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Estimating the Intensive Margin Effect in Panel Data Settings

Javier Viviens

TL;DR

This paper tackles the challenge of estimating the intensive-margin effect of policies in panel data when treatment is not randomly assigned. It embeds a Changes-in-Changes outcome model within Horowitz–Manski–Lee bounds to obtain partial identification of the average and distributional intensive-margin effects for Always-Observed (AO) units, introducing estimands $\text{ATT}_{\text{AO}}$ and $QTT_{AO}(q)$. The author extends the identification to multiple sources of sample selection, relaxing monotonicity under a multi-source framework, and develops consistent, asymptotically normal estimators with confidence intervals for the bounds. An empirical application to a Colombian job-training program demonstrates how the method yields bounds that can include zero—suggesting no intensive-margin effect in that setting—and reveals distributional heterogeneity across outcome quantiles, highlighting the policy-relevance of distributional insights beyond point estimates.

Abstract

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.

Estimating the Intensive Margin Effect in Panel Data Settings

TL;DR

This paper tackles the challenge of estimating the intensive-margin effect of policies in panel data when treatment is not randomly assigned. It embeds a Changes-in-Changes outcome model within Horowitz–Manski–Lee bounds to obtain partial identification of the average and distributional intensive-margin effects for Always-Observed (AO) units, introducing estimands and . The author extends the identification to multiple sources of sample selection, relaxing monotonicity under a multi-source framework, and develops consistent, asymptotically normal estimators with confidence intervals for the bounds. An empirical application to a Colombian job-training program demonstrates how the method yields bounds that can include zero—suggesting no intensive-margin effect in that setting—and reveals distributional heterogeneity across outcome quantiles, highlighting the policy-relevance of distributional insights beyond point estimates.

Abstract

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.

Paper Structure

This paper contains 35 sections, 26 theorems, 167 equations, 3 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Notation and set up
  3. Identification of intensive margin effects
  4. Identification of causal effects
  5. Identification of principal strata proportions
  6. Relaxing the monotonicity assumption
  7. Estimation and inference
  8. Estimation
  9. Asymptotic Normality
  10. Confidence Intervals
  11. Empirical application
  12. Conclusion and extensions
  13. Extensions and additional results
  14. Identification in the DiD setup
  15. Estimation and inference in the DiD setup
  16. ...and 20 more sections

Key Result

Proposition 1

Let $Y_{it}(1)$ and $Y_{it}(0)$ be continuous with compact support. Furthermore, let the support of $Y_{it}(0)$ for the treatment group be contained in the support of $Y_{it}(0)$ for the control group. Then, if Assumptions as:no_anti, as:abstate, as:rand_samp and as:outcome hold, then $\Lambda^{LB}( provided that If $F_{Y_{1}\mid G=0, S_2 = 1}\left(Q_{Y_{1}\mid G=1,S_2 = 1}(q\pi_1 + 1 - \pi_1)\ri

Figures (3)

  • Figure 1: Graphical intuition on Proposition \ref{['prop:bounds_qttao']}.
  • Figure 2: $\text{QTT}_{\text{AO}}(q)$. Outcome: log of salaried earnings
  • Figure A1: Graphical intuition on Proposition \ref{['prop:bounds_attao']}.

Theorems & Definitions (40)

  • Example 1
  • Proposition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 1
  • Remark 5
  • Remark 6
  • Proposition 2
  • ...and 30 more