Semiparametric Modeling of Multiple Quantiles

We develop a semiparametric model to track a large number of quantiles of a time series. The model satisfies the condition of non crossing quantiles and the defining property of fixed quantiles. A key feature of the specification is that the updating scheme for time varying quantiles at each probability level is based on the gradient of the check loss function, that forms a martingale difference sequence. Theoretical properties of the proposed model are derived, such as weak stationarity of the quantile process and consistency and asymptotic normality of the estimators of the fixed parameters. The model can be applied for filtering and prediction. We also illustrate a number of possible applications such as: i) semiparametric estimation of dynamic moments of the observables, ii) density prediction, and iii) quantile predictions.