[Paper Review] A Smoothed GMM for Dynamic Quantile Preferences Estimation

Xin Liu, Luciano I. de Castro+1 more|arXiv (Cornell University)|Jan 28, 2026

Financial Risk and Volatility Modeling0 citations

TL;DR

This paper develops a smoothed GMM framework to jointly estimate the structural quantile parameter tau (risk attitude) and other parameters in a dynamic quantile model with endogenous regressors, using multiple assets intertemporal consumption as an empirical example.

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.

Motivation & Objective

• Motivate the estimation of dynamic quantile preferences where tau is a structural parameter.
• Extend IVQR methods to allow endogenous regressors within a dynamic quantile Euler framework.
• Develop a two-step smoothed GMM estimator that achieves root-n consistency for tau and related parameters.
• Provide conditions for consistency and asymptotic normality under weak dependence and nonlinearity.
• Demonstrate finite-sample performance via simulations and an empirical application with multiple assets.

Proposed method

• Formulate a dynamic quantile Euler equation that generalizes the standard Euler equation under tau-quantile preferences.
• Represent the equilibrium conditions as nonlinear conditional quantile functions that can be estimated with instruments.
• Introduce a smoothed GMM objective by replacing the discontinuous indicator with a smooth function to improve computation.
• Implement a two-step GMM estimator with a smoothing-based first stage and an optimal weighting matrix for efficiency.
• Employ a global optimization (simulated annealing) to minimize a potentially non-convex GMM objective.
• Provide asymptotic results showing consistency and normality of the smoothed GMM estimator.

Experimental results

Research questions

• RQ1 How can the tau-quantile parameter (risk attitude) be identified and estimated jointly with other preference parameters in a dynamic quantile setting?
• RQ2 Can endogeneity be accommodated in the estimation via instrumental variables within a smoothed GMM framework for nonlinear quantile models?
• RQ3 What are the large-sample properties (consistency and asymptotic normality) of the smoothed GMM estimator when data may be weakly dependent and nonlinear?
• RQ4 How does the proposed method perform relative to grid-search approaches in estimating the quantile parameter?
• RQ5 What do empirical applications reveal about the interaction between risk attitude (tau) and the elasticity of intertemporal substitution (EIS) in intertemporal consumption with multiple assets?

Key findings

• The smoothed GMM estimator consistently recovers the quantile parameter tau along with other model parameters.
• The method allows endogenous regressors and nonlinear quantile restrictions within a GMM framework.
• Simulation evidence indicates the estimator is approximately unbiased in finite samples and that tau is well-estimated.
• Empirical application finds a slight risk-averse tau and an EIS near but slightly below one, aligning with related literature on preferences.
• The two-step smoothed GMM achieves root-n consistency and standard asymptotic normality under weak assumptions.