Quantile regression for the qualifying match of GEFCom2017 probabilistic load forecasting

We present a simple quantile regression-based forecasting method that was applied in the probabilistic load forecasting framework of the Global Energy Forecasting Competition 2017 (GEFCom2017). The hourly load data are log transformed and split into a long-term trend component and a remainder term. The key forecasting element is the quantile regression approach for the remainder term, which takes into account both weekly and annual seasonalities, such as their interactions. Temperature information is used only for stabilizing the forecast of the long-term trend component. Information on public holidays is ignored. However, the forecasting method still placed second in the open data track and fourth in the definite data track, which is remarkable given the simplicity of the model. The method also outperforms the Vanilla benchmark consistently.     

SCAD‐penalized quantile regression for high‐dimensional data analysis and variable selection

The present penalized quantile variable selection methods are only applicable to finite number of predictors or do not have oracle property associated with estimator. This technique is considered as an alternative to ordinary least squares regression in case of the outliers and the heavy‐tailed errors existing in linear models. The variable selection through quantile regression with diverging number of parameters is investigated in this paper. The convergence rate of estimator with smoothly clipped absolute deviation penalty function is also studied. Moreover, the oracle property with proper selection of tuning parameter for quantile regression under certain regularity conditions is also established. In addition, the rank correlation screening method is used to accommodate ultra‐high dimensional data settings. Monte Carlo simulations demonstrate finite performance of the proposed estimator. The results of real data reveal that this approach provides substantially more information as compared with ordinary least squares, conventional quantile regression, and quantile lasso.     

Probabilistic energy forecasting using the nearest neighbors quantile filter and quantile regression

Parametric quantile regression is a useful tool for obtaining probabilistic energy forecasts. Nonetheless, traditional quantile regressions may be complicated to obtain using complex data mining techniques (e.g., artificial neural networks), since they are trained using a non-differentiable cost function. This article presents a method that uses a new nearest neighbors quantile filter to obtain quantile regressions independently of the data mining technique utilized and without the non-differentiable cost function. This method is subsequently validated using the dataset from the 2014 Global Energy Forecasting Competition. The results show that the method presented here is able to solve the competition’s task with a similar accuracy to the competition’s winner and in a similar timeframe, but requiring a much less powerful computer. This property may be relevant in an online forecasting service for which the fast computation of probabilistic forecasts using less powerful machines is required.     

Bayesian analysis of a Tobit quantile regression model

This paper develops a Bayesian framework for Tobit quantile regression. Our approach is organized around a likelihood function that is based on the asymmetric Laplace distribution, a choice that turns out to be natural in this context. We discuss families of prior distributions on the quantile regression vector that lead to proper posterior distributions with finite moments. We show how the posterior distribution can be sampled and summarized by Markov chain Monte Carlo methods. A method for comparing alternative quantile regression models is also developed and illustrated. The techniques are illustrated with both simulated and real data. In particular, in an empirical comparison, our approach out-performed two other common classical estimators.     

Quantile cointegrating regression

Quantile regression has important applications in risk management, portfolio optimization, and asset pricing. The current paper studies estimation, inference and financial applications of quantile regression with cointegrated time series. In addition, a new cointegration model with quantile-varying coefficients is proposed. In the proposed model, the value of cointegrating coefficients may be affected by the shocks and thus may vary over the innovation quantile. The proposed model may be viewed as a stochastic cointegration model which includes the conventional cointegration model as a special case. It also provides a useful complement to cointegration models with (G)ARCH effects. Asymptotic properties of the proposed model and limiting distribution of the cointegrating regression quantiles are derived. In the presence of endogenous regressors, fully-modified quantile regression estimators and augmented quantile cointegrating regression are proposed to remove the second order bias and nuisance parameters. Regression Wald tests are constructed based on the fully modified quantile regression estimators. An empirical application to stock index data highlights the potential of the proposed method.     

Common correlated effects estimation of heterogeneous dynamic panel quantile regression models

This paper proposes a quantile regression estimator for a heterogeneous panel model with lagged dependent variables and interactive effects. The paper adopts the Common Correlated Effects (CCE) approach proposed in the literature and demonstrates that the extension to the estimation of dynamic quantile regression models is feasible under similar conditions to the ones used in the literature. The new quantile regression estimator is shown to be consistent and its asymptotic distribution is derived. Monte Carlo studies are carried out to study the small sample behavior of the proposed approach. The evidence shows that the estimator can significantly improve on the performance of existing estimators as long as the time series dimension of the panel is large. We present an application to the evaluation of Time-of-Use pricing using a large randomized control trial.     

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.     

