The Nonlinear Iterative Least Squares (NL-ILS) Estimator: An Application to Volatility Models

Abstract

The paper proposes a new robust estimator for GARCH-type models: the nonlinear iterative least squares (NL-ILS). This estimator is especially useful on specifications where errors have some degree of dependence over time (weak-GARCH) or when the conditional variance is misspecified. I illustrate the NL-ILS estimator by providing algorithms that consider the GARCH(1,1), weak-GARCH(1,1), and GARCH(1,1)-in-mean models. I establish the consistency and asymptotic distribution of the NL-ILS estimator for the GARCH(1,1) and weak-GARCH(1,1) models. A Monte Carlo study provides evidences that the NL-ILS estimator is consistent and outperforms the MLE benchmark in a variety of specifications. Moreover, when the conditional variance is misspecified, the MLE estimator delivers biased estimates of the parameters in the mean equation, whereas the NL-ILS estimator does not. The empirical application investigates the risk premium on the CRSP, S&P500 and S\&P100 indices. I document the risk premium parameter is significant and positive only for the CRSP index when using the robust NL-ILS estimator. This finding holds on daily, weekly and monthly frequencies and it is corroborated by a series of robustness checks.

Keywords: Risk premium; GARCH-type models; Iterative estimators; Contraction mapping

JEL Classification: C13, C15, C18, C22, G12