Economics Letters 195 (2020) 109402 Contents lists available at ScienceDirect Economics Letters journal homepage: www.elsevier.com/locate/ecolet Quantile selection in non-linear GMM quantile models Luciano de Castro a,d , Antonio F. Galvao b,∗ , Gabriel Montes-Rojas c a University of Iowa, Iowa City, USA b University of Arizona, Tucson, USA c Universidad de Buenos Aires, Ciudad Autónoma de Buenos Aires, Argentina d IMPA, Rio de Janeiro, Brazil article info Article history: Received 31 March 2020 Received in revised form 2 June 2020 Accepted 7 July 2020 Available online 16 July 2020 JEL classification: C31 C32 C36 E21 Keywords: Quantile regression Instrumental variables Quantile preferences Elasticity of intertemporal substitution Risk attitude abstract This note proposes a non-linear GMM quantile regression model to estimate the quantile as an additional parameter. The limiting distribution is studied. An empirical application to an intertemporal consumption model built on a structural dynamic quantile utility model illustrates the estimator. Using US data, it separately estimates the elasticity of intertemporal substitution and the risk attitude, which is captured by the estimated quantile. © 2020 Elsevier B.V. All rights reserved. 1. Introduction Since the seminal work of Koenker and Bassett (1978), quan- tile regression (QR) has attracted considerable interest in statis- tics and econometrics. QR estimates conditional quantile func- tions that provide insight into heterogeneous effects of variables of interest, indexed by the quantiles, τ ∈ (0, 1). Chernozhukov and Hansen (2005, 2006, 2008) extend QR methods and present results on identification, estimation, and inference for an instru- mental variables QR (IVQR) model that allows for endogenous regressors; see Chernozhukov et al. (2017) for an overview of IVQR. Kaplan and Sun (2017) and de Castro et al. (2019) de- velop estimators for QR models with instruments in a generalized method of moments (GMM) framework. In general, investigating the entire quantile process is of interest because one may be in- terested in either testing global hypotheses about conditional dis- tributions or making comparisons across different quantiles (for a discussion about inference in QR models see Koenker and Xiao, 2002). Nevertheless, in an attempt to provide the ‘most repre- sentative quantile’, Bera et al. (2016) consider a QR model within ∗ Corresponding author. E-mail addresses: luciano@impa.br (L. de Castro), agalvao@email.arizona.edu (A.F. Galvao), gabriel.montes@fce.uba.ar (G. Montes-Rojas). a quasi-maximum likelihood asymmetric Laplace framework and propose methods to estimate the quantile together with the pa- rameters of interest. 1 The estimated quantile captures a measure of asymmetry of the distribution of innovations, and it does not necessarily lead to the mode, but to a point estimate that is most probable, in the sense it maximizes the entropy. Recently, quantile preferences (QP) have attracted attention in modeling economic behavior in dynamic frameworks. 2 QP are an alternative to expected utility models with useful ad- vantages, such as, in dynamic models, allowing the separation between risk aversion and elasticity of intertemporal substitution (EIS), the ability to capture heterogeneity by offering a family of preferences indexed by the quantile index, τ ∈ (0, 1), dy- namic consistency and monotonicity. Giovannetti (2013) presents a two-period standard economy with one risky and one risk-free asset, where the agent has QP instead of the standard expected utility. de Castro and Galvao (2019a) develop a dynamic model of rational behavior under uncertainty, in which the agent max- imizes the stream of the future quantile utilities, and derive the 1 See also Machado (1993) and Koenker and Machado (1999) where the quantile appears naturally as a location parameter and the pioneering work of Yu and Moyeed (2001) and Yu and Zhang (2005). 2 Manski (1988) introduced QP, which have been subsequently axiomatized by Chambers (2009), Rostek (2010), and de Castro and Galvao (2019b). https://doi.org/10.1016/j.econlet.2020.109402 0165-1765/© 2020 Elsevier B.V. All rights reserved.