Simple GMM Estimation of the Semi-Strong GARCH(1,1) Model

Abstract

IV estimators for the semi-strong ARCH(1) model that rely on past squared residuals alone as instruments do not extend to the GARCH case. Efficient IV estimators of the semi-strong GARCH(1,1) model require the derivative of the conditional variance as well as both the third and fourth conditional moments to be included within the instrument vector. This paper proposes IV estimators for the semi-strong GARCH(1,1) model that only rely on past residuals and past squared residuals as instruments. These estimators are based on the autocovariances of squared residuals, as in the ARCH(1) case described above, as well as on the covariances between squared residuals and past residuals. These latter covariances are nonzero if the residuals are skewed. Jackknife GMM estimators and jackknife continuous updating estimators (CUE) eliminate the bias caused by many (weak) instruments. The jackknife CUE is new and applies to cases where the optimal weighting matrix is unavailable out of a concern over the existence of higher moments. In these cases, a robust analog to the variance-covariance matrix determines the weighting matrix. A Monte Carlo study shows that a CUE based on the optimal weighting matrix as well as the jackknife CUE outperforms QMLE in finite samples. An empirical application involving Australian Dollar and Japanese Yen spot returns is also included