Two‐Step Estimation for Time Varying Arch Models

A time varying autoregressive conditional heteroskedasticity (ARCH) model is proposed to describe the changing volatility of a financial return series over long time horizon, along with two‐step least squares and maximum likelihood estimation procedures. After preliminary estimation of the time varying trend in volatility scale, approximations to the latent stationary ARCH series are obtained, which are used to compute the least squares estimator (LSE) and maximum likelihood estimator (MLE) of the ARCH coefficients. Under elementary and mild assumptions, oracle efficiency of the two‐step LSE for ARCH coefficients is established, that is, the two‐step LSE is asymptotically as efficient as the infeasible LSE based on the unobserved ARCH series. As a matter of fact, the two‐step LSE deviates from the infeasible LSE by opn−1/2. The two‐step MLE, however, does not enjoy such efficiency, but n1/2 asymptotic normality is established for both the two‐step MLE as well as its deviation from the infeasible MLE. Simulation studies corroborate the asymptotic theory, and application to the S&P 500 index daily returns from 1950 to 2018 indicates significant change in volatility scale over time.