Expectile Regression Research Articles

Efficient Distributed Learning for Large-Scale Expectile Regression With Sparsity

High-dimensional datasets often display heterogeneity due to heteroskedasticity or other forms of non-location-scale covariance effects. When the size of datasets becomes very large, it may be infeasible to store all of the high-dimensional datasets on one machine, or at least to keep the datasets in memory. In this paper, we consider penalized expectile regression using smoothly clipped absolute deviation (SCAD) and adaptive LASSO penalties, which can effectively detect the heteroskedasticity of high-dimensional data. We propose a communication-efficient approach for distributed sparsity learning, where observations are randomly partitioned across machines. By selecting the appropriate tuning parameters, we show that the proposed estimators display oracle properties. Extensive numerical experiments on both synthetic and real data validate the theoretical results and demonstrate the superior performance of our proposed method.

M-Quantile Regression for Multivariate Longitudinal Data with an Application to the Millennium Cohort Study

Abstract Motivated by the analysis of data from the UK Millennium Cohort Study on emotional and behavioural disorders, we develop an M-quantile regression model for multivariate longitudinal responses. M-quantile regression is an appealing alternative to standard regression models; it combines features of quantile and expectile regression and it may produce a detailed picture of the conditional response variable distribution, while ensuring robustness to outlying data. As we deal with multivariate data, we need to specify what it is meant by M-quantile in this context, and how the structure of dependence between univariate profiles may be accounted for. Here, we consider univariate (conditional) M-quantile regression models with outcome-specific random effects for each outcome. Dependence between outcomes is introduced by assuming that the random effects in the univariate models are dependent. The multivariate distribution of the random effects is left unspecified and estimated from the observed data. Adopting this approach, we are able to model dependence both within and between outcomes. We further discuss a suitable model parameterisation to account for potential endogeneity of the observed covariates. An extended EM algorithm is defined to derive estimates under a maximum likelihood approach.

Prediction of Extremal Expectile Based on Regression Models With Heteroscedastic Extremes

Expectile recently receives much attention for its coherence as a tail risk measure. Estimation of conditional expectile at extremal tails is of great interest in quantitative risk management. Regression analysis is a convenient and useful way to quantify the conditional effect of some predictors or risk factors on an interesting response variable. However, when it comes to the estimation of extremal conditional expectile, the traditional inference methods may suffer from considerable variation due to a lack of sufficient samples on tail regions, which makes the prediction inaccurate. In this article, we study the estimation of extremal conditional expectile based on quantile regression and expectile regression models. We propose three methods to make extrapolation based on a second-order condition for a framework of the so-called conditionally heteroscedastic and unconditionally homoscedastic extremes. In addition, we establish the asymptotic properties of the proposed methods and show their empirical behaviors through simulation studies. Finally, data analysis is conducted to illustrate the applications of the proposed methods in real problems.

On expectile-assisted inverse regression estimation for sufficient dimension reduction

Moment-based sufficient dimension reduction methods such as sliced inverse regression may not work well in the presence of heteroscedasticity. We propose to first estimate the expectiles through kernel expectile regression, and then carry out dimension reduction based on random projections of the regression expectiles. Several popular inverse regression methods in the literature are extended under this general framework. The proposed expectile-assisted methods outperform existing moment-based dimension reduction methods in both numerical studies and an analysis of the Big Mac data.

Tail risks in large portfolio selection: penalized quantile and expectile minimum deviation models

Accurate estimation and optimal control of tail risk is important for building portfolios with desirable properties, especially when dealing with a large set of assets. In this work, we consider optimal asset allocation strategies based on the minimization of two asymmetric deviation measures, related to quantile and expectile regression, respectively. Their properties are discussed in relation with the ‘risk quadrangle’ framework introduced by Rockafellar and Uryasev [The fundamental risk quadrangle in risk management, optimization and statistical estimation. Surv. Oper. Res. Manag. Sci., 2013, 18(1–2), 33–53], and compared to traditional strategies, such as the mean–variance portfolio. In order to control estimation error and improve the out-of-sample performance of the proposed models, we include ridge and elastic-net regularization penalties. Finally, we propose quadratic programming formulations for the optimization problems. Simulations and real-world analyses on multiple datasets allow to discuss pros and cons of the different methods. The results show that the ridge and elastic-net allocations are effective in improving the out-of-sample performance, especially in large portfolios, compared to the un-penalized ones.

