Conditional Quantiles Research Articles

Copula‐based semiparametric analysis for time series data with detection limits

AbstractThe analysis of time series data with detection limits is challenging due to the high‐dimensional integral involved in the likelihood. Existing methods are either computationally demanding or rely on restrictive parametric distributional assumptions. We propose a semiparametric approach, where the temporal dependence is captured by parametric copula, while the marginal distribution is estimated non‐parametrically. Utilizing the properties of copulas, we develop a new copula‐based sequential sampling algorithm, which provides a convenient way to calculate the censored likelihood. Even without full parametric distributional assumptions, the proposed method still allows us to efficiently compute the conditional quantiles of the censored response at a future time point, and thus construct both point and interval predictions. We establish the asymptotic properties of the proposed pseudo maximum likelihood estimator, and demonstrate through simulation and the analysis of a water quality data that the proposed method is more flexible and leads to more accurate predictions than Gaussian‐based methods for non‐normal data. The Canadian Journal of Statistics 47: 438–454; 2019 © 2019 Statistical Society of Canada

Moving quantile regression

Quantile regression is a technique to estimate the conditional quantile. In this paper we propose a localized method for quantile regression, the regularized moving quantile regression, which can be used to analyze scattered data efficiently. We present a rigorous global error analysis in the learning theory framework. The main results include an inequality that bridges the gap between the global risk and local risk, a characterization of the approximation that shows the moving technique allows to approximate very complicated functions by simple function classes, and a learning rate analysis. These results indicate that the moving quantile regression method converges fast under mild conditions.

Distributionally Robust Conditional Quantile Prediction with Fixed Design

Conditional quantile prediction involves estimating/predicting the quantile of a response random variable conditioned on observed covariates. The existing literature assumes the availability of independent and identically distributed (i.i.d.) samples of both the covariates and the response variable. However, such an assumption often becomes restrictive in many real-world applications. By contrast, we consider a fixed-design setting of the covariates, under which neither the response variable nor the covariates have i.i.d. samples. The present study provides a new data-driven distributionally robust framework under a fixed-design setting. We propose a regress-then-robustify method by constructing a surrogate empirical distribution of the noise. The solution of our framework coincides with a simple yet practical method that involves only regression and sorting, therefore provides an explanation for its empirical success. Measure concentration results are obtained for the surrogate empirical distribution, which further lead to finite-sample performance guarantees and asymptotic consistency. Numerical experiments are conducted to demonstrate the advantages of our approach.

Quantile-based smooth transition value at risk estimation

SummaryValue at risk models are concerned with the estimation of conditional quantiles of a time series. Formally, these quantities are a function of conditional volatility and the respective quantile of the innovation distribution. The former is often subject to asymmetric dynamic behaviour, e.g., with respect to past shocks. In this paper, we propose a model in which conditional quantiles follow a generalised autoregressive process governed by two parameter regimes with their weights determined by a smooth transition function. We develop a two-step estimation procedure based on a sieve estimator, approximating conditional volatility by using composite quantile regression, which is then used in the generalised autoregressive conditional quantile estimation. We show that the estimator is consistent and asymptotically normal, and we complement the results with a simulation study. In our empirical application, we consider daily returns of the German equity index (DAX) and the USD/GBP exchange rate. Although only the latter follows a two-regime model, we find that our model performs well in terms of out-of-sample prediction in both cases.

Wind and solar power probability density prediction via fuzzy information granulation and support vector quantile regression

Uncertainty among wind and solar power affects the stability of power systems. In order to fully describe the uncertainty of wind and solar power, a probability density prediction model is proposed to predict the probability density function of wind and solar power. According to the wind and solar power time series, the original data is processed with fuzzy information granularity to eliminate the fluctuation and uncertainty of data. The Lagrange function is constructed by a support vector quantile regression model to get the quantile of wind and solar power at different points. The conditional quantiles are combined with the Epanechnikov kernel function to acquire complete probability density curves of forecasting results. In order to evaluate the performance of the output results, this paper analyzes the accuracy of the prediction results using point prediction error, prediction interval coverage probability and average bandwidth. The experimental data of wind and solar power with same temporal and spatial resolutions are taken into account. The results show that the method can effectively describe the uncertainty of wind and solar power, and also provide technical support for the safe and stable operation of the power system.

