Estimation and strict stationarity testing of ARCH processes based on weighted least squares

Abstract

This paper proposes a unified unconstrained two-stage weighted least squares estimate (2S-W LSE) theory for both stationary and nonstationary ARCH(1) processes. Without assuming strict stationarity, we show that the unconstrained 2S-W LSE of the conditional variance slope ARCH(1) parameter is consistent and asymptotically Gaussian and has the same asymptotic variance as its unconstrained quasi-maximum likelihood counterpart. Moreover, a consistent estimate of the asymptotic variance of the 2S-W LSE is provided irrespective of the stationarity requirement. As a result, strict stationarity testing of the ARCH process is considered. A numerical illustration on simulated and real data assesses the theory in finite samples.