Jackknife model averaging for expectile regressions in increasing dimension

Expectile regression is a useful tool for modeling data with heterogeneous conditional distributions. This paper develops the jackknife model averaging method for expectile regressions. The asymptotic properties of expectile estimator under misspecification with increasing dimension of parameters have been studied. The model averaging expectile estimator using the leave-one-out cross-validated weight is shown to be asymptotically optimal in the sense of out-of-sample final prediction error. Numerical results demonstrate the nice performance of the averaging estimators.

The consistency and asymptotic normality of the kernel type expectile regression estimator for functional data

The aim of this paper is to nonparametrically estimate the expectile regression in the case of a functional predictor and a scalar response. More precisely, we construct a kernel-type estimator of the expectile regression function. The main contribution of this study is the establishment of the asymptotic properties of the expectile regression estimator. Precisely, we establish the almost complete convergence with rate. Furthermore, we obtain the asymptotic normality of the proposed estimator under some mild conditions. We provide how to apply our results to construct the confidence intervals. The case of functional predictor is of particular interest and challenge, both from theoretical as well as practical point of view. We discuss the potential impacts of functional expectile regression in NFDA with a particular focus on the supervised classification, prediction and financial risk analysis problems. Finally, the finite-sample performances of the model and the estimation method are illustrated using the analysis of simulated data and real data coming from the financial risk analysis.

Mixed data sampling expectile regression with applications to measuring financial risk

To avoid information loss or measurement error in traditional methods dealing with mixed frequency data, we develop a novel mixed data sampling expectile regression (MIDAS-ER) model to measure financial risk. We construct the MIDAS-ER model by introducing a MIDAS structure into expectile regressions. This enables us to perform an expectile regression on raw mixed frequency data directly. We apply the proposed MIDAS-ER model to estimate two popular financial risk measures, namely, Value at Risk and Expected Shortfall, with both simulated data and four stock indices, and compare the model's performance with those of several popular models. The outstanding performance of our model demonstrates that high-frequency information helps to improve the accuracy of risk measurement. In addition, the numerical results also imply that our model can be a significant tool for risk-averse investors to control risk losses and for financial institutions to implement robust risk management.

KLERC: kernel Lagrangian expectile regression calculator

As a generalization to the ordinary least square regression, expectile regression, which can predict conditional expectiles, is fitted by minimizing an asymmetric square loss function on the training data. In literature, the idea of support vector machine was introduced to expectile regression to increase the flexibility of the model, resulting in support vector expectile regression (SVER). This paper reformulates the Lagrangian function of SVER as a differentiable convex function over the nonnegative orthant, which can be minimized by a simple iterative algorithm. The proposed algorithm is easy to implement, without requiring any particular optimization toolbox besides basic matrix operations. Theoretical and experimental analysis show that the algorithm converges r-linearly to the unique minimum point. The proposed method was compared to alternative algorithms on simulated data and real-world data, and we observe that the proposed method is much more computationally efficient while yielding similar prediction accuracy.

Distributed optimization and statistical learning for large-scale penalized expectile regression

Large-scale data from various research fields are not only heterogeneous and sparse but also difficult to store on a single machine. Expectile regression is a popular alternative for modeling heterogeneous data. In this paper, we devise a distributed optimization approach to SCAD and adaptive LASSO penalized expectile regression, where the observations are randomly partitioned across multiple machines. We construct a penalized communication-efficient surrogate loss (CSL) function. Computationally, our method based on the CSL function requires only the master machine to solve a regular M-estimation problem, while other worker machines compute the gradient of the loss function on local data. Our method matches the estimation error bound of the centralized method during consecutive rounds of communication. Under some mild assumptions, we establish the oracle properties of the SCAD and adaptive LASSO penalized expectile regression. We then develop a modified alternating direction method of multipliers (ADMM) algorithm for the implementation of the proposed estimator. A series of simulation studies are conducted to assess the finite-sample performance of the proposed estimator. Applications to an HIV study demonstrate the practicability of the proposed method.

