Bayesian Quantile Regression Research Articles

A Study of Bayesian Quantile Regression for Forecasting RMB Exchange Rates

Accurate forecasting of the RMB exchange rate is crucial for global financial market participants. This study proposes a Bayesian quantile regression approach to enhance the forecasting method. This paper uses RMB and US dollar exchange rate data from the State Administration of Foreign Exchange from 2018 to 2022 to build a Bayesian quantile regression model and empirically analyze the RMB exchange rate forecast. The results show that the proposed Bayesian quantile regression model yields accurate forecasts, with a root mean squared error (RMSE) of 1.8329 and a mean absolute error (MAE) of 1.2988. Furthermore, robustness and sensitivity analyses confirm the model's reliability. The findings of this study have practical implications for financial market participants and policymakers in managing and responding to foreign exchange risk.

Bayesian Quantile Regression Analysis for Bivariate Vector Autoregressive Models with an Application to Financial Time Series

Bayesian Quantile Regression Analysis for Bivariate Vector Autoregressive Models with an Application to Financial Time Series

Horseshoe prior Bayesian quantile regression

Abstract This paper extends the horseshoe prior to Bayesian quantile regression and provides a fast sampling algorithm for computation in high dimensions. Compared to alternative shrinkage priors, our method yields better performance in coefficient bias and forecast error, especially in sparse designs and in estimating extreme quantiles. In a high-dimensional growth-at-risk forecasting application, we forecast tail risks and complete forecast densities using a database covering over 200 macroeconomic variables. Quantile specific and density calibration score functions show that our method provides competitive performance compared to competing Bayesian quantile regression priors, especially at short- and medium-run horizons.

Bayesian mixed-frequency quantile vector autoregression: Eliciting tail risks of monthly US GDP

Timely characterizations of risks in economic and financial systems play an essential role in both economic policy and private sector decisions. However, the informational content of low-frequency variables and the results from conditional mean models provide only limited evidence to investigate this problem. We propose a novel mixed-frequency quantile vector autoregression (MF-QVAR) model to address this issue. Inspired by the univariate Bayesian quantile regression literature, the multivariate asymmetric Laplace distribution is exploited under the Bayesian framework to form the likelihood. A data augmentation approach coupled with a precision sampler efficiently estimates the missing low-frequency variables at higher frequencies under the state-space representation.The proposed methods allow us to analyse conditional quantiles for multiple variables of interest and to derive quantile-related risk measures at high frequency, thus enabling timely policy interventions. The main application of the model is to detect the vulnerability in the US economy and then to nowcast conditional quantiles of the US GDP, which is strictly related to the quantification of Value-at-Risk, the Expected Shortfall and distance among percentiles of real GDP nowcasts.

Effect of agricultural carbon emissions on farmers' health expenditure of China: evidence from the educational threshold effect.

Environmental pollution, especially agricultural carbon emissions (ACE), has led to public health problems to rural areas in China and accompanied by a heavy medical economic burden. However, most studies on carbon dioxide emissions and healthcare expenditures focused on the industrial sector, and the effect of ACE was overlooked. Therefore, studying the effect of ACE on rural residents' healthcare expenditures (NHCE) is not only conducive to accelerating the low-carbon transformation of agriculture but also has important implications for reducing healthcare expenditures. In addition, the effect of ACE on NHCE in different areas might be complex and nonlinear due to differences in years of schooling (EDU) leading to different awareness of environmental protection and health among farmers. Therefore, this paper used the Bayesian quantile regression (BQR) model and the panel threshold model to explore the effect of ACE on NHCE in different areas, based on the panel data of 31 provinces in China from 2007 to 2019. The results showed that ACE and NHCE experienced similar spatial distribution from 2007 to 2019. The BQR estimation results found that ACE had a significant positive effect on NHCE at different quantile levels during the sample period, public health concern, and thereby increasing the medical and economic burden of rural households. Meanwhile, ACE had a positive effect on NHCE with a significant single threshold effect from EDU. Specifically, farmers gradually realize the harm of environmental pollution to health with the continuous improvement of education level, and then ACE aggravated the improvement of NHCE after exceeding the threshold. EDU was more essential for farmers in contiguous poverty (CP) areas than in relatively developed (RD) areas and played an important role between ACE and NHCE. Furthermore, demographic structure, economic development, and public services were also positive driving factors for NHCE. The results of analysis provide a valuable reference for understanding the factors influencing NHCE and enable formulation of ACE emission reduction policies according to local conditions.

