Quantile Regions Research Articles

Distribution dynamics and quantile dynamic convergence of the digital economy: Prefecture-level evidence in China

The digital-substantive fusion is the realistic need to crack the problem of “out of the real to the virtual” in the real economy. Based on the panel data of 288 prefecture-level cities in China from 2011 to 2019, we use kernel density estimation and quantile convergence model to explore the distribution dynamics and convergence trend of the digital economy across the country and in the eastern, central, and western regions. In doing so, we unearth several key findings. First, the development of the digital economy in Chinese cities is on the rise, but absolute differences among cities still exist. Second, the overall development of the digital economy exhibits a dynamic q-σ convergence trend, but the development of the digital economy in the central region from the 0.75 to 0.95 quantiles shows a divergence trend. Third, our analysis reveals quantile heterogeneity and regional heterogeneity in the dynamic absolute q-β convergence of digital economy development. The tendency to converge is stronger in the extreme quantile regions than in other regions, while the convergence speed of central and western regions is higher than that of eastern regions. Finally, the dynamic conditional q-β convergence trend of digital economy development across the country and in the eastern, central, and western regions is stronger than the dynamic absolute q-β convergence, with the trend of convergence weakening only in the eastern high quantile regions. These findings provide empirical evidence for promoting regional coordinated high-quality development of digital economy, as well as theoretical references for vigorously promoting the construction of a modernized industrial system and accelerating the development of new-quality productive forces.

On extreme quantile region estimation under heavy-tailed elliptical distributions

Consider the estimation of an extreme quantile region corresponding to a very small probability. Estimation of extreme quantile regions is important but difficult since extreme regions contain only a few or no observations. In this article, we propose an affine equivariant extreme quantile region estimator for heavy-tailed elliptical distributions. The estimator is constructed by extending a well-known univariate extreme quantile estimator. Consistency of the estimator is proved under estimated location and scatter. The practicality of the developed estimator is illustrated with simulations and a real data example.

Co-benefits of the National Key Ecological Function Areas in China for carbon sequestration and environmental quality

IntroductionThe National Key Ecological Functional Areas (NKEFAs) are location-oriented ecological engineering of China, which rely on the main functional area planning. The co-benefits of ecological product supply and ecological environment improvement of NKEFAs has not been fully assessed in the literature.MethodsNKEFAs is considered a quasi-natural experiment, and the time-varying difference-in-differences (DID) model is used to assess the impact of NKEFAs on carbon sequestration (CS) and environmental quality (EQ) based on the panel data of 330 cities in China from 2001 to 2019. Then, we explore whether the co-benefits of ecological product supply and eco-environment protection can be achieved.Results and discussionNKEFAs can enhance CS and EQ and thus achieve co-benefits for both. NKEFAs can achieve the co-benefits of CS and EQ through territory spatial allocation and labor force aggregation, but industrial structure upgrading only positively mediates the impact of NKEFAs on CS. The co-benefits of CS and EQ are heterogeneous across functional area types, geospatial locations, and quantiles, while only CS at windbreak-sand fixation area, northwestern region, and low quantile regions is enhanced. This study makes a theoretical and methodological contribution to the existing literature on the policy effect assessment of ecological engineering. It also provides a comprehensive framework for evaluating the ecological effects of relevant policies in other countries by integrating the co-benefits of ecological products and eco-environment, analyzing regional heterogeneity, and exploring the underlying mechanisms.

From regression rank scores to robust inference for censored quantile regression

AbstractQuantile regression for right‐ or left‐censored outcomes has attracted attention due to its ability to accommodate heterogeneity in regression analysis of survival times. Rank‐based inferential methods have desirable properties for quantile regression analysis, but censored data poses challenges to the general concept of ranking. In this article, we propose a notion of censored quantile regression rank scores, which enables us to construct rank‐based tests for quantile regression coefficients at a single quantile or over a quantile region. A model‐based bootstrap algorithm is proposed to implement the tests. We also illustrate the advantage of focusing on a quantile region instead of a single quantile level when testing the effect of certain covariates in a quantile regression framework.

