Explore the vast range of titles available.

Since Quetelet's work in the nineteenth century, social science has iconified the average man, that hypothetical man without qualities who is comfortable with his head in the oven and his feet in a bucket of ice. Conventional statistical methods since Quetelet have sought to estimate the effects of policy treatments for this average man. However, such effects are often quite heterogeneous: Medical treatments may improve life expectancy but also impose serious short-term risks; reducing class sizes may improve the performance of good students but not help weaker ones, or vice versa. Quantile regression methods can help to explore these heterogeneous effects. Some recent developments in quantile regression methods are surveyed in this review.

We propose a new regression method to evaluate the impact of changes in the distribution of the explanatory variables on quantiles of the unconditional (marginal) distribution of an outcome variable. The proposed method consists of running a regression of the (recentered) influence function (RIF) of the unconditional quantile on the explanatory variables. The influence function, a widely used tool in robust estimation, is easily computed for quantiles, as well as for other distributional statistics. Our approach, thus, can be readily generalized to other distributional statistics.

This paper introduces a quantile regression estimator for panel data (QRPD) with nonadditive fixed effects, maintaining the nonseparable disturbance term commonly associated with quantile estimation. QRPD estimates the impact of exogenous or endogenous treatment variables on the outcome distribution using “within” variation in the instruments for identification purposes. Most quantile panel data estimators include additive fixed effects which separates the disturbance term and assumes the parameters vary based only on the time-varying components of the disturbance term. QRPD produces consistent estimates for small T. I estimate the effect of the 2008 tax rebates on the short-term household consumption distribution.

Ultra-high dimensional data often display heterogeneity due to either heteroscedastic variance or other forms of non-location-scale covariate effects. To accommodate heterogeneity, we advocate a more general interpretation of sparsity, which assumes that only a small number of covariates influence the conditional distribution of the response variable, given all candidate covariates; however, the sets of relevant covariates may differ when we consider different segments of the conditional distribution. In this framework, we investigate the methodology and theory of nonconvex, penalized quantile regression in ultra-high dimension. The proposed approach has two distinctive features: (1) It enables us to explore the entire conditional distribution of the response variable, given the ultra-high-dimensional covariates, and provides a more realistic picture of the sparsity pattern; (2) it requires substantially weaker conditions compared with alternative methods in the literature; thus, it greatly alleviates the difficulty of model checking in the ultra-high dimension. In theoretic development, it is challenging to deal with both the nonsmooth loss function and the nonconvex penalty function in ultra-high-dimensional parameter space. We introduce a novel, sufficient optimality condition that relies on a convex differencing representation of the penalized loss function and the subdifferential calculus. Exploring this optimality condition enables us to establish the oracle property for sparse quantile regression in the ultra-high dimension under relaxed conditions. The proposed method greatly enhances existing tools for ultra-high-dimensional data analysis. Monte Carlo simulations demonstrate the usefulness of the proposed procedure. The real data example we analyzed demonstrates that the new approach reveals substantially more information as compared with alternative methods. This article has online supplementary material.

In spite of the recent surge of interest in quantile regression, joint estimation of linear quantile planes remains a great challenge in statistics and econometrics. We propose a novel parameterization that characterizes any collection of noncrossing quantile planes over arbitrarily shaped convex predictor domains in any dimension by means of unconstrained scalar, vector and function valued parameters. Statistical models based on this parameterization inherit a fast computation of the likelihood function, enabling penalized likelihood or Bayesian approaches to model fitting. We introduce a complete Bayesian methodology by using Gaussian process prior distributions on the function valued parameters and develop a robust and efficient Markov chain Monte Carlo parameter estimation. The resulting method is shown to offer posterior consistency under mild tail and regularity conditions. We present several illustrative examples where the new method is compared against existing approaches and is found to offer better accuracy, coverage and model fit. Supplementary materials for this article are available online.

