Asymptotic efficiency of conditional least squares estimators for ARCH models

The conditional least squares (CL) estimators proposed by Tjostheim [1986. Estimation in nonlinear time series models. Stochastic Process. Appl. 21, 251-273] are important and fundamental. The CL estimator applied to the square-transformed ARCH model has an explicit form, which does not depend on the distribution of the innovation. Since the CLs are not asymptotically efficient in general, we give a necessary and sufficient condition that CL is asymptotically efficient based on the LAN approach. Next, a measure of efficiency for CL is introduced. Numerical evaluations of the measure of efficiency for various nonlinear time series models are given. They elucidate some interesting features of CL.