Measuring the risk of Chinese Fintech industry: evidence from the stock index

This study measures the risk of the emerging Fintech industry in China and identifies its influencing risk factors by calculating the tail risk of Fintech stock index. The expectile regression model that includes the lagged returns and macroeconomic risk factors is used to calculate the Expectile Value-at-Risk (EVaR). Based on the 1230 daily returns of Fintech index ranges from July 2, 2014, to September 10, 2019, the empirical results indicate that the Fintech industry possesses a higher risk, and is affected by both the past development and internal macroeconomic condition.

Estimating portfolio risk for tail risk protection strategies

AbstractWe forecast portfolio risk for managing dynamic tail risk protection strategies, based on extreme value theory, expectile regression, copula‐GARCH and dynamic generalized autoregressive score models. Utilizing a loss function that overcomes the lack of elicitability for expected shortfall, we propose a novel expected shortfall (and value‐at‐risk) forecast combination approach, which dominates simple and sophisticated standalone models as well as a simple average combination approach in modeling the tail of the portfolio return distribution. While the associated dynamic risk targeting or portfolio insurance strategies provide effective downside protection, the latter strategies suffer less from inferior risk forecasts, given the defensive portfolio insurance mechanics.

Efficient estimation in expectile regression using envelope models

As a generalization of the classical linear regression, expectile regression (ER) explores the relationship between the conditional expectile of a response variable and a set of predictor variables. ER with respect to different expectile levels can provide a comprehensive picture of the conditional distribution of the response variable given the predictors. We adopt an efficient estimation method called the envelope model ([8]) in ER, and construct a novel envelope expectile regression (EER) model. Estimation of the EER parameters can be performed using the generalized method of moments (GMM). We establish the consistency and derive the asymptotic distribution of the EER estimators. In addition, we show that the EER estimators are asymptotically more efficient than the ER estimators. Numerical experiments and real data examples are provided to demonstrate the efficiency gains attained by EER compared to ER, and the efficiency gains can further lead to improvements in prediction.

Large-Scale Expectile Regression With Covariates Missing at Random

Analysis of large volumes of data is very complex due to not only a high level of skewness and heteroscedasticity of variance but also the phenomenon of missing data. Expectile regression is a popular alternative method of analyzing heterogeneous data. In this paper, we consider fitting a linear expectile regression model for estimating conditional expectiles based on a large quantity of data with covariates missing at random. We construct a communication-efficient surrogate loss (CSL) function to estimate model parameters. The asymptotic normality of the proposed estimator is established. A proximal alternating direction method of multipliers (ADMM) algorithm is developed for distributed statistical optimization on a large quantity of data. Simulation studies are performed to assess the finite-sample performance of the proposed method. Survey data from the Behavioral Risk Factor Surveillance System (BRFSS) is used to demonstrate the utility of the proposed method in practice.

Tail Risks in Large Portfolio Selection: Penalized Quantile and Expectile Minimum Deviation Models

Accurate estimation and optimal control of tail risk is important for building portfolios with desirable properties, especially when dealing with a large set of assets. In this work, we consider optimal asset allocations strategies based on the minimization of two asymmetric deviation measures, related to quantile and expectile regression, respectively. Their properties are discussed in relation with the `risk quadrangle' framework introduced by Rockafellar et al (2013) and compared to traditional strategies, such as the mean-variance portfolio. In order to control estimation error and improve the out-of-sample performances of the proposed models, we include ridge and elastic-net regularization penalties. Finally, we propose quadratic programming formulations for the optimization problems. Simulations and real-world analyses on multiple datasets, allow to discuss pros and cons of the different methods. The results show that the ridge and elastic-net allocations are effective in improving the out-of-sample performances, especially in large portfolios, compared to the un-penalized ones.

