Estimating the Intensive Margin Effect in Panel Data Settings

Many policies operate through two different channels: the extensive margin (e.g., the decision to participate) and the intensive margin (e.g., the intensity of the response among participants). This paper develops a novel identification strategy to estimate the intensive margin effect in panel data settings. I adapt the Horowitz-Manski-Lee bounds to the Changes-in-Changes framework to partially identify both the average and quantile intensive margin treatment effects. Additionally, I explore how to leverage multiple sources of sample selection to relax the monotonicity assumption in the original Horowitz-Manski-Lee bounds, which may be of independent interest. Alongside the identification strategy, I present estimators and inference results. I illustrate the relevance of the proposed methodology by analyzing a job training program in Colombia.

💡 Research Summary

This paper tackles the problem of measuring policy effects that operate through two distinct channels: the extensive margin (whether an individual participates) and the intensive margin (the intensity of the outcome among participants). While the extensive margin is routinely estimated in panel and difference‑in‑differences settings, the intensive margin has largely been studied only under the assumption of unconfounded treatment, which is unrealistic in many observational panel contexts.

The author develops a novel identification strategy that adapts the Horowitz‑Manski‑Lee (HML) bounds—originally designed for binary selection problems—to the Changes‑in‑Changes (CiC) framework. The CiC model treats the distribution of a latent index (e.g., ability) as time‑invariant within each group, allowing the distribution to differ across treatment and control groups but not to evolve over time. Under this “single‑index invariance” assumption, the HML bounds can be used to partially identify two causal estimands for the “Always‑Observed” principal stratum: (i) the average treatment effect on the treated for the always‑observed group (ATT_AO) and (ii) the quantile treatment effect on the same group (QTT_AO).

A second major contribution is the relaxation of the monotonicity assumption that underlies the original HML bounds. By exploiting multiple sources of post‑treatment selection (for example, unemployment status and migration status), each of which can be assumed monotonic but possibly in opposite directions, the author shows that all four principal strata (Always‑Observed, Never‑Observed, Observed‑Only‑Control, Observed‑Only‑Treatment) can coexist. This multi‑source approach yields point identification of the principal‑strata proportions, a substantial improvement over the set‑identification that would result from a single selection variable.

The paper provides a full estimation procedure: non‑parametric CiC transformations recover the latent‑index distribution, bootstrap methods generate confidence intervals for the bounds, and extensions incorporate covariates, binary outcomes, and repeated cross‑sections. The author also discusses how the methodology can be applied to standard DiD designs (Appendix A.1) and how the identification results survive when the absorbing‑state assumption on missingness is dropped (Appendix A.2).

The empirical illustration uses data from a Colombian job‑training program (Attanasio et al., 2011). A naïve CiC estimator applied to observed wages suggests a 13 % increase in earnings due to training. In contrast, the proposed method, which isolates the intensive margin for the always‑observed workers, finds no statistically significant average effect (ATT_AO ≈ 0). The quantile bounds (QTT_AO) vary across the wage distribution, indicating possible positive effects in the upper tail and negative or null effects in the lower tail. This heterogeneity would be invisible to an average‑effect analysis and underscores the policy relevance of the intensive‑margin perspective.

Compared with recent work that extends HML bounds to DiD settings (Rathnayake et al., 2024; Shin, 2024), this paper’s use of the CiC model avoids the need for a parallel‑trends assumption in participation and guarantees that estimated stratum probabilities lie within the unit interval. Moreover, it is the first to provide partial identification of distributional intensive‑margin effects (QTT_AO), enabling researchers to assess not only mean impacts but also how policies reshape the entire outcome distribution.

In sum, the paper makes four key contributions: (1) it integrates HML bounds with the CiC framework to identify intensive‑margin effects in panel data; (2) it relaxes monotonicity by leveraging multiple selection mechanisms, achieving point identification of principal‑strata shares; (3) it extends the methodology to quantile effects and various practical data structures; and (4) it demonstrates that ignoring the intensive margin can lead to misleading conclusions, as illustrated by the Colombian training program. The approach promises broad applicability across labor, education, environmental, and health policy evaluations where selection and intensity both matter.