Expectile and quantile regression—David and Goliath?

Abstract

Recent interest in modern regression modelling has focused on extending available (mean) regression models by describing more general properties of the response distribution. An alternative approach is quantile regression where regression effects on the conditional quantile function of the response are assumed. While quantile regression can be seen as a generalization of median regression, expectiles as alternative are a generalized form of mean regression.Generally, quantiles provide a natural interpretation even beyond the 0.5 quantile, the median. A comparable simple interpretation is not available for expectiles beyond the 0.5 expectile, the mean. Nonetheless, expectiles have some interesting properties, some of which are discussed in this article. We contrast the two approaches and show how to get quantiles from a fine grid of expectiles. We compare such quantiles from expectiles with direct quantile estimates regarding efficiency. We also look at regression problems where both quantile and expectile curves have the undesirable property that neighbouring curves may cross each other. We propose a modified method to estimate non-crossing expectile curves based on splines. In an application, we look at the expected shortfall, a risk measure used in finance, which requires both expectiles and quantiles for estimation and which can be calculated easily with the proposed methods in the article.

Mincer–Zarnowitz quantile and expectile regressions for forecast evaluations under aysmmetric loss functions

“…Although both quantiles and expectiles are used in applications for similar purposes, quantiles are dominant in the literature. There has been some discussion in the literature on which one should be used (see, e.g., Kneib, ; Koenker, ; Waltrup, Sobotka, Kneib, and Gorän, ). In particular, Koenker () pointed out that expectiles were nonrobust: “although they seek to describe a local property of a distribution, they depend on global properties of that distribution” and “they are not equivariant to monotone transformations as are the quantiles.” On the other hand, some other scholars still argued that expectiles were an interesting alternative to quantiles and that their combined use was helpful in some applications (see Waltrup et al, ).…”

“…In particular, Koenker (2013) pointed out that expectiles were nonrobust: "although they seek to describe a local property of a distribution, they depend on global properties of that distribution" and "they are not equivariant to monotone transformations as are the quantiles." On the other hand, some other scholars still argued that expectiles were an interesting alternative to quantiles and that their combined use was helpful in some applications (see Waltrup et al, 2015). Whether or not to use the expectile approach, we will leave that decision to the readers.…”

Non-crossing nonlinear regression quantiles by monotone composite quantile regression neural network, with application to rainfall extremes

“…For s ¼ 0:5 and large a, the error function is symmetric and is, up to a constant scaling factor, equal to the LS error function. For s 6 ¼ 0:5, the asymmetric LS error function results in an estimate of the conditional expectile function (Newey and Powell 1987;Yao and Tong 1996;Waltrup et al 2015). Hence, depending on values of a and s, minimizing the approximate quantile regression error function can provide regression estimates for the conditional mean (a ) 0, s ¼ 0:5), median (a !…”

Expectile asymptotics

“…Expectile regression, that is, regression on a parameter that generalizes the mean and characterizes the tail behaviour of a distribution, has been introduced by Newey and Powell (1987) as an alternative to more standard quantile regression; Breckling and Chambers (1988) considered regression based on more general asymmetric M-estimators. For a recent comparison between quantile and expectile regression and references see Schulze-Waltrup et al (2014).…”