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A Smoothed GMM for Dynamic Quantile Preferences Estimation

Xin Liu, Luciano de Castro, Antonio F. Galvao

TL;DR

This paper develops a smoothed GMM framework to estimate dynamic quantile preferences, treating the quantile level $\tau$ as a structural parameter alongside other finite-dimensional parameters. By stacking multiple moment conditions from a quantile Euler equation and employing a smoothed indicator with bandwidth $h_n$, the authors allow for endogeneity and nonlinearity, achieving $\sqrt{n}$-consistency with efficient two-step GMM. Theoretical results establish consistency and asymptotic normality under weak assumptions, and the empirical intertemporal consumption application with multiple assets yields sensible estimates: $\tau \approx 0.40$ (slightly risk-averse) and an elasticity of intertemporal substitution near unity. Monte Carlo simulations corroborate good finite-sample performance, and the method extends IVQR techniques to a multi-choice, endogenous context with nontrivial dynamic structure, enhancing inference for risk attitudes in macro-finance settings.

Abstract

This paper suggests methods for estimation of the $τ$-quantile, $τ\in(0,1)$, as a parameter along with the other finite-dimensional parameters identified by general conditional quantile restrictions. We employ a generalized method of moments framework allowing for non-linearities and dependent data, where moment functions are smoothed to aid both computation and tractability. Consistency and asymptotic normality of the estimators are established under weak assumptions. Simulations illustrate the finite-sample properties of the methods. An empirical application using a quantile intertemporal consumption model with multiple assets estimates the risk attitude, which is captured by $τ$, together with the elasticity of intertemporal substitution.

A Smoothed GMM for Dynamic Quantile Preferences Estimation

TL;DR

This paper develops a smoothed GMM framework to estimate dynamic quantile preferences, treating the quantile level as a structural parameter alongside other finite-dimensional parameters. By stacking multiple moment conditions from a quantile Euler equation and employing a smoothed indicator with bandwidth , the authors allow for endogeneity and nonlinearity, achieving -consistency with efficient two-step GMM. Theoretical results establish consistency and asymptotic normality under weak assumptions, and the empirical intertemporal consumption application with multiple assets yields sensible estimates: (slightly risk-averse) and an elasticity of intertemporal substitution near unity. Monte Carlo simulations corroborate good finite-sample performance, and the method extends IVQR techniques to a multi-choice, endogenous context with nontrivial dynamic structure, enhancing inference for risk attitudes in macro-finance settings.

Abstract

This paper suggests methods for estimation of the -quantile, , as a parameter along with the other finite-dimensional parameters identified by general conditional quantile restrictions. We employ a generalized method of moments framework allowing for non-linearities and dependent data, where moment functions are smoothed to aid both computation and tractability. Consistency and asymptotic normality of the estimators are established under weak assumptions. Simulations illustrate the finite-sample properties of the methods. An empirical application using a quantile intertemporal consumption model with multiple assets estimates the risk attitude, which is captured by , together with the elasticity of intertemporal substitution.
Paper Structure (25 sections, 6 theorems, 56 equations, 1 figure, 4 tables)

This paper contains 25 sections, 6 theorems, 56 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. The Economic Model
  3. The Recursive Problem
  4. Assumptions
  5. Euler Equation
  6. Moment conditions representation
  7. The smoothed GMM estimators
  8. One-step estimator
  9. Two-step GMM estimator
  10. Large sample properties
  11. Assumptions
  12. Consistency
  13. Asymptotic normality
  14. Application
  15. Intertemporal consumption model with multiple assets
  16. ...and 10 more sections

Key Result

Theorem 1

Let Assumption ass:basic hold. Let $(x_t,y_t,e_t)_{t\in \mathbb{N}}$ be a sequence of states, optimal decisions and shocks, such that $(x_t,y_{t})$ are interior for all $t$. If $e_{t} \mapsto \frac{\partial u}{\partial x}\mathopen{}\mathclose{\left(x_{t} , y_{t} , e_{t} \right) \cdot \frac{\partial

Figures (1)

  • Figure 1: The two assets' SQFs and coefficient functions in simulation DGP 1 and DGP 2.

Theorems & Definitions (10)

  • Theorem 1: Euler Equation; deCastroGalvaoNunes25
  • Lemma 2
  • Theorem 3
  • Lemma 4
  • Theorem 5
  • Lemma 6: de Castro, Galvao, and Ota (2026)
  • proof : Proof of \ref{['lem:smooth-EMn-ULLN']}
  • proof : Proof of \ref{['thm:consistency']}
  • proof : Proof of \ref{['lem:Mn0-normality']}
  • proof : Proof of \ref{['thm:normality']}