Modelling firm‐size distribution using Box–Cox heteroscedastic regression

Using the Box–Cox regression model with heteroscedasticity (BCHR), we re‐examine the size distribution of the Portuguese manufacturing firms studied by Machado and Mata ( 2000 ) using the Box–Cox quantile regression (BCQR) method. We show that the BCHR model compares favourably against the BCQR method. In particular, the BCHR model can answer the key questions addressed by the BCQR method, with the advantage that the estimated quantile functions are monotonic. Furthermore, estimation of the BCHR model is straightforward and the confidence intervals of the BCHR regression quantiles are easy to compute. Copyright © 2006 John Wiley & Sons, Ltd.     

Posterior Inference in Bayesian Quantile Regression with Asymmetric Laplace Likelihood

The paper discusses the asymptotic validity of posterior inference of pseudo‐Bayesian quantile regression methods with complete or censored data when an asymmetric Laplace likelihood is used. The asymmetric Laplace likelihood has a special place in the Bayesian quantile regression framework because the usual quantile regression estimator can be derived as the maximum likelihood estimator under such a model, and this working likelihood enables highly efficient Markov chain Monte Carlo algorithms for posterior sampling. However, it seems to be under‐recognised that the stationary distribution for the resulting posterior does not provide valid posterior inference directly. We demonstrate that a simple adjustment to the covariance matrix of the posterior chain leads to asymptotically valid posterior inference. Our simulation results confirm that the posterior inference, when appropriately adjusted, is an attractive alternative to other asymptotic approximations in quantile regression, especially in the presence of censored data.     

Regression Revisited

Sir Francis Galton introduced median regression and the use of the quantile function to describe distributions. Very early on the tradition moved to mean regression and the universal use of the Normal distribution, either as the natural ‘error’ distribution or as one forced by transformation. Though the introduction of ‘quantile regression’ refocused attention on the shape of the variability about the line, it uses nonparametric approaches and so ignores the actual distribution of the ‘error’ term. This paper seeks to show how Galton's approach enables the complete regression model, deterministic and stochastic elements, to be modelled, fitted and investigated. The emphasis is on the range of models that can be used for the stochastic element. It is noted that as the deterministic terms can be built up from components, so to, using quantile functions, can the stochastic element. The model may thus be treated in both modelling and fitting as a unity. Some evidence is presented to justify the use of a much wider range of distributional models than is usually considered and to emphasize their flexibility in extending regression models.     

Quantile regression for dynamic panel data with fixed effects

This paper studies a quantile regression dynamic panel model with fixed effects. Panel data fixed effects estimators are typically biased in the presence of lagged dependent variables as regressors. To reduce the dynamic bias, we suggest the use of the instrumental variables quantile regression method of Chernozhukov and Hansen (2006) along with lagged regressors as instruments. In addition, we describe how to employ the estimated models for prediction. Monte Carlo simulations show evidence that the instrumental variables approach sharply reduces the dynamic bias, and the empirical levels for prediction intervals are very close to nominal levels. Finally, we illustrate the procedures with an application to forecasting output growth rates for 18 OECD countries.     

Bayesian median autoregression for robust time series forecasting

We develop a Bayesian median autoregressive (BayesMAR) model for time series forecasting. The proposed method utilizes time-varying quantile regression at the median, favorably inheriting the robustness of median regression in contrast to the widely used mean-based methods. Motivated by a working Laplace likelihood approach in Bayesian quantile regression, BayesMAR adopts a parametric model bearing the same structure as autoregressive models by altering the Gaussian error to Laplace, leading to a simple, robust, and interpretable modeling strategy for time series forecasting. We estimate model parameters by Markov chain Monte Carlo. Bayesian model averaging is used to account for model uncertainty, including the uncertainty in the autoregressive order, in addition to a Bayesian model selection approach. The proposed methods are illustrated using simulations and real data applications. An application to U.S. macroeconomic data forecasting shows that BayesMAR leads to favorable and often superior predictive performance compared to the selected mean-based alternatives under various loss functions that encompass both point and probabilistic forecasts. The proposed methods are generic and can be used to complement a rich class of methods that build on autoregressive models.     

基于双自适应Lasso惩罚的随机效应分位回归模型研究

研究目标：解决随机效应分位回归模型中固定效应和随机效应系数同时估计和选择问题。研究方法：对固定效应和随机效应系数同时实施自适应Lasso惩罚,并为参数估计设计交替迭代算法。研究发现：新方法不仅对随机误差分布具有较强的稳健性,而且在不同稀疏度模型下均有着良好的表现,尤其是在高维情形时。研究创新：本文提出的方法在对模型中重要自变量进行选择的同时能够充分考虑随机效应的影响;交替迭代算法不仅有效解决了需要选择两个惩罚参数的困境,而且收敛速度快。研究价值：为实际工作者对面板数据和纵向数据的分析提供了有效的建模方法。     

