Using quantile regression for duration analysis

Summary Quantile regression methods are emerging as a popular technique in econometrics and biometrics for exploring the distribution of duration data. This paper discusses quantile regression for duration analysis allowing for a flexible specification of the functional relationship and of the error distribution. Censored quantile regression addresses the issue of right censoring of the response variable which is common in duration analysis. We compare quantile regression to standard duration models. Quantile regression does not impose a proportional effect of the covariates on the hazard over the duration time. However, the method cannot take account of time-varying covariates and it has not been extended so far to allow for unobserved heterogeneity and competing risks. We also discuss how hazard rates can be estimated using quantile regression methods. This paper benefitted from the helpful comments by an anonymous referee. Due to space constraints, we had to omit the details of the empirical application. These can be found in the long version of this paper, Fitzenberger and Wilke (2005). We gratefully acknowledge financial support by the German Research Foundation (DFG) through the research project ‘Microeconometric modelling of unemployment durations under consideration of the macroeconomic situation’. Thanks are due to Xuan Zhang for excellent research assistance. All errors are our sole responsibility.     

Automatic Bayesian quantile regression curve fitting

Quantile regression, including median regression, as a more completed statistical model than mean regression, is now well known with its wide spread applications. Bayesian inference on quantile regression or Bayesian quantile regression has attracted much interest recently. Most of the existing researches in Bayesian quantile regression focus on parametric quantile regression, though there are discussions on different ways of modeling the model error by a parametric distribution named asymmetric Laplace distribution or by a nonparametric alternative named scale mixture asymmetric Laplace distribution. This paper discusses Bayesian inference for nonparametric quantile regression. This general approach fits quantile regression curves using piecewise polynomial functions with an unknown number of knots at unknown locations, all treated as parameters to be inferred through reversible jump Markov chain Monte Carlo (RJMCMC) of Green (Biometrika 82:711–732, 1995). Instead of drawing samples from the posterior, we use regression quantiles to create Markov chains for the estimation of the quantile curves. We also use approximate Bayesian factor in the inference. This method extends the work in automatic Bayesian mean curve fitting to quantile regression. Numerical results show that this Bayesian quantile smoothing technique is competitive with quantile regression/smoothing splines of He and Ng (Comput. Stat. 14:315–337, 1999) and P-splines (penalized splines) of Eilers and de Menezes (Bioinformatics 21(7):1146–1153, 2005).     

Variable selection in quantile regression when the models have autoregressive errors

This paper considers a problem of variable selection in quantile regression with autoregressive errors. Recently, Wu and Liu (2009) investigated the oracle properties of the SCAD and adaptive-LASSO penalized quantile regressions under non identical but independent error assumption. We further relax the error assumptions so that the regression model can hold autoregressive errors, and then investigate theoretical properties for our proposed penalized quantile estimators under the relaxed assumption. Optimizing the objective function is often challenging because both quantile loss and penalty functions may be non-differentiable and/or non-concave. We adopt the concept of pseudo data by Oh et al. (2007) to implement a practical algorithm for the quantile estimate. In addition, we discuss the convergence property of the proposed algorithm. The performance of the proposed method is compared with those of the majorization-minimization algorithm (Hunter and Li, 2005) and the difference convex algorithm (Wu and Liu, 2009) through numerical and real examples.     

Missing data in clinical trials: control‐based mean imputation and sensitivity analysis