Irrigation Increases Crop Species Diversity in Low-Diversity Farm Regions of Mexico

Although agricultural intensification generally has homogenizing effects on landscapes that reduce crop diversity, the specific effects of different input strategies on crop diversity are unclear. This study examines the effects of irrigation inputs on crop species diversity in Mexico. We assess the richness and evenness diversity of 297 crop species across 2455 municipalities while controlling for environmental and socioeconomic factors and farm structural and functional characteristics. Using a quantile regression approach, we assess relationships across conditional quantiles of low-, medium-, and high-diversity farm regions. Results show irrigation level (% cropland irrigated) is a strong positive predictor of crop species richness and evenness diversity across all quantile regions. Moreover, the quantile effects of irrigation on evenness diversity are five times greater in low-diversity rather than high-diversity regions. With implications for agricultural water policy in Mexico, this study illustrates the potential benefits of sustainable irrigation expansion in water-rich but irrigation-poor farming regions. Specifically, by enhancing crop species diversity, carefully targeted irrigation expansion can support the transition to sustainable intensification.

Inverted U-Shape Relationship Between Odor Concentration and Olfactory Short-Term Memory

We examined the return–volatility relationship for USO ETF oil price return and CBOE Crude Oil ETF Volatility Index, OVX. The data for the USO and OVX covers the period covering May 11, 2007 to February 28, 2013. Our OLS regression results suggest evidence of regular feedback and leverage effects. When we employ linear quantile regression techniques, we find evidence of regular and inverse feedback effects. The inverse feedback effects being noticeable in the upper quantile region of the oil return distribution. There is also support for a regular leverage effect in USO prices. We also examined the return–volatility relationship using quantile regression copula methods for measuring the degree of asymmetry in the relationships between the oil price return and implied volatility. The results of the analysis indicate, first, that there exists a negative relationship between contemporaneous oil VIX and USO ETF oil returns. Second, that the relationship between oil returns and implied volatilities depends on the quartile at which the relationship is being investigated. Third, there exists an inverted U-shaped dependency relationship between returns and implied volatilities across quantiles. Fourth, though an inverted U-shape exists, the shape is different from those observed in stock markets.

Functional, randomized and smoothed multivariate quantile regions

The mass transportation approach to multivariate quantiles in Chernozhukov et al. (2017) was modified in Faugeras and Rüschendorf (2017) by a two steps procedure. In the first step, a mass transportation problem from a spherical reference measure to the copula is solved and combined in the second step with a marginal quantile transformation in the sample space. Also, generalized quantiles given by suitable Markov morphisms are introduced there.In the present paper, this approach is further extended by a functional approach in terms of membership functions, and by the introduction of randomized quantile regions. In addition, in the case of continuous marginals, a smoothed version of the empirical quantile regions is obtained by smoothing the empirical copula. All three extended approaches give empirical quantile ares of exact level and improved stability. The resulting depth areas give a valid representation of the central quantile areas of a multivariate distribution and provide a valuable tool for their analysis.

Correction to: Estimation and uncertainty quantification for extreme quantile regions

Correction to: Estimation and uncertainty quantification for extreme quantile regions

Hypothesis testing of varying coefficients for regional quantiles

Testing the behavior of varying coefficients (VC) over a range of quantiles is important in the field of regression analysis. This study tests whether coefficient functions in varying quantile regression share common structural information across a certain range of quantile levels, even when linear combinations of covariates are unspecified in the null hypothesis. Our approach allows varying the coefficients, β(τ,t), as a function of the quantile level, τ∈Δ, and a random variable, t∈T, where Δ is the quantile region of interest and T is a domain of t. By incorporating an interval of quantiles in the inference, the proposed method can test whether the model can be justified by using a VC quantile model against other important reduced models, such as the linear quantile model, varying coefficient linear model, or linear model with homogeneous errors. We use bivariate B-splines to approximate the varying quantile functions, β(τ,t), and utilize composite quantile regression to estimate the parameters. Furthermore, we develop constrained composite quantile regression to provide a more efficient estimate in case the null hypothesis is not rejected. We show that the proposed test admits normal approximations. Using simulation and real data analysis, we demonstrate the superiority of the proposed test over other tests designed for a finite number of quantile levels.