Quantile models and estimators for data analysis

Quantile regression is used to estimate the cross sectional relationship between high school characteristics and student achievement as measured by ACT scores. The importance of school characteristics on student achievement has been traditionally framed in terms of the effect on the expected value. With quantile regression the impact of school characteristics is allowed to be different at the mean and quantiles of the conditional distribution. Like robust estimation, the quantile approach detects relationships missed by traditional data analysis. Robust estimates detect the influence of the bulk of the data, whereas quantile estimates detect the influence of co-variates on alternate parts of the conditional distribution. Since our design consists of multiple responses (individual student ACT scores) at fixed explanatory variables (school characteristics) the quantile model can be estimated by the usual regression quantiles, but additionally by a regression on the empirical quantile at each school. This is similar to least squares where the estimate based on the entire data is identical to weighted least squares on the school averages. Unlike least squares however, the regression through the quantiles produces a different estimate than the regression quantiles.     

ADMM for Penalized Quantile Regression in Big Data

Traditional linear programming algorithms for quantile regression, for example, the simplex method and the interior point method, work well for data of small to moderate sizes. However, these methods are difficult to generalize to high‐dimensional big data for which penalization is usually necessary. Further, the massive size of contemporary big data calls for the development of large‐scale algorithms on distributed computing platforms. The traditional linear programming algorithms are intrinsically sequential and not suitable for such frameworks. In this paper, we discuss how to use the popular ADMM algorithm to solve large‐scale penalized quantile regression problems. The ADMM algorithm can be easily parallelized and implemented in modern distributed frameworks. Simulation results demonstrate that the ADMM is as accurate as traditional LP algorithms while faster even in the nonparallel case.     

Counterfactual distributions with sample selection adjustments: Econometric theory and an application to the Netherlands

Several recent papers use the quantile regression decomposition method of Machado and Mata [Machado, J.A.F. and Mata, J. (2005). Counterfactual decomposition of changes in wage distributions using quantile regression, Journal of Applied Econometrics, 20, 445–65.] to analyze the gender gap across log wage distributions. In this paper, we prove that this procedure yields consistent and asymptotically normal estimates of the quantiles of the counterfactual distribution that it is designed to simulate. Since employment rates often differ substantially by gender, sample selection is potentially a serious issue for such studies. To address this issue, we extend the Machado–Mata technique to account for selection. We illustrate our approach to adjusting for sample selection by analyzing the gender log wage gap for full-time workers in the Netherlands.     

Asymptotics and smoothing parameter selection for penalized spline regression with various loss functions

Penalized splines are used in various types of regression analyses, including non‐parametric quantile, robust and the usual mean regression. In this paper, we focus on the penalized spline estimator with general convex loss functions. By specifying the loss function, we can obtain the mean estimator, quantile estimator and robust estimator. We will first study the asymptotic properties of penalized splines. Specifically, we will show the asymptotic bias and variance as well as the asymptotic normality of the estimator. Next, we will discuss smoothing parameter selection for the minimization of the mean integrated squares error. The new smoothing parameter can be expressed uniquely using the asymptotic bias and variance of the penalized spline estimator. To validate the new smoothing parameter selection method, we will provide a simulation. The simulation results show that the consistency of the estimator with the proposed smoothing parameter selection method can be confirmed and that the proposed estimator has better behavior than the estimator with generalized approximate cross‐validation. A real data example is also addressed.     

Quasi-maximum likelihood estimation for conditional quantiles

In this paper, we construct a new class of estimators for conditional quantiles in possibly misspecified nonlinear models with time series data. Proposed estimators belong to the family of quasi-maximum likelihood estimators (QMLEs) and are based on a new family of densities which we call ‘tick-exponential’. A well-known member of the tick-exponential family is the asymmetric Laplace density, and the corresponding QMLE reduces to the Koenker and Bassett's (Econometrica 46 (1978) 33) nonlinear quantile regression estimator. We derive primitive conditions under which the tick-exponential QMLEs are consistent and asymptotically normally distributed with an asymptotic covariance matrix that accounts for possible conditional quantile model misspecification and which can be consistently estimated by using the tick-exponential scores and Hessian matrix. Despite its non-differentiability, the tick-exponential quasi-likelihood is easy to maximize by using a ‘minimax’ representation not seen in the earlier work on conditional quantile estimation.     

Unit root quantile autoregression testing using covariates

This paper extends unit root tests based on quantile regression proposed by Koenker and Xiao [Koenker, R., Xiao, Z., 2004. Unit root quantile autoregression inference, Journal of the American Statistical Association 99, 775–787] to allow stationary covariates and a linear time trend. The limiting distribution of the test is a convex combination of Dickey–Fuller and standard normal distributions, with weight determined by the correlation between the equation error and the regression covariates. A simulation experiment is described, illustrating the finite sample performance of the unit root test for several types of distributions. The test based on quantile autoregression turns out to be especially advantageous when innovations are heavy-tailed. An application to the CPI-based real exchange rates using four different countries suggests that real exchange rates are not constant unit root processes.