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Distributional Instruments: Identification and Estimation with Quantile Least Squares

Rowan Cherodian, Guy Tchuente

TL;DR

The paper addresses econometric settings where policy variation reshapes the distribution of an endogenous variable rather than its mean, which can render mean-based IV diagnostics weak. It formalizes distributional relevance and shows identification of average structural effects in a nonseparable triangular model via a control function based on the conditional distribution $F_{X|Z}$, even when $ ext{Var}( ext{E}[X|Z])=0$. The authors introduce Quantile Least Squares (Q--LS), which constructs an optimal distributional instrument from conditional quantiles and integrates it into a standard IV framework, proving consistency, asymptotic normality, and validity of standard 2SLS SEs, with regularization to handle many quantiles. Monte Carlo simulations and an empirical Medicare Part D application demonstrate that Q--LS yields well-centered estimates and tighter confidence intervals when mean-based IV is weak but distributional shifts are strong, illustrating substantial gains in precision and credibility for studies of financial risk exposure and health outcomes. Overall, Q--LS complements existing IV tools by exploiting distributional variation to extract strong first stages from policy variation that primarily affects tails and dispersion, with clear guidance for diagnostics and practical implementation.

Abstract

We study instrumental-variable designs where policy reforms strongly shift the distribution of an endogenous variable but only weakly move its mean. We formalize this by introducing distributional relevance: instruments may be purely distributional. Within a triangular model, distributional relevance suffices for nonparametric identification of average structural effects via a control function. We then propose Quantile Least Squares (Q-LS), which aggregates conditional quantiles of X given Z into an optimal mean-square predictor and uses this projection as an instrument in a linear IV estimator. We establish consistency, asymptotic normality, and the validity of standard 2SLS variance formulas, and we discuss regularization across quantiles. Monte Carlo designs show that Q-LS delivers well-centered estimates and near-correct size when mean-based 2SLS suffers from weak instruments. In Health and Retirement Study data, Q-LS exploits Medicare Part D-induced distributional shifts in out-of-pocket risk to sharpen estimates of its effects on depression.

Distributional Instruments: Identification and Estimation with Quantile Least Squares

TL;DR

The paper addresses econometric settings where policy variation reshapes the distribution of an endogenous variable rather than its mean, which can render mean-based IV diagnostics weak. It formalizes distributional relevance and shows identification of average structural effects in a nonseparable triangular model via a control function based on the conditional distribution , even when . The authors introduce Quantile Least Squares (Q--LS), which constructs an optimal distributional instrument from conditional quantiles and integrates it into a standard IV framework, proving consistency, asymptotic normality, and validity of standard 2SLS SEs, with regularization to handle many quantiles. Monte Carlo simulations and an empirical Medicare Part D application demonstrate that Q--LS yields well-centered estimates and tighter confidence intervals when mean-based IV is weak but distributional shifts are strong, illustrating substantial gains in precision and credibility for studies of financial risk exposure and health outcomes. Overall, Q--LS complements existing IV tools by exploiting distributional variation to extract strong first stages from policy variation that primarily affects tails and dispersion, with clear guidance for diagnostics and practical implementation.

Abstract

We study instrumental-variable designs where policy reforms strongly shift the distribution of an endogenous variable but only weakly move its mean. We formalize this by introducing distributional relevance: instruments may be purely distributional. Within a triangular model, distributional relevance suffices for nonparametric identification of average structural effects via a control function. We then propose Quantile Least Squares (Q-LS), which aggregates conditional quantiles of X given Z into an optimal mean-square predictor and uses this projection as an instrument in a linear IV estimator. We establish consistency, asymptotic normality, and the validity of standard 2SLS variance formulas, and we discuss regularization across quantiles. Monte Carlo designs show that Q-LS delivers well-centered estimates and near-correct size when mean-based 2SLS suffers from weak instruments. In Health and Retirement Study data, Q-LS exploits Medicare Part D-induced distributional shifts in out-of-pocket risk to sharpen estimates of its effects on depression.
Paper Structure (51 sections, 11 theorems, 145 equations, 2 figures, 10 tables)

This paper contains 51 sections, 11 theorems, 145 equations, 2 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Literature Review
  4. Weak identification in triangular and linear IV models
  5. Quantile regression and endogeneity
  6. Existing ways to circumvent weak identification
  7. Linear IV setup and distributional relevance
  8. Linear IV model
  9. Distributional relevance
  10. The Q--LS estimator in the linear model
  11. Conditional quantiles and quantile-aggregated instruments
  12. Optimal Q--LS and relevance under distributional relevance
  13. Finite-grid implementation
  14. Discrete Q--LS weights.
  15. Second stage.
  16. ...and 36 more sections

Key Result

Lemma 1

Suppose Assumption def:dist-relevance and eq:first-stage-quantile hold, and let $h_{\mathrm{opt}}$ be defined as in Definition def:opt-QLS. Then and either or In particular, whenever $h_{\mathrm{opt}}$ is non-degenerate it is a relevant instrument for $X$.

Figures (2)

  • Figure 1: Mean out-of-pocket medical spending and depression prevalence among HRS respondents aged 65+. Notes: The figure plots mean annual out-of-pocket (OOP) medical spending (thousands of dollars, left axis) and the share of respondents with any depressive symptoms (CESD indicator, right axis) by survey year for HRS respondents aged 65 and older. The vertical line at 2006 marks the introduction of Medicare Part D.
  • Figure 2: Mean and upper quantiles of out-of-pocket medical spending by year (HRS 65+). Notes: The figure plots the mean, median (P50), and upper quantiles (P75, P90, P95) of annual OOP medical spending for HRS respondents aged 65 and older, from 2000 to 2020. The vertical line at 2006 marks the introduction of Medicare Part D. While the median and P75 remain relatively flat over time, the upper quantiles rise in the early 2000s and then flatten or decline after 2006, indicating a post–Part D compression of the right tail of the OOP spending distribution.

Theorems & Definitions (31)

  • Definition 1: Mean relevance
  • Definition 2: Distributional relevance
  • Definition 3: Purely distributional instruments
  • Remark 1: Illustrative example
  • Definition 4: Optimal Q--LS
  • Lemma 1: Distributional relevance and optimal Q--LS
  • proof
  • Proposition 1: Consistency of the Q--LS estimator
  • proof
  • Proposition 2: Asymptotic normality of the Q--LS estimator
  • ...and 21 more