Bayesian density regression for discrete outcomes

We develop Bayesian models for density regression with emphasis on discrete outcomes. The problem of density regression is approached by considering methods for multivariate density estimation of mixed scale variables, and obtaining conditional densities from the multivariate ones. The approach to multivariate mixed scale outcome density estimation that we describe represents discrete variables, either responses or covariates, as discretised versions of continuous latent variables. We present and compare several models for obtaining these thresholds in the challenging context of count data analysis where the response may be over‐ and/or under‐dispersed in some of the regions of the covariate space. We utilise a nonparametric mixture of multivariate Gaussians to model the directly observed and the latent continuous variables. The paper presents a Markov chain Monte Carlo algorithm for posterior sampling, sufficient conditions for weak consistency, and illustrations on density, mean and quantile regression utilising simulated and real datasets.     

Forecasting the Distribution of Economic Variables in a Data-Rich Environment

This article investigates the relevance of considering a large number of macroeconomic indicators to forecast the complete distribution of a variable. The baseline time series model is a semiparametric specification based on the quantile autoregressive (QAR) model that assumes that the quantiles depend on the lagged values of the variable. We then augment the time series model with macroeconomic information from a large dataset by including principal components or a subset of variables selected by LASSO. We forecast the distribution of the h-month growth rate for four economic variables from 1975 to 2011 and evaluate the forecast accuracy relative to a stochastic volatility model using the quantile score. The results for the output and employment measures indicate that the multivariate models outperform the time series forecasts, in particular at long horizons and in tails of the distribution, while for the inflation variables the improved performance occurs mostly at the 6-month horizon. We also illustrate the practical relevance of predicting the distribution by considering forecasts at three dates during the last recession.     

分位数回归技术综述

普通最小二乘回归建立了在自变量X=x下因变量Y的条件均值与X的关系的线性模型。而分位数回归（Quantile Regression）则利用自变量X和因变量y的条件分位数进行建模。与普通的均值回归相比,它能充分反映自变量X对于因变量y的分布的位置、刻度和形状的影响,有着十分广泛的应用,尤其是对于一些非常关注尾部特征的情况。文章介绍了分位数回归的概念以及分位数回归的估计、检验和拟合优度,回顾了分位数回归的发展过程以及其在一些经济研究领域中的应用,最后做了总结。     

Shrinkage estimation of varying covariate effects based on quantile regression

Varying covariate effects often manifest meaningful heterogeneity in covariate-response associations. In this paper, we adopt a quantile regression model that assumes linearity at a continuous range of quantile levels as a tool to explore such data dynamics. The consideration of potential non-constancy of covariate effects necessitates a new perspective for variable selection, which, under the assumed quantile regression model, is to retain variables that have effects on all quantiles of interest as well as those that influence only part of quantiles considered. Current work on l ₁-penalized quantile regression either does not concern varying covariate effects or may not produce consistent variable selection in the presence of covariates with partial effects, a practical scenario of interest. In this work, we propose a shrinkage approach by adopting a novel uniform adaptive LASSO penalty. The new approach enjoys easy implementation without requiring smoothing. Moreover, it can consistently identify the true model (uniformly across quantiles) and achieve the oracle estimation efficiency. We further extend the proposed shrinkage method to the case where responses are subject to random right censoring. Numerical studies confirm the theoretical results and support the utility of our proposals.     

Consistent model identification of varying coefficient quantile regression with BIC tuning parameter selection

Quantile regression provides a flexible platform for evaluating covariate effects on different segments of the conditional distribution of response. As the effects of covariates may change with quantile level, contemporaneously examining a spectrum of quantiles is expected to have a better capacity to identify variables with either partial or full effects on the response distribution, as compared to focusing on a single quantile. Under this motivation, we study a general adaptively weighted LASSO penalization strategy in the quantile regression setting, where a continuum of quantile index is considered and coefficients are allowed to vary with quantile index. We establish the oracle properties of the resulting estimator of coefficient function. Furthermore, we formally investigate a Bayesian information criterion (BIC)-type uniform tuning parameter selector and show that it can ensure consistent model selection. Our numerical studies confirm the theoretical findings and illustrate an application of the new variable selection procedure.     

Estimation of general semi-parametric quantile regression

Quantile regression introduced by Koenker and Bassett (1978) produces a comprehensive picture of a response variable on predictors. In this paper, we propose a general semi-parametric model of which part of predictors are presented with a single-index, to model the relationship of conditional quantiles of the response on predictors. Special cases are single-index models, partially linear single-index models and varying coefficient single-index models. We propose the qOPG, a quantile regression version of outer-product gradient estimation method (OPG, Xia et al., 2002) to estimate the single-index. Large-sample properties, simulation results and a real-data analysis are provided to examine the performance of the qOPG.     

