Quantiles via moments☆
Introduction
We study the conditions under which it is possible to estimate regression quantiles by estimating conditional means. We focus on the conditional location-scale model considered, among others, by Koenker and Bassett Jr. (1982), Gutenbrunner and Jurečková (1992), Koenker and Zhao (1994), He (1997), and Zhao (2000), and propose an estimator of the conditional quantiles obtained by combining estimates of the location and scale functions, both of which are identified by conditional expectations of appropriately defined variables.
The advantage of our approach is that it allows the use of methods that are only valid in the estimation of conditional means, such as differencing out individual effects in panel data models, while providing information on how the regressors affect the entire conditional distribution. These informational gains are perhaps the most attractive feature of quantile regression (see, e.g., the influential papers by Chamberlain, 1994, and Buchinsky, 1994) and were emphasized, for example, in the surveys by Koenker and Hallock (2001), Cade and Noon (2003), and Bassett Jr and Koenker (2018). Besides greatly facilitating the estimation of complex models, our approach also leads to estimates of the regression quantiles that do not cross, a crucial requisite often ignored in empirical applications (see also He, 1997, and Chernozhukov et al., 2010).
Because our estimator is based on conditional means, it does not share some of the robustness properties of the seminal quantile regression estimator of Koenker and Bassett Jr. (1978), which is based on the check function. For example, our estimator requires stronger assumptions on the existence of moments than those needed for the validity of Koenker and Bassett Jr.’s (1978) estimator. However, under the appropriate conditions, our estimator identifies the same conditional quantiles, the optimal predictors under the usual asymmetric loss function, and these are inherently robust.
The setup we consider is restrictive in that we need to assume that the covariates only affect the distribution of interest through known location and scale functions.1 However, practitioners are often prepared to make even stronger assumptions,2 and we will argue that in spite of its assumptions our approach can be useful in many empirical applications. Importantly, although we do not develop such tests here, it is possible to test the assumption that the covariates only affect the location and scale functions, and therefore it is possible to check whether or not our approach is suitable in a particular application.
The approach we propose is not meant to replace the well-established and very attractive estimation methods based on the check-function. Instead, we see our estimator as an additional tool that can complement those techniques and allow the estimation of regression quantiles in settings where otherwise that would be difficult or even impossible. For example, our approach is attractive when panel data are available and the researcher wants to estimate regression quantiles including individual effects.
Quantile regressions with individual effects suffer from the incidental parameters problem (see, e.g., Neyman and Scott, 1948, and Lancaster, 2000), and there is now a substantial literature dealing with the challenges posed by these models (see, e.g., Koenker, 2004, Lamarche, 2010, Canay, 2011, Galvão, 2011, Kato et al., 2012, Galvão and Wang, 2015, Galvão and Kato, 2016, and Powell, 2016). However, none of these methods gained widespread popularity, either because of their computational complexity or because they rely on very restrictive assumptions on how the fixed effects affect the quantiles. Albeit also based on a somewhat restrictive (but testable) assumption, our approach has the advantage of being very easy to implement even in very large problems and it allows the individual effects to affect the entire distribution, rather than being just location shifters as in, e.g., Koenker (2004), Lamarche (2010), and Canay (2011).3
Our approach can also be adapted to the estimation of cross-sectional models with endogenous variables as, for example, in Abadie et al. (2002) and in Chernozhukov and Hansen, 2005, Chernozhukov and Hansen, 2006, Chernozhukov and Hansen, 2008. Strictly speaking, in this context our approach is not based on the estimation of conditional means, but on moment conditions that under exogeneity identify conditional means. The proposed estimator is closely related to that of Chernozhukov and Hansen (2008) in the sense that under suitable regularly conditions it identifies the same structural quantile function, but has the advantage of being applicable to non-linear models and being computationally much simpler, especially in models with multiple endogenous variables.
The remainder of the paper is organized as follows. Section 2 introduces our approach to the estimation of regression quantiles in location-scale models. Section 3 considers the application of our approach in the context of a panel data model with fixed effects. In Section 4 we consider estimation with cross-sectional data when some of the variables of the model are endogenous. Section 5 presents the results of a small simulation study and Section 6 illustrates the application of the proposed methods with two empirical examples. Section 7 concludes and an Appendix collects the more technical details.
Section snippets
The basic idea
The rationale of the proposed estimator can be introduced in a simple setup. We are interested in estimating the conditional quantiles of a random variable whose distribution conditional on a -vector of covariates belongs to the location-scale family and can be expressed as where:
- •
are unknown parameters;4
- •
is a -vector of known differentiable (with probability ) transformations of the
Linear models
The estimation of linear regression quantiles for longitudinal data was seminally considered by Koenker (2004). To mitigate the effects of the incidental parameters problem, Koenker considers a model where the individual effects only cause parallel (location) shifts of the distribution of the response variable (see also Lamarche, 2010, Canay, 2011, and Galvão, 2011). We also start by considering a linear specification, but allow the individual effects to affect the entire distribution, as in
Endogenous regressors
We explore now the application of the MM-QR estimator to cross-sectional models with endogenous explanatory variables. Consider a scalar random variable that is related to an unobserved scalar random variable satisfying (2) and to a vector of observed random variables , , (with dimensions , , , respectively, and ), by the following structural relationship where ,
Simulation evidence
This section presents the results of two small simulation exercises illustrating the performance of the methods proposed in Sections 3 Panel data with fixed effects, 4 Endogenous regressors.
Illustrative applications
In this section we present two examples illustrating that the proposed methods lead to results that are comparable to those obtained with approaches that are computationally much more demanding. To facilitate the comparison of our results with those in the extant literature, we only consider linear specifications of the conditional quantiles.
Conclusions
In a conditional location-scale model, the information provided by the conditional mean and the conditional scale function is equivalent to the information provided by regression quantiles in the sense that these functions completely characterize how the regressors affect the conditional distribution. This is the result we use to estimate quantiles from estimates of the conditional mean and of the conditional scale function. Our approach is more restrictive than the traditional quantile
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We are grateful to the Editor Xuming He and to two anonymous referees for constructive comments and advice that helped to substantially improve the paper. We also thank Geert Dhaene and Paulo Parente for useful comments and advice. The usual disclaimer applies.