Title: Adaptive Student's t-distribution with method of moments moving estimator for nonstationary time series Show Chinese title

Title: 具有动窗矩估计方法的自适应Student's t分布用于非平稳时间序列

Abstract: The real life time series are usually nonstationary, bringing a difficult question of model adaptation. Classical approaches like ARMA-ARCH assume arbitrary type of dependence. To avoid their bias, we will focus on recently proposed agnostic philosophy of moving estimator: in time $t$ finding parameters optimizing e.g. $F_t=\sum_{\tau<t} (1-\eta)^{t-\tau} \ln(\rho_\theta (x_\tau))$ moving log-likelihood, evolving in time. It allows for example to estimate parameters using inexpensive exponential moving averages (EMA), like absolute central moments $m_p=E[|x-\mu|^p]$ evolving for one or multiple powers $p\in\mathbb{R}^+$ using $m_{p,t+1} = m_{p,t} + \eta (|x_t-\mu_t|^p-m_{p,t})$. Application of such general adaptive methods of moments will be presented on Student's t-distribution, popular especially in economical applications, here applied to log-returns of DJIA companies. While standard ARMA-ARCH approaches provide evolution of $\mu$ and $\sigma$, here we also get evolution of $\nu$ describing $\rho(x)\sim |x|^{-\nu-1}$ tail shape, probability of extreme events - which might turn out catastrophic, destabilizing the market. Show Chinese abstract

Abstract: 现实生活中的时间序列通常是非平稳的，这带来了模型适应的难题。经典的ARMA-ARCH方法假设任意类型的依赖关系。为了避免它们的偏差，我们将关注最近提出的移动估计器的无偏哲学：在时间$t$处找到优化例如$F_t=\sum_{\tau < t} (1-\eta)^{t-\tau} \ln(\rho_\theta (x_\tau))$移动对数似然性的参数，该参数随时间演变。这允许使用廉价的指数移动平均（EMA），比如绝对中心矩$m_p=E[|x-\mu|^p]$，它随一个或多个幂$p\in\mathbb{R}^+$使用$m_{p,t+1} = m_{p,t} + \eta (|x_t-\mu_t|^p-m_{p,t})$演化。这种一般自适应矩方法的应用将在学生 t 分布上展示，尤其是在经济应用中特别受欢迎的学生 t 分布，这里应用于道琼斯工业平均指数（DJIA）公司的对数收益率。 虽然标准的ARMA-ARCH方法提供了$\mu$和$\sigma$的演变，但在这里我们还得到了描述$\rho(x)\sim |x|^{-\nu-1}$尾部形状的$\nu$的演变，以及极端事件的概率——这些事件可能会导致灾难，从而破坏市场稳定。