# CASE STUDY: Expectile (XVaR) Estimation by Three Variants of Regression

CASE STUDY: Expectile (XVaR) Estimation by Three Variants of Regression

Instructions for optimization with PSG Run-File, PSG MATLAB Toolbox, PSG MATLAB Subroutines and PSG R.

PROBLEM 1: problem_xvar_err_Regression
Minimize piecewise linear Expectile Error

parameter values K = 1/6; 1/18

Dataset 4 1,264 4.8801e-03; 7.5379e-03 0.867; 0.861 0.01; 0.01 # of Variables # of Scenarios Objective Value Pseudo R2 Solving Time, PC 2.50GHz (sec) Environments Run-File Problem Statement Data Solution

PROBLEM 2: problem_xvar_dev_Regression
Minimize Expectile Deviation (xvar_dev) from Expectile Quadrangle with piecewise linear error
Calculate intercept = Expectile (xvar)

parameter values q = 0.875; 0.95

Dataset 4 1,264 4.8801e-03; 7.5379e-03 0.01; 0.01 # of Variables # of Scenarios Objective Value Solving Time, PC 2.50GHz (sec) Environments Run-File Problem Statement Data Solution

PROBLEM 3a: problem_xvar_L_err_Regression
Minimize piecewise linear Expectile Error
Calculate asymmetric variance Expectile error

parameter values K = 1/6; K = 1/18 (corresponding q=0.875; q=0.95)

Dataset 6 12,690 0.38874; 0.58018 0.06642; 0.03366 0.01; 0.01 # of Variables # of Scenarios Objective Value Calculated Value Solving Time, PC 2.50GHz (sec) Environments Run-File Problem Statement Data Solution

PROBLEM 4a: problem_xvar_Q_err_Regression
Minimize asymmetric variance Expectile error
Calculate piecewise linear Expectile Error

parameter values q=0.875; q=0.95 (corresponding K = 1/6; K = 1/18)

Dataset 6 12,690 0.05988; 0.03139 0.41072; 0.59501 0.01; 0.01 # of Variables # of Scenarios Objective Value Calculated Value Solving Time, PC 2.50GHz (sec) Environments Run-File Problem Statement Data Solution

PROBLEM 3b: problem_xvar_L0_err_Regression
Minimize piecewise linear Expectile error
Constraint: polynom_abs ≤ 0
polynom_abs
————————————————————
polynom_abs = Polynomial Absolute
————————————————————

parameter values K = 1/6 and K = 1/18

Dataset 6 12,690 0.443727; 0.622792 0.663506; 0.842571 0.01; 0.01 # of Variables # of Scenarios Objective Value Intercept Solving Time, PC 2.50GHz (sec) Environments Run-File Problem Statement Data Solution

PROBLEM 4b: problem_xvar_Q0_err_Regression
Minimize asymmetric variance Expectile error
Constraint: polynom_abs ≤ 0
————————————————————
polynom_abs = Polynomial Absolute
————————————————————

parameter values q=0.875; q=0.95

Dataset 6 12,690 0.064710; 0.032870 0.663506; 0.842571 0.01; 0.01 # of Variables # of Scenarios Objective Value Intercept Solving Time, PC 2.50GHz (sec) Environments Run-File Problem Statement Data Solution

CASE STUDY SUMMARY

Rockafellar and Uryasev (2013) developed a paradigm called Fundamental Risk Quadrangle which links Risk Management, Reliability, Statistics and Stochastic Optimization theories. Risk Quadrangle includes four stochastic functions and statistic: Risk R(X), Deviation D(X), Error E(X), Regret V(X), Statistic S(X). Kuzmenko, Malandii, Uryasev (2023) studied piecewise linear Expectile Quadrangle with Risk and Statistic equal to Expectile and two other Quadrangles where Expectile is statistic only. One of these two quadrangles has asymmetric variance error suggested by Newey (1987) . This Case Study estimates Expectile by minimizing piecewise linear and asymmetric variance Erros. Alternatively, Expectile regression is done in two steps (Rockafellar et al (2013)): Step 1. Minimize Expectile Deviation from Expectile Quadrangle with the residual depending only on loading factors. Step 2. Calculate intercept = Expectile for the optimal value of variables from Step1

References

Newey W.K., Powell J.L. (1987). Asymmetric least squares estimation and testing. Econometrica. 55 (4). P. 819-847. https://doi.org/10.2307/1911031

Rockafellar R.T., Uryasev S. (2013). The Fundamental Risk Quadrangle in Risk Management, Optimization and Statistical Estimation. Surveys in Operations Research and Management Science. 18 (1). P. 33-53. https://doi.org/10.1016/j.sorms.2013.03.001

Rockafellar R.T., Royset J.O., Miranda S.I. (2014). Superquantile regression with applications to buffered reliability, uncertainty quantification and conditional value-at-risk. European J. Operations Research. 234 (1). P. 140-154. https://doi.org/10.1016/j.ejor.2013.10.046

Kuzmenko V. (2020). A New Family of Expectiles and its Properties. Cybernetics and Computer Technologies. No. 3. P. 43-58. https://doi.org/10.34229/2707-451X.20.3.5

Kuzmenko, Malandii, Uryasev (2023). Expectile Risk Quadrangle and Applications. Working Paper.