Comparing Possibly Misspecified Forecasts

Recent work has emphasized the importance of evaluating estimates of a statistical functional (such as a conditional mean, quantile, or distribution) using a loss function that is consistent for the functional of interest, of which there is an infinite number. If forecasters all use correctly specified models free from estimation error, and if the information sets of competing forecasters are nested, then the ranking induced by a single consistent loss function is sufficient for the ranking by any consistent loss function. This article shows, via analytical results and realistic simulation-based analyses, that the presence of misspecified models, parameter estimation error, or nonnested information sets, leads generally to sensitivity to the choice of (consistent) loss function. Thus, rather than merely specifying the target functional, which narrows the set of relevant loss functions only to the class of loss functions consistent for that functional, forecast consumers or survey designers should specify the single specific loss function that will be used to evaluate forecasts. An application to survey forecasts of U.S. inflation illustrates the results.

Quantile Forecast via Serial Dependence Learning

This paper develops a conditional quantile model that can learn long term and short term memories of sequential data. It builds on sequential neural networks and yet outputs interpretable dynamics. We apply the model to asset return time series across eleven asset classes using historical data from the 1960s to 2018. Our results reveal that it extracts not only the serial dependence structure in conditional volatility but also the memories buried deep in the tails of historical prices. We further evaluate its Value-at-Risk forecasts against a wide range of prevailing models. Our model outperforms the GARCH family as well as models using filtered historical simulation, conditional extreme value theory, and dynamic quantile regression. These studies indicate that conditional quantiles of asset return have persistent sources of risk that are not coming from those responsible for volatility clustering. These findings could have important implications for risk management in general and tail risk forecasts in particular.

Analysis and Application of Log-Linear and Quantile Regression Models to Predict Bus Dwell Times

Understanding the key factors that contribute to transit travel times and travel-time variability is an essential part of transit planning and research. Delay that occurs when buses service bus stops, dwell time, is one of the main sources of travel-time variability and has therefore been the subject of ongoing research to identify and quantify its determinants. Previous research has focused on testing new variables using linear regressions that may be added to models to improve predictions. An important assumption of linear regression models used in past research efforts is homoscedasticity or the equal distribution of the residuals across all values of the predicted dwell times. The homoscedasticity assumption is usually violated in linear regression models of dwell time and this can lead to inconsistent and inefficient estimations of the independent variable coefficients. Log-linear models can sometimes correct for the lack of homoscedasticity, that is, for heteroscedasticity in the residual distribution. Quantile regressions, which predict the conditional quantiles, rather than the conditional mean, are non-parametric and therefore more robust estimators in the presence of heteroscedasticity. This research furthers the understanding of established dwell determinants using these novel approaches to estimate dwell and provides a relatively simple approach to improve existing models at bus stops with low average dwell times.

Measuring electronic communication networks in virtual care teams using electronic health records access-log data

ObjectiveTo develop methods for measuring electronic communication networks in virtual care teams using electronic health records (EHR) access-log data. MethodsFor a convenient sample of 100 surgical colorectal cancer patients, we used time-stamped EHR access-log data extracted from an academic medical center’s EHR system to construct communication networks among healthcare professionals (HCPs) in each patient’s virtual care team. We measured communication linkages between HCPs using the inverse of the average time between access events in which the source HCPs sent information to and the destination HCPs retrieved information from the EHR system. Social network analysis was used to examine and visualize communication network structures, identify principal care teams, and detect meaningful structural differences across networks. We conducted a non-parametric multivariate analysis of variance (MANOVA) to test the association between care teams’ communication network structures and patients’ cancer stage and site. ResultsThe 100 communication networks showed substantial variations in size and structures. Principal care teams, the subset of HCPs who formed the core of the communication networks, had higher proportions of nurses, physicians, and pharmacists and a lower proportion of laboratory medical technologists than the overall networks. The distributions of conditional uniform graph quantiles suggested that our network-construction technique captured meaningful underlying structures that were different from random unstructured networks. MANOVA results found that the networks’ topologies were associated with patients’ cancer stage and site. ConclusionsThis study demonstrates that it is feasible to use EHR access-log data to measure and examine communication networks in virtual care teams. The proposed methods captured salient communication patterns in care teams that were associated with patients’ clinical differences.