Bayesian quantile regression for parametric nonlinear mixed effects models

We propose quantile regression (QR) in the Bayesian framework for a class of nonlinear mixed effects models with a known, parametric model form for longitudinal data. Estimation of the regression quantiles is based on a likelihood-based approach using the asymmetric Laplace density. Posterior computations are carried out via Gibbs sampling and the adaptive rejection Metropolis algorithm. To assess the performance of the Bayesian QR estimator, we compare it with the mean regression estimator using real and simulated data. Results show that the Bayesian QR estimator provides a fuller examination of the shape of the conditional distribution of the response variable. Our approach is proposed for parametric nonlinear mixed effects models, and therefore may not be generalized to models without a given model form.     

Penalized quantile regression for dynamic panel data

This paper studies penalized quantile regression for dynamic panel data with fixed effects, where the penalty involves l₁ shrinkage of the fixed effects. Using extensive Monte Carlo simulations, we present evidence that the penalty term reduces the dynamic panel bias and increases the efficiency of the estimators. The underlying intuition is that there is no need to use instrumental variables for the lagged dependent variable in the dynamic panel data model without fixed effects. This provides an additional use for the shrinkage models, other than model selection and efficiency gains. We propose a Bayesian information criterion based estimator for the parameter that controls the degree of shrinkage. We illustrate the usefulness of the novel econometric technique by estimating a “target leverage” model that includes a speed of capital structure adjustment. Using the proposed penalized quantile regression model the estimates of the adjustment speeds lie between 3% and 44% across the quantiles, showing strong evidence that there is substantial heterogeneity in the speed of adjustment among firms.     

Composite quantile regression for varying-coefficient single-index models

ABSTRACT

The varying-coefficient single-index model (VCSIM) is a very general and flexible tool for exploring the relationship between a response variable and a set of predictors. Popular special cases include single-index models and varying-coefficient models. In order to estimate the index-coefficient and the non parametric varying-coefficients in the VCSIM, we propose a two-stage composite quantile regression estimation procedure, which integrates the local linear smoothing method and the information of quantile regressions at a number of conditional quantiles of the response variable. We establish the asymptotic properties of the proposed estimators for the index-coefficient and varying-coefficients when the error is heterogeneous. When compared with the existing mean-regression-based estimation method, our simulation results indicate that our proposed method has comparable performance for normal error and is more robust for error with outliers or heavy tail. We illustrate our methodologies with a real example.     

Efficient Robust Estimation for Linear Models with Missing Response at Random

Coefficient estimation in linear regression models with missing data is routinely carried out in the mean regression framework. However, the mean regression theory breaks down if the error variance is infinite. In addition, correct specification of the likelihood function for existing imputation approach is often challenging in practice, especially for skewed data. In this paper, we develop a novel composite quantile regression and a weighted quantile average estimation procedure for parameter estimation in linear regression models when some responses are missing at random. Instead of imputing the missing response by randomly drawing from its conditional distribution, we propose to impute both missing and observed responses by their estimated conditional quantiles given the observed data and to use the parametrically estimated propensity scores to weigh check functions that define a regression parameter. Both estimation procedures are resistant to heavy‐tailed errors or outliers in the response and can achieve nice robustness and efficiency. Moreover, we propose adaptive penalization methods to simultaneously select significant variables and estimate unknown parameters. Asymptotic properties of the proposed estimators are carefully investigated. An efficient algorithm is developed for fast implementation of the proposed methodologies. We also discuss a model selection criterion, which is based on an IC_Q ‐type statistic, to select the penalty parameters. The performance of the proposed methods is illustrated via simulated and real data sets.     

Central quantile subspace

Quantile regression (QR) is becoming increasingly popular due to its relevance in many scientific investigations. There is a great amount of work about linear and nonlinear QR models. Specifically, nonparametric estimation of the conditional quantiles received particular attention, due to its model flexibility. However, nonparametric QR techniques are limited in the number of covariates. Dimension reduction offers a solution to this problem by considering low-dimensional smoothing without specifying any parametric or nonparametric regression relation. The existing dimension reduction techniques focus on the entire conditional distribution. We, on the other hand, turn our attention to dimension reduction techniques for conditional quantiles and introduce a new method for reducing the dimension of the predictor $$\mathbf {X}$$. The novelty of this paper is threefold. We start by considering a single index quantile regression model, which assumes that the conditional quantile depends on $$\mathbf {X}$$ through a single linear combination of the predictors, then extend to a multi-index quantile regression model, and finally, generalize the proposed methodology to any statistical functional of the conditional distribution. The performance of the methodology is demonstrated through simulation examples and real data applications. Our results suggest that this method has a good finite sample performance and often outperforms the existing methods.     

Influence Measures in Quantile Regression Models

In this article, we use the asymmetric Laplace distribution to define a new method to determine the influence of a certain observation in the fit of quantile regression models. Our measure is based on the likelihood displacement function and we propose two types of measures in order to determine influential observations in a set of conditional quantiles conjointly or in each conditional quantile of interest. We verify the validity of our average measure in a simulated data set as well in an illustrative example with data about air pollution.     