Center-outward quantiles and the measurement of multivariate risk

All multivariate extensions of the univariate theory of risk measurement run into the same fundamental problem of the absence, in dimension d>1, of a canonical ordering of Rd. Based on measure transportation ideas, several attempts have been made recently in the statistical literature to overcome that conceptual difficulty. In Hallin (2017), the concepts of center-outward distribution and quantile functions are developed as generalizations of the classical univariate concepts of distribution and quantile functions, along with their empirical versions. The center-outward distribution function F± is a cyclically monotone mapping from Rd to the open unit ball Bd, while its empirical counterpart F±(n) is obtained as a cyclically monotone mapping from the sample to a regular grid over Bd; in dimension d=1, F± reduces to 2F−1. Based on the concept of Moreau envelope, a smooth interpolation F˜± of F± has been proposed in del Barrio et al. (2018). Here, we suggest to adapt the definition of the empirical F±(n) so as to relax the presence of ties, which is impractical in the context of risk measurement and propose a class of smooth approximations Fn,ξ (ξ a smoothness index) of F±(n) as an alternative to the F˜± interpolation. Associated with the concepts of center-outward distribution and quantile functions and the associated convex potentials, we construct measures of risk of the maximum correlation type and their estimators based on F±(n) and Fn,ξ. We also discuss the use of the volumes of the resulting empirical quantile regions. Some simulations and applications to case studies illustrate the value of the approach.

Estimation and uncertainty quantification for extreme quantile regions

Estimation of extreme quantile regions, spaces in which future extreme events can occur with a given low probability, even beyond the range of the observed data, is an important task in the analysis of extremes. Existing methods to estimate such regions are available, but do not provide any measures of estimation uncertainty. We develop univariate and bivariate schemes for estimating extreme quantile regions under the Bayesian paradigm that outperforms existing approaches and provides natural measures of quantile region estimate uncertainty. We examine the method’s performance in controlled simulation studies. We illustrate the applicability of the proposed method by analysing high bivariate quantiles for pairs of pollutants, conditionally on different temperature gradations, recorded in Milan, Italy.

Geographical diversification, firm size and profitability in Malaysia: A quantile regression approach

The relationship between geographical diversification (GDI) and profitability (ROA) has yielded mixed findings across various developed countries. This study re-examined the relationship using data of public firms listed on the main market of Bursa Malaysia for the period of 2010–2014 using quantile regression approach. The firms are categorised into small firms and large firms based on the firm size median value. The empirical results show that GDI affects ROA heterogeneously in various quantile levels of the ROA for all firms, small firms and large firms. GDI significantly (positive relationship) influences ROA in the middle quantile region (from quantile 0.25 to 0.75) for all firms, in the low quantile region (from quantile 0.1 to 0.5) for the sample of small firms and in the high quantile region (from quantile 0.5 to 0.9) for the sample of large firms. Therefore, GDI activities could benefit firms, provided that the activities are conducted wisely by taking into account the profitability levels of firms as well as the size of firms. This study contributes to literature on geographical diversification by providing empirical support in the context of an emerging market.

Interquantile shrinkage in additive models

ABSTRACTIn this paper, we investigate the commonality of nonparametric component functions among different quantile levels in additive regression models. We propose two fused adaptive group Least Absolute Shrinkage and Selection Operator penalties to shrink the difference of functions between neighbouring quantile levels. The proposed methodology is able to simultaneously estimate the nonparametric functions and identify the quantile regions where functions are unvarying, and thus is expected to perform better than standard additive quantile regression when there exists a region of quantile levels on which the functions are unvarying. Under some regularity conditions, the proposed penalised estimators can theoretically achieve the optimal rate of convergence and identify the true varying/unvarying regions consistently. Simulation studies and a real data application show that the proposed methods yield good numerical results.

Directional quantile regression in Octave (and MATLAB)

Although many words have been written about two recent directional (regression) quantile concepts, their applications, and the algorithms for computing associated (regression) quantile regions, their software implementation is still not widely available, which, of course, severely hinders the dissemination of both methods. Wanting to partly fill in the gap here, we provide all the codes needed for computing and plotting the multivariate (regression) quantile regions in Octave and MATLAB, describe their use in detail, and explain their output with a few carefully designed examples.