Copula-based semiparametric models for spatiotemporal data.

The joint analysis of spatial and temporal processes poses computational challenges due to the data's high dimensionality. Furthermore, such data are commonly non-Gaussian. In this paper, we introduce a copula-based spatiotemporal model for analyzing spatiotemporal data and propose a semiparametric estimator. The proposed algorithm is computationally simple, since it models the marginal distribution and the spatiotemporal dependence separately. Instead of assuming a parametric distribution, the proposed method models the marginal distributions nonparametrically and thus offers more flexibility. The method also provides a convenient way to construct both point and interval predictions at new times and locations, based on the estimated conditional quantiles. Through a simulation study and an analysis of wind speeds observed along the border between Oregon and Washington, we show that our method produces more accurate point and interval predictions for skewed data than those based on normality assumptions.

Semiparametric quantile averaging in the presence of high-dimensional predictors

The paper proposes a method for forecasting conditional quantiles. In practice, one often does not know the “true” structure of the underlying conditional quantile function, and in addition, we may have a large number of predictors. Focusing on such cases, we introduce a flexible and practical framework based on penalized high-dimensional quantile averaging. In addition to prediction, we show that the proposed method can also serve as a predictor selector. We conduct extensive simulation experiments to asses its prediction and variable selection performances for nonlinear and linear time series model designs. In terms of predictor selection, the approach tends to select the true set of predictors with minimal false positives. With respect to prediction accuracy, the method competes well even with the benchmark/oracle methods that know one or more aspects of the underlying quantile regression model. We further illustrate the merit of the proposed method by providing an application to the out-of-sample forecasting of U.S. core inflation using a large set of monthly macroeconomic variables based on FRED-MD database. The application offers several empirical findings.

Quantile regression and empirical likelihood for the analysis of longitudinal data with monotone missing responses due to dropout, with applications to quality of life measurements from clinical trials.

The analysis of quality of life (QoL) data can be challenging due to the skewness of responses and the presence of missing data. In this paper, we propose a new weighted quantile regression method for estimating the conditional quantiles of QoL data with responses missing at random. The proposed method makes use of the correlation information within the same subject from an auxiliary mean regression model to enhance the estimation efficiency and takes into account of missing data mechanism. The asymptotic properties of the proposed estimator have been studied and simulations are also conducted to evaluate the performance of the proposed estimator. The proposed method has also been applied to the analysis of the QoL data from a clinical trial on early breast cancer, which motivated this study.

Asymptotic inference for the constrained quantile regression process

I investigate the asymptotic distribution of linear quantile regression coefficient estimates when the parameter lies on the boundary of the parameter space. In order to allow for inferences made across many conditional quantiles, I provide a uniform characterization of constrained quantile regression estimates as a stochastic process over an interval of quantile levels. To do this I pose the process of estimates as solutions to a parameterized family of constrained optimization problems, parameterized by quantile level. A uniform characterization of the dual solution to these problems – the so-called regression rankscore process – is also derived, which can be used for score-type inference in quantile regression. The asymptotic behavior of quasi-likelihood ratio, Wald and regression rankscore processes for inference when the null hypothesis asserts that the parameters lie on a boundary follows from the features of the constrained solutions.

Estimation of extreme conditional quantiles under a general tail-first-order condition

We consider the estimation of an extreme conditional quantile. In a first part, we propose a new tail condition in order to establish the asymptotic distribution of an extreme conditional quantile estimator. Next, a general class of estimators is introduced, which encompasses, among others, kernel or nearest neighbors types of estimators. A unified theorem of the asymptotic normality for this general class of estimators is provided under the new tail condition and illustrated on the different well-known examples. A comparison between different estimators belonging to this class is provided on a small simulation study and illustrated on a real dataset on earthquake magnitudes.

Semi-parametric copula-based models under non-stationarity

In this paper, we consider non-stationary random vectors, where the marginal distributions and the associated copula may be time-dependent. We propose estimators for the unknown parameters and we establish the limiting distribution of the estimators of the copula and the conditional copula, together with a parametric bootstrap method for constructing confidence bands around the estimator and for testing the adequacy of the model. We also consider three examples of functionals of the copula-based model under non-stationarity: conditional quantiles, conditional means, and conditional expected shortfalls. The asymptotic distribution of the estimation errors is shown to be Gaussian, and bootstrapping methods are proposed to estimate their asymptotic variances. The finite-sample performance of our estimators is investigated through Monte Carlo experiments, and we give three examples of implementation of the proposed methodology.

