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The real life time series are usually nonstationary, bringing a difficultquestion of model adaptation. Classical approaches like ARMA-ARCH assumearbitrary type of dependence. To avoid such bias, we will focus on recentlyproposed agnostic philosophy of moving estimator: in time $t$ findingparameters 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 toestimate parameters using inexpensive exponential moving averages (EMA), likeabsolute 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 onStudent's t-distribution, popular especially in economical applications, hereapplied to log-returns of DJIA companies. While standard ARMA-ARCH approachesprovide 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.

Adaptive Student's t-distribution with method of moments moving estimator for nonstationary time series