Estimation of High Conditional Quantiles for Heavy-Tailed Distributions

Quasi-maximum likelihood estimation for conditional quantiles

Estimation of low or high conditional quantiles is called for in many applications, but commonly encountered data sparsity at the tails of distributions makes this a challenging task. We develop a Bayesian joint-quantile regression method to borrow information across tail quantiles through a linear approximation of quantile coefficients. Motivated by a working likelihood linked to the asymmetric Laplace distributions, we propose a new Bayesian estimator for high quantiles by using a delayed rejection and adaptive Metropolis and Gibbs algorithm. We demonstrate through numerical studies that the proposed estimator is generally more stable and efficient than conventional methods for estimating tail quantiles, especially at small and modest sample sizes.

GARCH models play a fundamental role in modeling time-varying volatility in financial return series. In practice, financial returns are also well known to exhibit heavy-tailed distributions, which naturally motivates the use of quasi-maximum exponential likelihood estimation (QMELE) for accurately capturing tail behavior and risk measures such as Value-at-Risk. At the same time, the increasing availability of intraday high-frequency data has led to the development of high-frequency augmented GARCH models, which incorporate intraday information into conventional low-frequency volatility frameworks. By exploiting transaction-level data recorded at very fine time scales, these models are able to capture intraday volatility dynamics and market microstructure effects that are not reflected in standard low-frequency observations. Against this background, this paper studies conditional quantile estimation for high-frequency augmented GARCH models. We develop QMELE-based estimators for both model parameters and conditional quantiles, and construct an adjusted test statistic for assessing model adequacy. The asymptotic properties of the proposed estimators and test statistic are established, and their finite-sample performance is examined through extensive simulation studies. Empirical applications to three major stock indices demonstrate that augmenting GARCH models with high-frequency information leads to substantial improvements in conditional quantile estimation compared with traditional low-frequency approaches.

Estimation of conditional quantiles from data with additional measurement errors

Statistical Inference of Conditional Quantiles in Nonlinear Time Series Models

Quantile regression is an important tool for estimation of conditional quantiles of a response Y given a vector of covariates X. It can be used to measure the effect of covariates not only in the center of a distribution, but also in the upper and lower tails. This paper develops a theory of quantile regression in the tails. Specifically, it obtains the large sample properties of extremal (extreme order and intermediate order) quantile regression estimators for the linear quantile regression model with the tails restricted to the domain of minimum attraction and closed under tail equivalence across regressor values. This modeling setup combines restrictions of extreme value theory with leading homoscedastic and heteroscedastic linear specifications of regression analysis. In large samples, extreme order regression quantiles converge weakly to arg min functionals of stochastic integrals of Poisson processes that depend on regressors, while intermediate regression quantiles and their functionals converge to normal vectors with variance matrices dependent on the tail parameters and the regressor design.

The paper examines the relationship between powerful CEO and firm performance using the estimation of quantile regression which models heterogeneous effects of a variable along different points of an outcome distribution. This paper targets the nonlinear and size effects of CEO power on firm performance and explores the possibility that CEO power and firm performance are endogenously determined. The results indicate that CEOs tend to possess more power in the early stage of a company's lifecycle. They acquire more ownership, expert, and prestige power following poor market returns. There is an inverted U-shaped relation between CEO ownership and accounting performance in low quantiles of firm performance distribution, while the impact on performance becomes negative in high quantiles. Moreover, the size effect shows that the positive impact of ownership decreases when firm size grows. CEOs' expert power improves firm performance, although the effect turns negative when tenure exceeds 15 years. The negative impact of expert power is more significant in small firms. Finally, CEOs' prestige power enhances firm performance in low quantiles and hurts performance in high quantiles.

The second-order bias of quantile estimators

We consider regression problems under asymmetric or/and non-constant variance error. We see this problem in several fields such as insurance premium estimation, medical cost analysis, etc. Applying the method of Least Squares (LS) to this problem yields unstable solution because of outliers that appears on one side of regression surfaces. Conventional robust techniques to deal with outliers, which intend to discard or down-weight the outliers equally from both sides of regression surfaces, does not help for asymmetric error. In this paper, we propose an robust regression estimator (an estimator of the conditional mean) under asymmetric or/and non-constant variance error by simultaneously training conditional quantiles in multi-layer perceptron (MLP). This is considered as a kind of learning from hint or multitask learning approach, i.e., we train the conditional quantile estimator as hints or extra tasks to improve generalization properties of the conditional mean estimator. Numerical experiments and an application to medical cost estimation problem have shown that our proposal has robustness and good generalization properties.