In some randomized (drug versus placebo) clinical trials, the estimand of interest is the between‐treatment difference in population means of a clinical endpoint that is free from the confounding effects of “rescue” medication (e.g., HbA1c change from baseline at 24 weeks that would be observed without rescue medication regardless of whether or when the assigned treatment was discontinued). In such settings, a missing data problem arises if some patients prematurely discontinue from the trial or initiate rescue medication while in the trial, the latter necessitating the discarding of post‐rescue data. We caution that the commonly used mixed‐effects model repeated measures analysis with the embedded missing at random assumption can deliver an exaggerated estimate of the aforementioned estimand of interest. This happens, in part, due to implicit imputation of an overly optimistic mean for “dropouts” (i.e., patients with missing endpoint data of interest) in the drug arm. We propose an alternative approach in which the missing mean for the drug arm dropouts is explicitly replaced with either the estimated mean of the entire endpoint distribution under placebo (primary analysis) or a sequence of increasingly more conservative means within a tipping point framework (sensitivity analysis); patient‐level imputation is not required. A supplemental “dropout = failure” analysis is considered in which a common poor outcome is imputed for all dropouts followed by a between‐treatment comparison using quantile regression. All analyses address the same estimand and can adjust for baseline covariates. Three examples and simulation results are used to support our recommendations.     

Bayesian quantile regression for hierarchical linear models

The paper proposes a Bayesian quantile regression method for hierarchical linear models. Existing approaches of hierarchical linear quantile regression models are scarce and most of them were not from the perspective of Bayesian thoughts, which is important for hierarchical models. In this paper, based on Bayesian theories and Markov Chain Monte Carlo methods, we introduce Asymmetric Laplace distributed errors to simulate joint posterior distributions of population parameters and across-unit parameters and then derive their posterior quantile inferences. We run a simulation as the proposed method to examine the effects on parameters induced by units and quantile levels; the method is also applied to study the relationship between Chinese rural residents' family annual income and their cultivated areas. Both the simulation and real data analysis indicate that the method is effective and accurate.     

Box–Cox power transformation unconditional quantile regressions with an application on wage inequality

This study proposes a semi-parametric estimation method, Box–Cox power transformation unconditional quantile regression, to estimate the impact of changes in the distribution of the explanatory variables on the unconditional quantile of the outcome variable. The proposed method consists of running a nonlinear regression of the recentered influence function (RIF) of the outcome variable on the explanatory variables. We also show the asymptotic properties of the proposed estimator and apply the estimation method to address an existing puzzle in labor economics–why the 50th/10th percentile wage gap has been falling in the USA since the late 1980s. Our results show that declining unionization can explain approximately 10% of the decline in the 50/10 wage gap in 1990–2000 and 23% in 2000–2010.     

Online learning for quantile regression and support vector regression

We consider for quantile regression and support vector regression a kernel-based online learning algorithm associated with a sequence of insensitive pinball loss functions. Our error analysis and derived learning rates show quantitatively that the statistical performance of the learning algorithm may vary with the quantile parameter τ$\tau$. In our analysis we overcome the technical difficulty caused by the varying insensitive parameter introduced with a motivation of sparsity.     

Noncrossing structured additive multiple-output Bayesian quantile regression models

Quantile regression models are a powerful tool for studying different points of the conditional distribution of univariate response variables. Their multivariate counterpart extension though is not straightforward, starting with the definition of multivariate quantiles. We propose here a flexible Bayesian quantile regression model when the response variable is multivariate, where we are able to define a structured additive framework for all predictor variables. We build on previous ideas considering a directional approach to define the quantiles of a response variable with multiple outputs, and we define noncrossing quantiles in every directional quantile model. We define a Markov chain Monte Carlo (MCMC) procedure for model estimation, where the noncrossing property is obtained considering a Gaussian process design to model the correlation between several quantile regression models. We illustrate the results of these models using two datasets: one on dimensions of inequality in the population, such as income and health; the second on scores of students in the Brazilian High School National Exam, considering three dimensions for the response variable.     

Bayesian Semiparametric Modelling in Quantile Regression

Abstract. We propose a Bayesian semiparametric methodology for quantile regression modelling. In particular, working with parametric quantile regression functions, we develop Dirichlet process mixture models for the error distribution in an additive quantile regression formulation. The proposed non‐parametric prior probability models allow the shape of the error density to adapt to the data and thus provide more reliable predictive inference than models based on parametric error distributions. We consider extensions to quantile regression for data sets that include censored observations. Moreover, we employ dependent Dirichlet processes to develop quantile regression models that allow the error distribution to change non‐parametrically with the covariates. Posterior inference is implemented using Markov chain Monte Carlo methods. We assess and compare the performance of our models using both simulated and real data sets.     

