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.