Quasi-maximum likelihood estimation for conditional quantiles

In this paper, we construct a new class of estimators for conditional quantiles in possibly misspecified nonlinear models with time series data. Proposed estimators belong to the family of quasi-maximum likelihood estimators (QMLEs) and are based on a new family of densities which we call 'tick-exponential'. A well-known member of the tick-exponential family is the asymmetric Laplace density, and the corresponding QMLE reduces to the Koenker and Bassett's (Econometrica 46 (1978) 33) nonlinear quantile regression estimator. We derive primitive conditions under which the tick-exponential QMLEs are consistent and asymptotically normally distributed with an asymptotic covariance matrix that accounts for possible conditional quantile model misspecification and which can be consistently estimated by using the tick-exponential scores and Hessian matrix. Despite its non-differentiability, the tick-exponential quasi-likelihood is easy to maximize by using a 'minimax' representation not seen in the earlier work on conditional quantile estimation.