Two-stage weighted least squares estimator of the conditional mean of observation-driven time series models

General parametric forms are assumed for the conditional mean lambda t(theta 0) and variance upsilon t of a time series. These conditional moments can for instance be derived from count time series, Autoregressive Conditional Duration or Generalized Autoregressive Score models. In this paper, our aim is to estimate the conditional mean parameter theta 0, trying to be as agnostic as possible about the conditional distribution of the observations. QuasiMaximum Likelihood Estimators (QMLEs) based on the linear exponential family fulfill this goal, but they may be inefficient and have complicated asymptotic distributions when theta 0 contains boundary coefficients. We thus study alternative Weighted Least Square Estimators (WLSEs), which enjoy the same consistency property as the QMLEs when the conditional distribution is misspecified, but have simpler asymptotic distributions when components of theta 0 are null and gain in efficiency when upsilon t is well specified. We compare the asymptotic properties of the QMLEs and WLSEs, and determine a data driven strategy for finding an asymptotically optimal WLSE. Simulation experiments and illustrations on realized volatility forecasting are presented. (c) 2021 Elsevier B.V. All rights reserved.