Optimal estimating function for weak location-scale dynamic models

Estimating functions provide a very general framework for the statistical inference of dynamic models under weak assumptions. We consider a class of time series models consisting in the parametrization of the first two conditional moments which – by contrast with classical location-scale dynamic models – do not impose further constraints on the conditional distribution/moments. Quasi-likelihood estimators (QLE) are obtained by solving estimating equations deduced from those two conditional moments. Conditions ensuring the existence and asymptotic properties (consistency and asymptotic normality) of such estimators are provided. We propose optimal QLEs in Godambe's sense, deduced from a condition obtained by Chandra and Taniguchi (2001, Annals of the Institute of Statistical Mathematics 53, 125–141). The particular case of the quasi-maximum likelihood estimators is considered. For pure location models, a data-driven procedure for optimally choosing the QLE is proposed. Our results are illustrated via Monte Carlo experiments and real financial data.