Improved estimation of dynamic models of conditional means and variances

Abstract

Using ‘working’ assumptions on conditional third and fourth moments of errors, we propose a method of moments estimator that can have improved efficiency over the popular Gaussian quasi‐maximum likelihood estimator (GQMLE). Higher‐order moment assumptions are not needed for consistency – we only require the first two conditional moments to be correctly specified – but the optimal instruments are derived under these assumptions. The working assumptions allow both asymmetry in the distribution of the standardized errors as well as fourth moments that can be smaller or larger than that of the Gaussian distribution. The approach is related to the generalized estimation equations (GEE) approach – which seeks the improvement of estimators of the conditional mean parameters by making working assumptions on the conditional second moments. We derive the asymptotic distribution of the new estimator and show that it does not depend on the estimators of the third and fourth moments. A simulation study shows that the efficiency gains over the GQMLE can be non‐trivial.