Efficient Sequential Extremum Estimation and a Comparison with Maximization by Parts

This research consider the problem of efficiently estimating a parameter of interest when the model is complicated by a vector of nuisance parameters. If the model is nonadaptive we must often resort to full information estimation to gain an efficient estimator for the parameter of interest. In certain cases full information estimation can be computationally intensive and lead to poor finite sample properties, Fan, Pastorello and Renault(2012)[9]. To avoid such complications Fan, Pastorello and Renault(2012)[9](FPR hereafter) derive algorithms, known as maximization by parts(MBP), which yield iterative estimators for the parameter of interest that converge to the full information estimates as the number of iterations goes to infinity. However, these iterative estimators are only applicable when a set of technical conditions are satisfied. As an alternative to the MBP algorithms we derive consistent and efficient sequential extremum estimators for the parameter of interest. Unlike MBP these sequential estimators only rely on a standard set of regularity conditions, which are common across extremum estimators. To compare the computational cost of the sequential and iterative estimators we derive Newton-Raphson(NR) updating rules for the sequential estimators and compare these to the known NR updating rules for the iterative estimators. We show that within the specific case of separable log-likelihood functions the sequential estimators are strictly preferable as they achieve consistency and efficiency in two steps whereas the iterative estimators are only consistent after two steps. We demonstrate the applicability of this method by applying the sequential estimators to the stochastic volatility model of Taylor(1994)[26] and the affine term structure models of Dai and Singleton(2000)[7] Applying the sequential estimation methodology to the generalized method of moments(GMM) leads to a dual representation of the sequential GMM estimator as a minimum chi-squared estimator. Thus, if we use GMM as a unifying estimation framework the sequential estimators have a dual representation as minimum chi-squared estimators.