Functional quantile autoregression

This paper proposes a new class of time series models, the functional quantile autoregression (FQAR) models, in which the conditional distribution of the observation at the current time point is affected by its past distributional information, and is expressed as a functional of the past conditional quantile functions. Different from the conventional functional time series models which are based on functionally observed data, the proposed FQAR method studies functional dynamics in traditional time series data. We propose a sieve estimator for the model. Asymptotic properties of the estimators are derived. Numerical investigations are conducted to highlight the proposed method.