354–373. DOI: 10.1177/1536867X1401400307.

Borusyak K., Jaravel X., Spiess J. (2024). Revisiting event-study designs: Robust and efficient estimation. Review of Economic Studies, 91 (6), 3253–3285. DOI: 10.1093/restud/rdae007.

Callaway B., Goodman-Bacon A., Sant’Anna P. H. (2024a). Data and code for: Event-studies with a continuous treatment. Nashville, TN: American Economic Association. DOI: 10.3886/E201785V1.

Callaway B., Goodman-Bacon A., Sant’Anna P. H. (2024b). Difference-in-differences with a continuous treatment. Working Paper 32117. National Bureau of Economic Research.

Callaway B., Goodman-Bacon A., Sant’Anna P. H. (2024c). Event studies with a continuous treatment. AEA Papers and Proceedings, 114, 601–605. DOI: 10.1257/pandp.20241047.

Callaway B., Sant’Anna P. H. (2021). Difference-in-differences with multiple time periods. Journal of Econometrics, 225 (2), 200–230.

Cameron A. C., Trivedi P. K. (2005). Microeconometrics: Methods and applications. Cambridge University Press.

Cerulli G. (2015). Econometric evaluation of socio-economic programs. Advanced Studies in Theoretical and Applied Econometrics Series, 49. Springer.

Chernozhukov V., Demirer M., Duflo E., Fernandez-Val I. (2018). Generic machine learning inference on heterogeneous treatment effects in randomized experiments, with an application to immunization in India. Working Paper 24678. National Bureau of Economic Research.

Chernozhukov V., Hansen C., Kallus N., Spindler M., Syrgkanis V. (2024). Applied causal inference powered by ML and AI. https://causalml-book.org/.

De Chaisemartin C., D’Haultfoeuille X. (2020). Two-way fixed effects estimators with heterogeneous treatment effects. American Economic Review, 110 (9), 2964–2996. DOI: 10.1257/aer.20181169.

DML for CATE (2025). https://colab.research.google.com/github/CausalAIBook/MetricsMLNotebooks/blob/main/T/dml-for-conditional-average-treatment-effect.irnb.

Flores C. A., Flores-Lagunes A., Gonzalez A., Neumann T. C. (2012). Estimating the effects of length of exposure to instruction in a training program: The case of job corps. Review of Economics and Statistics, 94 (1), 153–171. DOI: 10.1162/REST_a_00177.

Galagate D. (2016). Causal inference with a continuous treatment and outcome: Alternative estimators for parametric dose-response functions with applications. Ph.D. Dissertation. University of Maryland, College Park. https://www.proquest.com/openview/888e50198b2563222f8e2030e5a8c9e7/1?pq-origsite=gscholar&cbl=18750.

Galagate D., Schafer J. L. (2022). Estimating average dose response functions using the R package causaldrf. https://cran.r-project.org/web/packages/causaldrf/vignettes/Using_causaldrf.pdf.

Goodman-Bacon A. (2021), Difference-in-differences with variation in treatment timing. Journal of Econometrics, 225, 254–277. DOI: 10.1016/j.jeconom.2021.03.014.

Guardabascio B., Ventura M. (2014). Estimating the dose–response function through a generalized linear model approach. The Stata Journal, 14 (1), 141–158. DOI: 10.1177/1536867X1401400110.

Hirano K., Imbens G. W. (2004) The propensity score with continuous treatments. In: Applied Bayesian Modeling and Causal Inference from Incomplete-Data Perspectives: An Essential Journey with Donald Rubin’s Statistical Family, 73–84. New York: Wiley.

Hünermund P., Louw B., Caspi I. (2023). Double machine learning and automated confounder selection: A cautionary tale. Journal of Causal Inference, 11 (1), 20220078.

Ichimura H., Newey W. K. (2022). The influence function of semiparametric estimators. Quantitative Economics, 13 (1), 29–61. DOI: 10.3982/QE826.

Imbens G. W. (2000). The role of the propensity score in estimating dose-response functions. Biometrika, 87 (3), 706–710.

Kennedy E. H., Ma Z., McHugh M. D., Small D. S. (2017). Non-parametric methods for doubly robust estimation of continuous treatment effects. Journal of the Royal Statistical Society Series B: Statistical Methodology, 79 (4), 1229–1245. DOI: 10.1111/rssb.12212.

King G., Nielsen R. (2019). Why propensity scores should not be used for matching. Political Analysis, 27 (4), 435–454.

Kotyrlo E. (2024). The impact of anti-COVID‑19 restrictions and transitory unemployment insurance policies on unemployment in Russia. Journal of Economic Studies, 51 (1), 111–125. DOI: 10.1108/JES-12-2022-0615.

Mattei A., Bia M. (2009). DoseResponse: Stata module to estimate dose-response function through adjustment for the generalized propensity score. Statistical Software Components S457096. Boston College Department of Economics.

Rosenbaum P. R., Rubin D. B. (1983). The central role of the propensity score in observational studies for causal effects. Biometrika, 70, 41–55.

Roth J., Sant’Anna P. H., Bilinski A., Poe J. (2023). What’s trending in difference-in-differences? A synthesis of the recent econometrics literature. Journal of Econometrics, 235 (2), 2218–2244. DOI: 10.1016/j.jeconom.2023.03.008.

Semenova V., Chernozhukov V. (2021). Debiased machine learning of conditional average treatment effects and other causal functions. The Econometrics Journal, 24 (2), 264–289. DOI: 10.1093/ectj/utaa027.

Shalev-Shwartz S., Ben-David S. (2014). Understanding machine learning: From theory to algorithms. Cambridge University Press.

Sostero M., Milasi S., Hurley J., Fernandez-Macías E., Bisello M. (2020). Teleworkability and the COVID‑19 crisis: A new digital divide? JRC Working Papers Series on Labour, Education and Technology, 2020/05.

Wasserman L. (2006) All of nonparametric statistics. New York: Springer.

Wooldridge J. M. (2023). Simple approaches to nonlinear difference-in-differences with panel data. The Econometrics Journal, 26 (3), C31–C66.

Zhao S., van Dyk D. A., Imai K. (2020). Propensity score-based methods for causal inference in observational studies with non-binary treatments. Statistical Methods in Medical Research, 29 (3), 709–727. DOI: 10.1177/0962280219888745.

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Published

2025-04-15

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How to Cite

Kotyrlo E. Evaluation of continuous treatment // Applied Econometrics. 2025. Vol. 80, no. 4. P. 93–116. https://doi.org/10.22394/1993-7601-2025-80-93-116