B1035
Title: On expectiles and quantiles
Authors: Fabian Sobotka - University Oldenburg (Germany) [presenting]
Goeran Kauermann - LMU Munich (Germany)
Thomas Kneib - University of Goettingen (Germany)
Linda Schulze Waltrup - LMU Munich (Germany)
Abstract: While quantile regression can be seen as a generalization of median regression, expectiles are a generalized form of the mean. Quantile regression estimates are acquired by minimizing the asymmetrically weighted sum of the absolute residuals while an expectile is computed from the least asymmetrically weighted squares (LAWS) of the residuals. This means, instead of a linear programming problem, we face an easier optimization problem with an objective function being convex and differentiable. Despite this ease in computation, expectiles lack the intuitive interpretation of quantiles. We contrast the two approaches and show how to get quantiles from a fine grid of expectiles. This has the additional advantage that we can directly compare quantiles and expectiles. We compare such quantiles from expectiles with native quantile estimates regarding efficiency on an asymptotic and an empirical basis. We also look at regression problems where both, quantile and expectile curves, have the undesirable property that individually estimated neighboring curves may cross each other. We show empirically that crossing curves are more frequent in quantile regression and we propose a 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 can be calculated easily with the proposed methods.
