Quasi-Maximum Likelihood Estimation for Conditional Quantiles

In this paper we derive the asymptotic distribution of a new class of quasi-maximum likelihood estimators (QMLE) based on a ‘tick-exponential’ family of densities. We show that the ‘tick-exponential’ assumption is a necessary and sufficient condition for a QMLE to be consistent for the parameters of a correctly specified model of a given conditional quantile. Hence, the role of this family of densities in the conditional quantile estimation is analog to the role of the linear-exponential family in the conditional mean estimation. The ‘tick-exponential’ QMLEs are shown to be asymptotically normal with an asymptotic covariance matrix that has a novel form, not seen in earlier work, and which accounts for possible model misspecification. For practical purposes, we show that the maximization of the ‘tick-exponential’ (quasi) log-likelihood can conveniently be carried out by using standard gradient-based optimization techniques. More importantly, we provide a consistent estimator for the asymptotic covariance matrix based on the “scores” of the log-likelihood, which allows us to compute the conditional quantile confidence intervals