This paper examines the evolution of the returns to education in Portugal over the 1980s and early 1990s. The main findings indicate that the returns to education have increased, particularly after joining the European Union in 1986. Since this occurred along with an increase in the level of education within the labour force, the process is most likely demand driven. The results also indicate that modelling on average (i.e. OLS) misses important features of the wage structure. Quantile regression (QR) analysis reveals that the effect of education is not constant across the conditional wage distribution. They are higher for those at higher quantiles in the conditional wage distribution. Wage inequality expanded in Portugal over the 1980s and the returns to education had an important role in this process.

In this article, we focus on the efficient estimators of the derivative of the nonparametric function in the nonparametric quantile regression model. We develop two ways of combining quantile regression information to derive the estimators. One is the weighted composite quantile regression estimator based on the quantile weighted loss function; the other is the weighted quantile average estimator based on the weighted average of quantile regression estimators at a single quantile. Furthermore, by minimizing the asymptotic variance, the optimal weight vector is computed, and consequently, the optimal estimator is obtained. Furthermore, we conduct some simulations to evaluate the performance of our proposed estimators under different symmetric error distributions. Simulation studies further illustrate that both estimators work better than the local linear least square estimator for all the symmetric errors considered except the normal error, and the weighted quantile average estimator performs better than the weighted composite quantile regression estimator in most situations.

Fine particulate matter(PM2.5) pollution will affect people’s well-being and cause economic losses. It is of great value to study the impact of PM2.5 on the real estate market. While previous studies have examined the effects of PM2.5 pollution on urban housing prices, there has been little in-depth research on these effects, which are spatially heterogeneous at different conditional quantiles. To address this issue, this study employs quantile regression (QR) and geographically weighted quantile regression (GWQR) models to obtain a full account of asymmetric and spatial non-stationary effects of PM2.5 pollution on urban housing prices through 286 Chinese prefecture-level cities for 2005–2013. Considerable differences in the data distributions and spatial characteristics of PM2.5 pollution and urban housing prices are found, indicating the presence of asymmetric and spatial non-stationary effects. The quantile regression results show that the negative influences of PM2.5 pollution on urban housing prices are stronger at higher quantiles and become more pronounced with time. Furthermore, the spatial relationship between PM2.5 pollution and urban housing prices is spatial non-stationary at most quantiles for the study period. A negative correlation gradually dominates in most of the study areas. At higher quantiles, PM2.5 pollution is always negatively correlated with urban housing prices in eastern coastal areas and is stable over time. Based on these findings, we call for more targeted approaches to regional real estate development and environmental protection policies.

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. 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 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 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 a real data application. Our results suggest that this method has a good finite sample performance and often outperforms existing methods.

We propose a regression method for the estimation of conditional quantiles of a continuous response variable given a set of covariates when the data are dependent. Along with fixed regression coefficients, we introduce random coefficients which we assume to follow a form of multivariate Laplace distribution. In a simulation study, the proposed quantile mixed-effects regression is shown to model the dependence among longitudinal data correctly and estimate the fixed effects efficiently. It performs similarly to the linear mixed model at the central location when the regression errors are symmetrically distributed, but provides more efficient estimates when the errors are over-dispersed. At the same time, it allows the estimation at different locations of conditional distribution, which conveys a comprehensive understanding of data. We illustrate an application to clinical data where the outcome variable of interest is bounded within a closed interval.

This study aimed at investigating the nexus between government debt and its determinants in the context of sustainable economic growth for 97 countries over the period from 2004 to 2018 by employing a panel data quantile regression and comparing it across a fixed-effects model and a generalized method of moments (GMM) model. The different touches of debt on economic growth were analysed. The quantile regression estimates revealed that government debt has various impacts on economic growth, relying on the degree of economic performance. Government debt negatively affects economic growth across all the quantiles. More interestingly, the quantile regression estimates indicated that in countries with a very low level of real GDP per capita growth, government debt has a stronger pernicious effect on economic growth compared to upper-middle and high-income countries. Hence, our results provided insightful information for policy makers in designing appropriate fiscal policies particularly for low- and lower- middle-income countries to mitigate the negative effect on economic growth, and it should be maintained at a low and reasonable level to promote sustainable economic growth. JEL Classification: E61, E62, E63, H63, O4, O40