Quasi-Maximum Likelihood Estimation for Conditional Expectiles

We characterize the quasi-likelihood functions that may elicit expectiles and find that the family has a unique representation under standard conditions for linear regression. The only distribution that elicits expectiles as its quasi-maximum likelihood estimator under general conditions is an asymmetric normal distribution. Next, we analyze the quasi maximum likelihood estimator and give conditions for consistency, asymptotic normality, and efficiency. The estimator is unique up to the choice of weights on individual observations and nests the usual GLS estimator. We give the asymptotic MVUE and a uniform Cramer-Rao theorem for expectile regression