Comparison of Bayesian and frequentist quantile regressions in studying the trend of discharge changes in several hydrometric stations of the Gorganroud basin in Iran

Abstract This research utilized Bayesian and quantile regression techniques to analyze trends in discharge levels across various seasons for three stations in the Gorganroud basin of northern Iran. The study spanned a period of 50 years (1966–2016). Results indicate a decrease in high discharge rates during springtime for the Arazkouseh and Galikesh stations, with a steep slope of −0.31 m3/s per year for Arazkouseh and −0.19 and −0.17 for Galikesh. Furthermore, Tamar station experienced an increase in very high discharge during summer, with a slope of 0.12 m3/s per year. However, low discharge rates remained relatively unchanged. Arazkouseh station showed a higher rate of decreasing discharge levels and this trend was most prominent during spring. Additionally, the Bayesian quantile regression model proved to be more accurate and reliable than the frequency-oriented quantile regression model. These findings suggest that quantile regression models are a valuable tool for predicting and managing extremely high and low discharge changes, ultimately reducing the risk of flood and drought damage.

Not your average firm: A quantile regression approach to firm‐level investment in the United States

AbstractA significant portion of the work published on firm investment adapts models that operate on an “average firm” assumption, which is different from the investment behavior of a modal firm. This study employs a Bayesian quantile regression model to explore the investment rates in the United States and finds, first, that the firms with higher investment rates have a higher responsiveness to the valuation ratio and lower responsiveness to the profit rate, and, second, that there is a decline in the responsiveness of firm investment to these factors in recent years. The paper also emphasizes the role of autonomous investments in determining firm‐level investment rates, based on differing sectoral factors.

Hypothalamus volumes in adolescent Myalgic Encephalomyelitis/Chronic Fatigue Syndrome (ME/CFS): impact of self-reported fatigue and illness duration

Adolescent Myalgic Encephalomyelitis/Chronic Fatigue Syndrome (ME/CFS) is a complex illness of unknown aetiology. Emerging theories suggest ME/CFS may reflect a progressive, aberrant state of homeostasis caused by disturbances within the hypothalamus, yet few studies have investigated this using magnetic resonance imaging in adolescents with ME/CFS. We conducted a volumetric analysis to investigate whether whole and regional hypothalamus volumes in adolescents with ME/CFS differed compared to healthy controls, and whether these volumes were associated with fatigue severity and illness duration. 48 adolescents (25 ME/CFS, 23 controls) were recruited. Lateralised whole and regional hypothalamus volumes, including the anterior–superior, superior tubular, posterior, anterior-inferior and inferior tubular subregions, were calculated from T1-weighted images. When controlling for age, sex and intracranial volume, Bayesian linear regression models revealed no evidence for differences in hypothalamus volumes between groups. However, in the ME/CFS group, a weak linear relationship between increased right anterior–superior volumes and fatigue severity was identified, which was absent in controls. In addition, Bayesian quantile regression revealed a likely-positive association between illness duration and right superior tubular volumes in the ME/CFS group. While these findings suggest overall comparability in regional and whole hypothalamus volumes between adolescents with ME/CFS and controls, preliminary evidence was identified to suggest greater fatigue severity and longer illness duration were associated with greater right anterior–superior and superior-tubular volumes, respectively. These regions contain the anterior and superior divisions of the paraventricular nucleus, involved in the neuroendocrine response to stress, suggesting involvement in ME/CFS pathophysiology. However, replication in a larger, longitudinal cohort is required.

