Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

Difference-in-Differences with multiple time periods

To estimate the dynamic effects of an absorbing treatment, researchers often use two-way fixed effects regressions that include leads and lags of the treatment. We show that in settings with variation in treatment timing across units, the coefficient on a given lead or lag can be contaminated by effects from other periods, and apparent pretrends can arise solely from treatment effects heterogeneity. We propose an alternative estimator that is free of contamination, and illustrate the relative shortcomings of two-way fixed effects regressions with leads and lags through an empirical application.

Estimating Dynamic Treatment Effects in Event Studies with Heterogeneous Treatment Effects

Probably because of their interpretability and transparent nature, synthetic controls have become widely applied in empirical research in economics and the social sciences. This article aims to provide practical guidance to researchers employing synthetic control methods. The article starts with an overview and an introduction to synthetic control estimation. The main sections discuss the advantages of the synthetic control framework as a research design, and describe the settings where synthetic controls provide reliable estimates and those where they may fail. The article closes with a discussion of recent extensions, related methods, and avenues for future research.

https://pubs.aeaweb.org/doi/pdf/10.1257/jel.20191450

Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects

This note revisits the role of time-invariant observed covariates in the Synthetic Control (SC) method. We first derive conditions under which the original result of Abadie et al (2010) regarding the bias of the SC estimator remains valid when we relax the assumption of a perfect match on observed covariates and assume only a perfect match on pre-treatment outcomes. We then show that, even when the conditions for the first result are valid, a perfect match on pre-treatment outcomes does not generally imply an approximate match for all covariates. This will only be true for those that are both relevant and whose effects (over time) are not collinear with the effects of other observed and unobserved covariates. Taken together, our results show that a perfect match on covariates should not be required for the SC method, as long as there is a perfect match on a long set of pre-treatment outcomes.

/pdf/on-the-role-of-covariates-in-the-synthetic-control-method-2qvgje9drg.pdf

On the Role of Covariates in the Synthetic Control Method

Summary
This paper considers the difference-in-differences (DID) method when the data come from repeated cross-sections and the treatment status is observed either before or after the implementation of a program. We propose a new method that point-identifies the average treatment effect on the treated (ATT) via a DID method when there is at least one proxy variable for the latent treatment. Key assumptions are the stationarity of the propensity score conditional on the proxy and an exclusion restriction that the proxy must satisfy with respect to the change in average outcomes over time conditional on the true treatment status. We propose a generalized method of moments estimator for the ATT and we show that the associated overidentification test can be used to test our key assumptions. The method is used to evaluate JUNTOS, a Peruvian conditional cash transfer program. We find that the program significantly increased the demand for health inputs among children and women of reproductive age.

Difference‐in‐differences when the treatment status is observed in only one period

Nonparametric heteroskedasticity in persistent panel processes: An application to earnings dynamics

In nonlinear panel models with fixed effects and fixed-T, the incidental parameter problem poses identification difficulties for structural parameters and partial effects. Existing solutions are model-specific, likelihood-based, impose time homogeneity, or restrict the distribution of unobserved heterogeneity. We provide new identification results for the large class of Fixed Effects Linear Transformation (FELT) models with unknown, time-varying, weakly monotone transformation functions. Our results accommodate continuous and discrete outcomes and covariates, require only two time periods and no parametric distributional assumptions. First, we provide a systematic solution to the incidental parameter problem in FELT via binarization, which transforms FELT into many binary choice models. Second, we identify the distribution of counterfactual outcomes and a menu of time-varying partial effects. Third, we obtain new results for nonlinear difference-in-differences with discrete and censored outcomes, and for FELT with random coefficients. Finally, we propose rank- and likelihood-based estimators that achieve vn rate of convergence.

/pdf/binarization-for-panel-models-with-fixed-effects-4aqjqsc3bo.pdf

Irene Botosaru

Papers

On the Role of Covariates in the Synthetic Control Method

On the Role of Covariates in the Synthetic Control Method

Difference‐in‐differences when the treatment status is observed in only one period

Nonparametric heteroskedasticity in persistent panel processes: An application to earnings dynamics

Binarization for panel models with fixed effects