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 such 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 $E[|x-\mu|^p]$ 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.

翻译：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。诸如ARMA-ARCH之类的经典方法假设了任意类型的依赖关系。为避免这种偏差，我们将专注于近期提出的不可知论移动估计器哲学：在时间$t$找到优化例如$F_t=\sum_{\tau<t} (1-\eta)^{t-\tau} \ln(\rho_\theta (x_\tau))$移动对数似然的参数，这些参数随时间演变。这使得我们可以使用廉价指数移动平均（EMA）来估计参数，例如绝对值中心矩$E[|x-\mu|^p]$，它通过$m_{p,t+1} = m_{p,t} + \eta (|x_t-\mu_t|^p-m_{p,t})$针对一个或多个幂次$p\in\mathbb{R}^+$进行演变。这种通用自适应矩方法的应用将展示在学生t分布上，该分布在经济学应用中尤其流行，此处应用于道琼斯工业平均指数公司对数收益率。虽然标准ARMA-ARCH方法提供了$\mu$和$\sigma$的演变，但我们还得到了描述$\rho(x)\sim |x|^{-\nu-1}$尾部形状的$\nu$的演变——这是极端事件的概率，而极端事件可能具有灾难性，破坏市场稳定。