Separating Hydraulic Fracturing Microseismicity From Induced Seismicity by Bayesian Inference of Non‐Linear Pressure Diffusivity

AbstractMicroseismic data acquired during hydraulic stimulations is routinely used to characterize hydraulic fracture (HF) propagation away from a wellbore. When the data include induced seismicity (IS) related to induced fault slip, separating HF‐related and IS‐related events is essential for HF treatment optimization and IS mitigation. Linear and non‐linear analytical diffusivity models can be used to interpret microseismic data and quantify the propagation of HFs, but their accuracy reduces when significant induced microseismicity is present. A Bayesian quantile regression is used to extend these existing diffusivity models to data contaminated with IS. A plausible ellipsoid filters events that are clearly anomalous prior to the quantile regression. The regression effectively separates HF‐related microseismic events from induced events in a case study for all stages without interpretative bias. This reveals faults that are directed connected to HFs, as well as those solely related with induced seismicity.

Quantile regression based on the skewed exponential power distribution

Bayesian quantile regression generally relies on the asymmetric Laplace distribution (ALD) as the error distribution. We consider methods for Lp -quantile regression based on the skewed exponential power distribution (SEPD). Both Bayesian and frequentist estimation procedures are outlined and compared with previous work based on the SEPD. We find that our proposed methods greatly outperform a previous method in terms of quantile estimation. Further, compared with standard quantile regression, we find that our proposed methods generally perform better in terms of root mean square error (RMSE). Empirical evidence of the statistical properties of the proposed models is provided through a simulation study. Further, a real data application illustrates their performance.

Exact convergence analysis for metropolis–hastings independence samplers in Wasserstein distances

AbstractUnder mild assumptions, we show that the exact convergence rate in total variation is also exact in weaker Wasserstein distances for the Metropolis–Hastings independence sampler. We develop a new upper and lower bound on the worst-case Wasserstein distance when initialized from points. For an arbitrary point initialization, we show that the convergence rate is the same and matches the convergence rate in total variation. We derive exact convergence expressions for more general Wasserstein distances when initialization is at a specific point. Using optimization, we construct a novel centered independent proposal to develop exact convergence rates in Bayesian quantile regression and many generalized linear model settings. We show that the exact convergence rate can be upper bounded in Bayesian binary response regression (e.g. logistic and probit) when the sample size and dimension grow together.

A novel approach to recognize the long-term spatial-temporal pattern of dry and wet years over Iran

More studies in estimating rainfall variations have focused on examining the change in the series' mean/median, which does not lead to the analysis of wet and dry conditions. Variations in wet and dry years, equivalent to changes in the upper and lower quantiles, have critical effects on human, environmental, and ecological activities. This study investigated variations in wet and dry years compared to the normal years in Iran using long-term data (1966–2019) from 41 synoptic stations. The Bayesian quantile regression model was used to study variations in annual rainfall quantiles. In addition, the spatial-temporal distribution of variations was analyzed. The results reveal that the intensities of wet years (Q90) variations are significantly higher than dry years (Q10) and normal years (Q50). In addition, wet and dry years show decreasing and increasing behaviors in most parts of Iran. Wet and dry years variations in the western, north-western, and northern regions of Iran have been more severe than the rest of the country. Moreover, the results reveal that disregarding variations in wet and dry years can lead to inappropriate inferences about water-related activities.

Flexible Bayesian quantile regression for nonlinear mixed effects models based on the generalized asymmetric Laplace distribution

We propose flexible Bayesian quantile regression for a class of parametric nonlinear mixed effects models for longitudinal data based on the generalized asymmetric Laplace distribution, which exhibits more flexibility in skewness, mode and tail behaviour than the frequently used asymmetric Laplace distribution in quantile regression. An efficient Markov chain Monte Carlo procedure based on the adaptive random walk Metropolis-within-Gibbs sampling algorithm is derived for posterior inference. We demonstrate through simulation studies and empirical analysis that the proposed method could provide more accurate parameter estimation and better model fit than the existing methods.