The Time-varying Risk of Italian GDP

The uncertainty surrounding economic forecasts is generally related to multiple sources of risks, of domestic and foreign origin. This paper studies the predictive distribution of Italian GDP growth as a function of selected risk indicators, related to both financial and real economic developments. The conditional distribution is characterized by means of expectile regressions. Expectiles are closely related to the Expected Shortfall, a well-known measure of risk with desirable properties. Here a decomposition of Expected Shortfall in terms of contributions of different indicators is proposed, which allows to track over time the main drivers of risk. Our analysis of the predictive distribution of GDP confirms that financial conditions are relevant for the left tail of the distribution but it also highlights that indicators of global trade and uncertainty have strong explanatory power for both left and right tail. Their usefulness is supported also in a pseudo real-time predictive context. Overall, our findings suggest that Italian GDP risks have been mostly driven by foreign developments around the Great Recession, by domestic financial conditions at the time of the Sovereign Debt Crisis and by economic policy uncertainty in more recent years.

Expectile Regression on Distributed Large-Scale Data

Large-scale data presents great challenges to data analysis due to the limited computer storage capacity and the heterogeneous data structure. In this article, we propose a distributed expectile regression model to resolve the challenges of large-scale data by designing a surrogate loss function and using the Iterative Local Alternating Direction Method of the Multipliers (IL-ADMM) algorithm, which is developed for the calculation of the proposed estimator. To obtain nice performance only after fewer rounds of communications, the proposed method only needs to solve an M-estimation problem on the master machine while the other working machines only to compute the gradients based on local data. Moreover, we show the consistency and the asymptotic normality of the proposed estimator, and illustrate the efficient proof by numerical simulations and positive analysis on the superconductor data.

The kth power expectile regression

Check functions of least absolute deviation make sure quantile regression methods are robust, while squared check functions make expectiles more sensitive to the tails of distributions and more effective for the normal case than quantiles. In order to balance robustness and effectiveness, we adopt a loss function, which falls in between the above two loss functions, to introduce a new kind of expectiles and develop an asymmetric least kth power estimation method that we call the kth power expectile regression, k larger than 1 and not larger than 2. The asymptotic properties of the corresponding estimators are provided. Simulation results show that the asymptotic efficiency of the kth power expectile regression is higher than those of the common quantile regression and expectile regression in some data cases. A primary procedure of choosing satisfactory k is presented. We finally apply our method to the real data.

Domestic and Global Determinants of Inflation: Evidence from Expectile Regression

The paper investigates the role of domestic and global determinants of euro area core inflation. We analyse the entire conditional distribution of inflation by estimating a Phillips curve type relationship using an expectile regression approach, extended to capture time-varying effects. The main findings are as follows. First, both the domestic and foreign output gap are significant drivers of euro area core inflation, once external demand pressures are properly orthogonalized in a modified measure of domestic gap. However, the inflationary impact of the domestic component is relatively stronger. Second, the domestic output gap has a bigger influence in the right tail of the conditional distribution of inflation. Third, adding international price pressures in the regression weakens the link between inflation and the foreign output gap. Fourth, in a time- varying perspective, there is an increase in the response of inflation to the domestic gap in the last decade but only at the lower quantiles. Overall, the evidence on the so-called “globalization hypothesis” is mixed: while the pass-through to inflation of foreign prices and the exchange rate increased over time at all quantiles, the impact of global slack remained broadly stable, particularly in the central part of the distribution.

Modeling physical activity data using L0 -penalized expectile regression.

In recent years accelerometers have become widely used to objectively assess physical activity. Usually intensity ranges are assigned to the measured accelerometer counts by simple cut points, disregarding the underlying activity pattern. Under the assumption that physical activity can be seen as distinct sequence of distinguishable activities, the use of hidden Markov models (HMM) has been proposed to improve the modeling of accelerometer data. As further improvement we propose to use expectile regression utilizing a Whittaker smoother with an L0 -penalty to better capture the intensity levels underlying the observed counts. Different expectile asymmetries beyond the mean allow the distinction of monotonous and more variable activities as expectiles effectively model the complete distribution of the counts. This new approach is investigated in a simulation study, where we simulated 1,000 days of accelerometer data with 1 and 5 s epochs, based on collected labeled data to resemble real-life data as closely as possible. The expectile regression is compared to HMMs and the commonly used cut point method with regard to misclassification rate, number of identified bouts and identified levels as well as the proportion of the estimate being in the range of of the true activity level. In summary, expectile regression utilizing a Whittaker smoother with an L0 -penalty outperforms HMMs and the cut point method and is hence a promising approach to model accelerometerdata.