Multistage weighted least squares estimation of ARCH processes in the stable and unstable cases

Abstract

For an ARCH model, we propose a multistage weighted least squares (WLS) estimate which consists of repeated WLS procedures until the corresponding asymptotic variance equals that of the quasi-maximum likelihood estimate (QMLE). At every stage, the current estimate is of a WLS type weighted by the squared conditional variance evaluated at the estimate of the previous stage. Initially, the weighting parameter is any fixed and known value in the parameter space. The procedure provides, without any moment requirement, an asymptotically Gaussian estimate having the same asymptotic distribution as the QMLE even in the unstable case. Apart from the initialization stage, two additional stages are required in the stable case to obtain the same asymptotic distribution as the QMLE, while in the unstable case only one stage is enough. So in all, the proposed procedure involves three stages WLS in the stable case and two stages WLS in the unstable case.