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.

中文翻译：

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时变拱模型的两步估计

提出了一种时变自回归条件异方差 (ARCH) 模型来描述长期范围内金融回报序列的变化波动性，以及两步最小二乘法和最大似然估计程序。在对波动率尺度的时变趋势进行初步估计后，获得了潜在平稳 ARCH 序列的近似值，用于计算 ARCH 系数的最小二乘估计量 (LSE) 和最大似然估计量 (MLE)。在基本和温和的假设下，建立了 ARCH 系数的两步 LSE 的预言效率，即两步 LSE 与基于未观察到的 ARCH 级数的不可行 LSE 渐近有效。事实上，两步 LSE 与不可行 LSE 偏离了 opn-1/2。两步 MLE，然而，它并不享有这样的效率，但是对于两步 MLE 以及它与不可行 MLE 的偏差都建立了 n1/2 渐近正态性。模拟研究证实了渐近理论，将 1950 年至 2018 年的标准普尔 500 指数日收益率应用到波动率规模随着时间的推移发生显着变化。