Bayesian variable selection in quantile regression with random effects: an application to Municipal Human Development Index

According to the Atlas of Human Development in Brazil, the income dimension of Municipal Human Development Index (MHDI-I) is an indicator that shows the population''s ability in a municipality to ensure a minimum standard of living to provide their basic needs, such as water, food and shelter. In public policy, one of the research objectives is to identify social and economic variables that are associated with this index. Due to the income inequality, evaluate these associations in quantiles, instead of the mean, could be more interest. Thus, in this paper, we develop a Bayesian variable selection in quantile regression models with hierarchical random effects. In particular, we assume a likelihood function based on the Generalized Asymmetric Laplace distribution, and a spike-and-slab prior is used to perform variable selection. The Generalized Asymmetric Laplace distribution is a more general alternative than the Asymmetric Laplace one, which is a common approach used in quantile regression under the Bayesian paradigm. The performance of the proposed method is evaluated via a comprehensive simulation study, and it is applied to the MHDI-I from municipalities located in the state of Rio de Janeiro.     

Bayesian variable selection and estimation in maximum entropy quantile regression

Quantile regression has gained increasing popularity as it provides richer information than the regular mean regression, and variable selection plays an important role in the quantile regression model building process, as it improves the prediction accuracy by choosing an appropriate subset of regression predictors. Unlike the traditional quantile regression, we consider the quantile as an unknown parameter and estimate it jointly with other regression coefficients. In particular, we adopt the Bayesian adaptive Lasso for the maximum entropy quantile regression. A flat prior is chosen for the quantile parameter due to the lack of information on it. The proposed method not only addresses the problem about which quantile would be the most probable one among all the candidates, but also reflects the inner relationship of the data through the estimated quantile. We develop an efficient Gibbs sampler algorithm and show that the performance of our proposed method is superior than the Bayesian adaptive Lasso and Bayesian Lasso through simulation studies and a real data analysis.     

居民生活用电特征与影响机理

近年来我国家庭生活领域碳排放增长迅速,其中电力消费增长是导致家庭碳排放增加的重要原因。本文利用在上海地区开展的居民生活碳消费调查中的居住用电数据,分析了上海市常住居民家庭用电的特征和影响机理。样本中,上海户均年用电量为2184.6kWh,标准差为1398.5kWh,用电基尼系数为0.32,此外生活用电量呈现冬夏高、春秋低,且冬夏两季用电量的离散程度高于春秋两季的现象。回归模型显示上海居民生活用电受到人口规模、收入水平、居住面积、低碳态度和用能习惯的显著影响,且不同用电量家庭的用电影响因素种类和作用效果都存在变化：低用电家庭的生活用电受到人口规模、低碳态度和用能行为的影响,中等用电家庭的生活用电显著影响因素为人口规模、收入水平、低碳态度和用能行为,高用电家庭的生活用电受到人口规模、用能习惯和居住面积的影响;并且随着用电分布从低向高移动,各影响因素的作用效果或增高或降低,呈现不同的变化趋势。通过研究不同用电量家庭用电影响因素的变化,有利于更加深入地了解不同群体生活用电影响因素和完善生活领域电力消费的约束措施。     

Variable selection in additive quantile regression using nonconcave penalty

This paper considers variable selection in additive quantile regression based on group smoothly clipped absolute deviation (gSCAD) penalty. Although shrinkage variable selection in additive models with least-squares loss has been well studied, quantile regression is sufficiently different from mean regression to deserve a separate treatment. It is shown that the gSCAD estimator can correctly identify the significant components and at the same time maintain the usual convergence rates in estimation. Simulation studies are used to illustrate our method.     