Causal inference by quantile regression kink designs

The quantile regression kink design (QRKD) is proposed by empirical researchers, but its causal interpretation remains unknown. We show that the QRKD estimand measures a weighted average of heterogeneous marginal effects at respective conditional quantiles of outcome given a designed kink point. We also derive limit processes for the QRKD estimator to conduct statistical inference on heterogeneous treatment effects using the QRKD. Applying our methods to the Continuous Wage and Benefit History Project (CWBH) data, we find heterogeneous positive causal effects of unemployment insurance benefits on unemployment durations. These effects are larger for individuals with longer unemployment durations.

Analysis for instability of parameter in quantile regression with Lagrange multiplier: Is the dependent and independent variable relationships have changed?

Quantile regression is a statistical method to estimate the relationship between the dependent variable and independent variables on particular conditional quantile functions. Quantile regression method approach is done by separating the data into two or more groups which are suspected to have a different estimated value of a certain quantile. Test parameter instability and the structural break is a relevant issue. Some tests that have been carried out to verify the existence of structural break have largely focused only on conditional mean, while a structural break that occurs in the conditional distribution or in the conditional quantiles is an important key. This article aims to provide an insight into the test of a structural break in quantile regression with the Lagrange Multiplier. Explanation will be accompanied by the identification of structural break in the analysis of the relationship between household food expenditure and household income in the Belgian community household. The result shows that a structural break occurs on the 10th and 25thquantiles.

Joint estimation of conditional quantiles in multivariate linear regression models with an application to financial distress

This paper proposes a maximum likelihood approach to jointly estimate marginal conditional quantiles of multivariate response variables in a linear regression framework. We consider a slight reparameterization of the multivariate asymmetric Laplace distribution proposed by Kotz et al. (2001) and exploit its location–scale mixture representation to implement a new EM algorithm for estimating model parameters. The idea is to extend the link between the asymmetric Laplace distribution and the well-known univariate quantile regression model to a multivariate context, i.e., when a multivariate dependent variable is concerned. The approach accounts for association among multiple responses and studies how the relationship between responses and explanatory variables can vary across different quantiles of the marginal conditional distribution of the responses. A penalized version of the EM algorithm is also presented to tackle the problem of variable selection. The validity of our approach is analyzed in a simulation study, where we also provide evidence on the efficiency gain of the proposed method compared to estimation obtained by separate univariate quantile regressions. A real data application examines the main determinants of financial distress in a sample of Italian firms.

Quantile effect of educational facilities on housing price: Do homebuyers of higher-priced housing pay more for educational resources?

Educational facilities are valued highly by citizens and play an important part in improving housing prices. Most existing studies applied the hedonic price model to obtain the average effects of educational facilities on housing prices. However, these effects may be heterogeneous in different conditional quantiles of housing prices. Aside from the average effects, scholars are also interested in the different behaviors of high-, medium-, and low-priced homebuyers, behaviors that cannot be simply explained by ordinary least-squares estimation without sample truncation. To fill this gap, this study utilizes quantile and spatial quantile regression models to obtain a complete description of the entire distribution. On the basis of 2015 housing transaction data in Hangzhou, substantial differences are identified in the effects of educational facilities on housing price across the conditional distribution. In particular, primary and secondary schools considerably improve housing prices over the entire conditional distribution, whereas kindergartens are only valued by homebuyers of lower- and higher-priced housing. Furthermore, only households of high-priced housing are willing to pay a premium for the accessibility of senior high schools and universities. The effect of spatial dependence on higher-priced housing is more pronounced than that on lower-priced housing, and the consideration of spatial attributes improves the explanatory ability. Empirical results indicate that the average effects of educational facilities on housing prices obtained by existing studies that used the traditional hedonic price model is far from adequate in shedding light on this real-world phenomenon. In particular, the quantile effect should be further revealed to obtain an in-depth understanding.

Bayesian non-crossing quantile regression for regularly varying distributions

ABSTRACTQuantile 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.