Bayesian information criterion (BIC) is known to identify the true model consistently as long as the predictor dimension is finite. Recently, its moderate modifications have been shown to be consistent in model selection even when the number of variables diverges. Those works have been done mostly in mean regression, but rarely in quantile regression. The best-known results about BIC for quantile regression are for linear models with a fixed number of variables. In this article, we investigate how BIC can be adapted to high-dimensional linear quantile regression and show that a modified BIC is consistent in model selection when the number of variables diverges as the sample size increases. We also discuss how it can be used for choosing the regularization parameters of penalized approaches that are designed to conduct variable selection and shrinkage estimation simultaneously. Moreover, we extend the results to structured nonparametric quantile models with a diverging number of covariates. We illustrate our theoretical results via some simulated examples and a real data analysis on human eye disease. Supplementary materials for this article are available online.

Data subject to heavy-tailed errors are commonly encountered in various scientific fields. To address this problem, procedures based on quantile regression and least absolute deviation regression have been developed in recent years. These methods essentially estimate the conditional median (or quantile) function. They can be very different from the conditional mean functions, especially when distributions are asymmetric and heteroscedastic. How can we efficiently estimate the mean regression functions in ultrahigh dimensional settings with existence of only the second moment? To solve this problem, we propose a penalized Huber loss with diverging parameter to reduce biases created by the traditional Huber loss. Such a penalized robust approximate (RA) quadratic loss will be called the RA lasso. In the ultrahigh dimensional setting, where the dimensionality can grow exponentially with the sample size, our results reveal that the RA lasso estimator produces a consistent estimator at the same rate as the optimal rate under the light tail situation. We further study the computational convergence of the RA lasso and show that the composite gradient descent algorithm indeed produces a solution that admits the same optimal rate after sufficient iterations. As a by-product, we also establish the concentration inequality for estimating the population mean when there is only the second moment. We compare the RA lasso with other regularized robust estimators based on quantile regression and least absolute deviation regression. Extensive simulation studies demonstrate the satisfactory finite sample performance of the RA lasso.

We propose a notion of conditional vector quantile function and a vector quantile regression. A conditional vector quantile function (CVQF) of a random vector Y, taking values in ℝd given covariates Z = z, taking values in ℝk, is a map u ↦ QY|Z(u, z), which is monotone, in the sense of being a gradient of a convex function, and such that given that vector U follows a reference non-atomic distribution FU, for instance uniform distribution on a unit cube in ℝd, the random vector QY|Z(U, z) has the distribution of Y conditional on Z = z. Moreover, we have a strong representation, Y = Qy|z(U, Z) almost surely, for some version of U. The vector quantile regression (VQR) is a linear model for CVQF of Y given Z. Under correct specification, the notion produces strong representation, Y = β(U)T f(Z), for f(Z) denoting a known set of transformations of Z, where u ↦ β(u)T f(Z) is a monotone map, the gradient of a convex function and the quantile regression coefficients u ↦ β(u) have the interpretations analogous to that of the standard scalar quantile regression. As f (Z) becomes a richer class of transformations of Z, the model becomes nonparametric, as in series modelling. A key property of VQR is the embedding of the classical Monge-Kantorovich's optimal transportation problem at its core as a special case. In the classical case, where Y is scalar, VQR reduces to a version of the classical QR, and CVQF reduces to the scalar conditional quantile function. An application to multiple Engel curve estimation is considered.

We analyze gender differences in expected starting salaries along the wage expectations distribution of prospective university students in Germany, using elicited beliefs about both own salaries and salaries for average other students in the same field. Unconditional and conditional quantile regressions show 5–15% lower wage expectations for females. At all percentiles considered, the gender gap is more pronounced in the distribution of expected own salary than in the distribution of wages expected for average other students. Decomposition results show that biased beliefs about the own earnings potential relative to others and about average salaries play a major role in explaining the gender gap in wage expectations for oneself.

The event study model is a powerful econometric tool used for the purpose of estimating dynamic treatment effects. One of its most appealing features is that it provides a built-in graphical summary of results, which can reveal rich patterns of behavior. Another value of the picture is the estimated pre-event pseudo-\"effects\", which provide a type of placebo test. In this essay I aim to provide a framework for a shared understanding of these models. There are several (sometimes subtle) decisions and choices faced by users of these models, and I offer guidance for these decisions.