Estimation of Extreme Depth-Based Quantile Regions

Summary Consider the extreme quantile region induced by the half-space depth function HD of the form Q={x∈Rd:HD(x,P)⩽β}, such that PQ=p for a given, very small p > 0. Since this involves extrapolation outside the data cloud, this region can hardly be estimated through a fully non-parametric procedure. Using extreme value theory we construct a natural semiparametric estimator of this quantile region and prove a refined consistency result. A simulation study clearly demonstrates the good performance of our estimator. We use the procedure for risk management by applying it to stock market returns.

An inverted U-shaped crude oil price return-implied volatility relationship

We examined the return–volatility relationship for USO ETF oil price return and CBOE Crude Oil ETF Volatility Index, OVX. The data for the USO and OVX covers the period covering May 11, 2007 to February 28, 2013. Our OLS regression results suggest evidence of regular feedback and leverage effects. When we employ linear quantile regression techniques, we find evidence of regular and inverse feedback effects. The inverse feedback effects being noticeable in the upper quantile region of the oil return distribution. There is also support for a regular leverage effect in USO prices. We also examined the return–volatility relationship using quantile regression copula methods for measuring the degree of asymmetry in the relationships between the oil price return and implied volatility. The results of the analysis indicate, first, that there exists a negative relationship between contemporaneous oil VIX and USO ETF oil returns. Second, that the relationship between oil returns and implied volatilities depends on the quartile at which the relationship is being investigated. Third, there exists an inverted U-shaped dependency relationship between returns and implied volatilities across quantiles. Fourth, though an inverted U-shape exists, the shape is different from those observed in stock markets.

Herd Behavior in Emerging Equity Markets: Evidence from Vietnam

This study examines the herding behavior of investors in the Vietnam stock market using a sample of 299 companies listed on the Ho Chi Minh City Stock Exchange covering the time period 2005-2015. Employing the least squares estimation method, we find the evidence of herding in both rising and falling market. Further analysis by applying quantile regression to estimate the herding equation, we find supporting evidence of herding during the whole period studied as well as when splitting the market into two sub-periods; however, the level of this trend is somewhat different conditional on quantile region.

Estimating extreme bivariate quantile regions

When simultaneously monitoring two possibly dependent, positive risks one is often interested in quantile regions with very small probability p. These extreme quantile regions contain hardly any or no data and therefore statistical inference is difficult. In particular when we want to protect ourselves against a calamity that has not yet occurred, we need to deal with probabilities p < 1/n, with n the sample size. We consider quantile regions of the form {(x, y) ∈ (0, ∞ )2: f(x, y) ≤ β}, where f, the joint density, is decreasing in both coordinates. Such a region has the property that it consists of the less likely points and hence that its complement is as small as possible. Using extreme value theory, we construct a natural, semiparametric estimator of such a quantile region and prove a refined form of consistency. A detailed simulation study shows the very good statistical performance of the estimated quantile regions. We also apply the method to find extreme risk regions for bivariate insurance claims.

Computing multiple-output regression quantile regions from projection quantiles

In the multiple-output regression context, Hallin et al. (Ann Statist 38:635–669, 2010) introduced a powerful data-analytical tool based on regression quantile regions. However, the computation of these regions, that are obtained by considering in all directions an original concept of directional regression quantiles, is a very challenging problem. Paindaveine and Siman (Comput Stat Data Anal 2011b) described a first elegant solution relying on linear programming techniques. The present paper provides another solution based on the fact that the quantile regions can also be computed from a competing concept of projection regression quantiles, elaborated in Kong and Mizera (Quantile tomography: using quantiles with multivariate data 2008) and Paindaveine and Siman (J Multivar Anal 2011a). As a by-product, this alternative solution further provides various characteristics useful for statistical inference. We describe in detail the algorithm solving the parametric programming problem involved, and illustrate the resulting procedure on simulated data. We show through simulations that the Matlab implementation of the algorithm proposed in this paper is faster than that from Paindaveine and Siman (Comput Stat Data Anal 2011b) in various cases.

Computing multiple-output regression quantile regions

A procedure relying on linear programming techniques is developed to compute (regression) quantile regions that have been defined recently. In the location case, this procedure allows for computing halfspace depth regions even beyond dimension two. The corresponding algorithm is described in detail, and illustrations are provided both for simulated and real data. The efficiency of a Matlab implementation of the algorithm11The code can be downloaded from http://homepages.ulb.ac.be/~dpaindav. is also investigated through extensive simulations.