Surface roughness prediction using multi-source heterogeneous data and Bayesian quantile regression in milling process

To quantify the influence of cutting parameters on surface roughness and improve the prediction accuracy, a novel method is proposed for surface roughness prediction in milling process. A quantile regression (QR) model is constructed using multi-source heterogeneous (MSH) data comprising monitoring signals and cutting parameters within a Bayesian framework. The model parameters are then estimated using Gibbs sampling, and the predicted roughness and confidence interval (CI) can be obtained using the proposed Bayesian quantile model (BQR). Experimental results show that the proposed method has a prediction error of 15.05 %, which is 13.11 %, 0.01 %, and 1.7 % lower than least squares regression (LSR), back propagation network (BPN), and Bayesian linear regression (BLR), respectively. Through the comparison between predictions and experimental measurements, the proposed method has no great advantages over BPN in terms of prediction error, and is even 2.39 % higher than the prediction error of Support Vector Machine (SVM). However, the advantage of the proposed method compared with the above methods is that it can provide the analysis results of the influence of cutting parameters on surface roughness, which has practical significance in providing guidance for operators to reduce surface roughness in the milling process.

Embracing asymmetry in nature: How to account for skewness in ecological data

Asymmetric regression is an alternative to conventional linear regression that allows us to model the relationship between predictor variables and the response variable while accommodating skewness. Advantages of asymmetric regression include incorporating realistic ecological patterns observed in data, robustness to model misspecification and less sensitivity to outliers. Bayesian asymmetric regression relies on asymmetric distributions such as the asymmetric Laplace (ALD) or asymmetric normal (AND) in place of the normal distribution used in classic linear regression models. Asymmetric regression concepts can be used for process and parameter components of hierarchical Bayesian models and have a wide range of applications in data analyses. In particular, asymmetric regression allows us to fit more realistic statistical models to skewed data and pairs well with Bayesian inference. We first describe asymmetric regression using the ALD and AND. Second, we show how the ALD and AND can be used for Bayesian quantile and expectile regression for continuous response data. Third, we consider an extension to generalize Bayesian asymmetric regression to survey data consisting of counts of objects. Fourth, we describe a regression model using the ALD, and show that it can be applied to add needed flexibility, resulting in better predictive models compared to Poisson or negative binomial regression. We demonstrate concepts by analyzing a data set consisting of counts of Henslow’s sparrows following prescribed fire and provide annotated computer code to facilitate implementation. Our results suggest Bayesian asymmetric regression is an essential component of a scientist’s statistical toolbox.

Bayesian Conditional Transformation Models

Recent developments in statistical regression methodology shift away from pure mean regression towards distributional regression models. One important strand thereof is that of conditional transformation models (CTMs). CTMs infer the entire conditional distribution directly by applying a transformation function to the response conditionally on a set of covariates towards a simple log-concave reference distribution. Thereby, CTMs allow not only variance, kurtosis or skewness but the complete conditional distribution to depend on the explanatory variables. We propose a Bayesian notion of conditional transformation models (BCTMs) focusing on exactly observed continuous responses, but also incorporating extensions to randomly censored and discrete responses. Rather than relying on Bernstein polynomials that have been considered in likelihood-based CTMs, we implement a spline-based parametrization for monotonic effects that are supplemented with smoothness priors. Furthermore, we are able to benefit from the Bayesian paradigm via easily obtainable credible intervals and other quantities without relying on large sample approximations. A simulation study demonstrates the competitiveness of our approach against its likelihood-based counterpart but also Bayesian additive models of location, scale and shape and Bayesian quantile regression. Two applications illustrate the versatility of BCTMs in problems involving real world data, again including the comparison with various types of competitors.

Bayesian scale mixtures of normals linear regression and Bayesian quantile regression with big data and variable selection

Quantile regression, which estimates various conditional quantiles of a response variable, including the median (0.5th quantile), is particularly useful when the conditional distribution is asymmetric or heterogeneous or fat-tailed or truncated. Bayesian methods for the inference of quantile regression have been receiving increasing attention from both theoretical and empirical viewpoints but facing the challenge of scaling up when the data are too large to be processed by a single machine under many big data environments nowadays. In this paper, we develop a structure link between Bayesian scale mixtures of normals linear regression and Bayesian quantile regression (BQR) via normal-inverse-gamma (NIG) distribution type of likelihood function, prior distribution and posterior distribution. We further explore the detailed methods of BQR for big data, variable selection and posterior prediction. The performance of the proposed techniques is evaluated via simulation studies and a real data analysis.