Comparison of designs for multivariate generalized linear models

The purpose of this paper is to discuss response surface designs for multivariate generalized linear models (GLMs). Such models are considered whenever several response variables can be measured for each setting of a group of control variables, and the response variables are adequately represented by GLMs. The mean-squared error of prediction (MSEP) matrix is used to assess the quality of prediction associated with a given design. The MSEP incorporates both the prediction variance and the prediction bias, which results from using maximum likelihood estimates of the parameters of the fitted linear predictor. For a given design, quantiles of a scalar-valued function of the MSEP are obtained within a certain region of interest. The quantiles depend on the unknown parameters of the linear predictor. The dispersion of these quantiles over the space of the unknown parameters is determined and then depicted by the so-called quantile dispersion graphs. An application of the proposed methodology is presented using the special case of the bivariate binary distribution.     

Bayesian non-crossing quantile regression for regularly varying distributions

Quantile regression is a very important statistical tool for predictive modelling and risk assessment. For many applications, conditional quantile at different levels are estimated separately. Consequently the monotonicity of conditional quantiles can be violated when quantile regression curves cross each other. In this paper, we propose a new Bayesian multiple quantile regression based on heavy tailed distribution for non-crossing. We consider a linear quantile regression model for simultaneous Bayesian estimation of multiple quantiles based on a regularly varying assumptions. The numerical and competitive performance of the proposed method is illustrated by simulation.     

Extreme Quantile Estimation for Autoregressive Models

ABSTRACT

A quantile autoregresive model is a useful extension of classical autoregresive models as it can capture the influences of conditioning variables on the location, scale, and shape of the response distribution. However, at the extreme tails, standard quantile autoregression estimator is often unstable due to data sparsity. In this article, assuming quantile autoregresive models, we develop a new estimator for extreme conditional quantiles of time series data based on extreme value theory. We build the connection between the second-order conditions for the autoregression coefficients and for the conditional quantile functions, and establish the asymptotic properties of the proposed estimator. The finite sample performance of the proposed method is illustrated through a simulation study and the analysis of U.S. retail gasoline price.     

Variable selection in heteroscedastic single-index quantile regression

We propose a new algorithm for simultaneous variable selection and parameter estimation for the single-index quantile regression (SIQR) model . The proposed algorithm, which is non iterative , consists of two steps. Step 1 performs an initial variable selection method. Step 2 uses the results of Step 1 to obtain better estimation of the conditional quantiles and , using them, to perform simultaneous variable selection and estimation of the parametric component of the SIQR model. It is shown that the initial variable selection method consistently estimates the relevant variables , and the estimated parametric component derived in Step 2 satisfies the oracle property.     

Regularized Bayesian quantile regression

A number of nonstationary models have been developed to estimate extreme events as function of covariates. A quantile regression (QR) model is a statistical approach intended to estimate and conduct inference about the conditional quantile functions. In this article, we focus on the simultaneous variable selection and parameter estimation through penalized quantile regression. We conducted a comparison of regularized Quantile Regression model with B-Splines in Bayesian framework. Regularization is based on penalty and aims to favor parsimonious model, especially in the case of large dimension space. The prior distributions related to the penalties are detailed. Five penalties (Lasso, Ridge, SCAD0, SCAD1 and SCAD2) are considered with their equivalent expressions in Bayesian framework. The regularized quantile estimates are then compared to the maximum likelihood estimates with respect to the sample size. A Markov Chain Monte Carlo (MCMC) algorithms are developed for each hierarchical model to simulate the conditional posterior distribution of the quantiles. Results indicate that the SCAD0 and Lasso have the best performance for quantile estimation according to Relative Mean Biais (RMB) and the Relative Mean-Error (RME) criteria, especially in the case of heavy distributed errors. A case study of the annual maximum precipitation at Charlo, Eastern Canada, with the Pacific North Atlantic climate index as covariate is presented.     

On the uth geometric conditional quantile

Motivated by Chaudhuri's work [1996. On a geometric notion of quantiles for multivariate data. J. Amer. Statist. Assoc. 91, 862–872] on unconditional geometric quantiles, we explore the asymptotic properties of sample geometric conditional quantiles, defined through kernel functions, in high-dimensional spaces. We establish a Bahadur-type linear representation for the geometric conditional quantile estimator and obtain the convergence rate for the corresponding remainder term. From this, asymptotic normality including bias on the estimated geometric conditional quantile is derived. Based on these results, we propose confidence ellipsoids for multivariate conditional quantiles. The methodology is illustrated via data analysis and a Monte Carlo study.     

On multivariate quantile regression

To detect the dependence on the covariates in the lower and upper tails of the response distribution, regression quantiles are very useful tools in linear model problems with univariate response. We consider here a notion of regression quantiles for problems with multivariate responses. The approach is based on minimizing a loss function equivalent to that in the case of univariate response. To construct an affine equivariant notion of multivariate regression quantiles, we have considered a transformation retransformation procedure based on ‘data-driven coordinate systems’. We indicate some algorithm to compute the proposed estimates and establish asymptotic normality for them. We also, suggest an adaptive procedure to select the optimal data-driven coordinate system. We discuss the performance of our estimates with the help of a finite sample simulation study and to illustrate our methodology, we analyzed an interesting data-set on blood pressures of a group of women and another one on the dependence of sales performances on creative test scores.