Local Linear Additive Quantile Regression

Abstract.  We consider non-parametric additive quantile regression estimation by kernel-weighted local linear fitting. The estimator is based on localizing the characterization of quantile regression as the minimizer of the appropriate 'check function'. A backfitting algorithm and a heuristic rule for selecting the smoothing parameter are explored. We also study the estimation of average-derivative quantile regression under the additive model. The techniques are illustrated by a simulated example and a real data set.     

分位数回归与上证综指VaR研究

把极端分位数所具有的行为特征应用到VaR的研究中，建立上海股市收益率的条件分位数回归模型，描述其在极端分位数下的变化趋势。同时选取适当的尾部模型，并在此基础之上应用外推法预测非常极端分位数下的条件VaR，并与直接由分位数回归模型预测的结果进行比较。结果表明：两种方法得到的结果变化趋势都是一致的，由外推法预测的结果相对小一些。     

Common threshold in quantile regressions with an application to pricing for reputation

The paper develops a systematic estimation and inference procedure for quantile regression models where there may exist a common threshold effect across different quantile indices. We first propose a sup-Wald test for the existence of a threshold effect, and then study the asymptotic properties of the estimators in a threshold quantile regression model under the shrinking threshold effect framework. We consider several tests for the presence of a common threshold value across different quantile indices and obtain their limiting distributions. We apply our methodology to study the pricing strategy for reputation through the use of a data set from Taobao.com. In our economic model, an online seller maximizes the sum of the profit from current sales and the possible future gain from a targeted higher reputation level. We show that the model can predict a jump in optimal pricing behavior, which is considered as “reputation effect” in this paper. The use of threshold quantile regression model allows us to identify and explore the reputation effect and its heterogeneity in data. We find both reputation effects and common thresholds for a range of quantile indices in seller’s pricing strategy in our application.     

Quantile regression in longitudinal studies with dropouts and measurement errors

ABSTRACT

Quantile regression models, as an important tool in practice, can describe effects of risk factors on the entire conditional distribution of the response variable with its estimates robust to outliers. However, there is few discussion on quantile regression for longitudinal data with both missing responses and measurement errors, which are commonly seen in practice. We develop a weighted and bias-corrected quantile loss function for the quantile regression with longitudinal data, which allows both missingness and measurement errors. Additionally, we establish the asymptotic properties of the proposed estimator. Simulation studies demonstrate the expected performance in correcting the bias resulted from missingness and measurement errors. Finally, we investigate the Lifestyle Education for Activity and Nutrition study and confirm the effective of intervention in producing weight loss after nine month at the high quantile.     

Combining forecast quantiles using quantile regression: Investigating the derived weights, estimator bias and imposing constraints

SUMMARY A novel proposal for combining forecast distributions is to use quantile regression to combine quantile estimates. We consider the usefulness of the resultant linear combining weights. If the quantile estimates are unbiased, then there is strong intuitive appeal for omitting the constant and constraining the weights to sum to unity in the quantile regression. However, we show that suppressing the constant renders one of the main attractive features of quantile regression invalid. We establish necessary and sufficient conditions for unbiasedness of a quantile estimate, and show that a combination with zero constant and weights that sum to unity is not necessarily unbiased.     

Bayesian quantile regression and variable selection for partial linear single-index model: Using free knot spline

Partial linear single-index model (PLSIM) has both the flexibility of nonparametric treatment and interpretability of linear term, yet existing literatures about it mainly focused on mean regression, and quantile regression analysis is scarce. Based on free knot spline approximation, we apply asymmetric Laplace distribution to implement Bayesian quantile regression, and perform variable selection in linear term and index vector via binary indicators. Our approach is exempt from regularity conditions in frequentist method, and could execute variable selection and quantile regression under mutual posterior correction, which is also the first work to implement them jointly for PLSIM in fully Bayesian framework. The numerical simulation manifests the superiority of our approach to previous methods, which embodied in better efficiency of variable selection, index vector estimates and link function approximation with different error distributions. For illustration of its application, we build a power consumption model of A₂/O process in wastewater treatment and emphatically analyze the impact of water quality factors.