How spatial patterns affect urban green space equity at different equity levels: A Bayesian quantile regression approach

In many cities, urban green space (UGS) is not evenly distributed, and marginalized communities often lack UGS due to legacies of disinvestment. Existing research mainly focused on the effect of socioeconomic factors to understand the disparity in access to UGS. However, little is known on how spatial patterns affect UGS equality at the regional and subregional levels. Moreover, the potential quantile effects in the UGS pattern and equality relationship remain unclear. Here, we explored how UGS equality varied among UGS spatial patterns and socioeconomic gradients in seven counties (1,227 census tracts) in Southeast Michigan. First, we quantified UGS equality of each tract with a spatially explicit Gini coefficient and UGS spatial patterns via landscape metrics. Then, we applied a Bayesian quantile regression model to investigate the nonlinear relationship between spatial pattern and UGS equality in the whole study area and its three subregions of different population density. Our results showed that spatial patterns had quantile effects on UGS equality. At the regional scale, patch density and the largest patch index had significant positive effects on UGS equality at all equality levels. The mean patch shape index positively correlated with UGS equality in areas with higher equality level (Gini coefficient: 0.52–0.92). At the subregional level, patch density most efficiently predicted UGS equality in densely populated areas, while the largest patch index also affected UGS equality in areas with low population density. We suggest policymakers should focus on areas with extremely unequal access to UGS to increase the number rather than the cumulative area of UGS. Besides, more linear and irregular UGS patches with longer perimeters have potential to serve more communities.

Leading pedestrian intervals – Yay or Nay? A Before-After evaluation of multiple conflict types using an enhanced Non-Stationary framework integrating quantile regression into Bayesian hierarchical extreme value analysis

A pedestrian was estimated to be killed every 85 min and injured every 7 min on US roads in 2019. Targeted safety treatments are particularly required at urban intersections where pedestrians regularly conflict with turning vehicles. Leading Pedestrian Intervals (LPIs) are an innovative, low-cost treatment where the pedestrian and vehicle usage of the potential conflict area (a crosswalk) is staggered in time to give the pedestrians a head start of a few seconds and reduce the “element of surprise” for right-turning vehicles. The effectiveness of LPI treatment on pedestrian safety is mixed, and most importantly, its effect on vehicle-vehicle conflicts is unknown. This study investigates the before-after effects of LPI treatments on vehicle–pedestrian and vehicle-vehicle crash risk by applying traffic conflict techniques. In particular, this study has developed a quantile regression technique within the extreme value model to estimate and compare crash risks before and after the installation of the LPI treatment. The before-after traffic movement video data (504 h in total) were collected from three signalized intersections in the City of Bellevue, Washington. The recorded movements were analyzed using Microsoft’s proprietary computer vision platform, Edge Video Service, and Advanced Mobility Analytics Group’s cloud-based SMART SafetyTM platform to automatedly extract traffic conflicts by analyzing road user trajectories. The treatment effect was measured using a Bayesian hierarchical extreme value model with the peak-over threshold approach. For the extreme value model, a Bayesian quantile regression analysis was conducted to estimate the conflict thresholds corresponding to a high (95th) quantile. Odds ratios were estimated for both conflict types using untreated crossing as a control group. Results indicate that the LPI treatment reduces the crash risk of pedestrians as measured by the reduction in extreme vehicle–pedestrian conflicts by about 42%. The LPI treatment has also been found not to negatively affect rear-end conflicts along the approaches leading to the LPI-treated pedestrian crossing at the signalized intersections. The findings of this study further emphasize the effectiveness of video analytics in proactive safety evaluations of engineering treatments.

A Bayesian approach for more reliable tail risk forecasts

This paper demonstrates that existing quantile regression models used for jointly forecasting Value-at-Risk (VaR) and expected shortfall (ES) are sensitive to initial conditions. Given the importance of these measures in financial systems, this sensitivity is a critical issue. A new Bayesian quantile regression approach is proposed for estimating joint VaR and ES models. By treating the initial values as unknown parameters, sensitivity issues can be dealt with. Furthermore, new additive-type models are developed for the ES component that are more robust to initial conditions. A novel approach using the open-faced sandwich (OFS) method is proposed which improves uncertainty quantification in risk forecasts. Simulation and empirical results highlight the improvements in risk forecasts ensuing from the proposed methods.