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

The real life time series are usually nonstationary, bringing a difficult question of model adaptation. Classical approaches like GARCH assume arbitrary type of dependence. To prevent 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 with $m_{p,t+1} = m_{p,t} + \eta (|x_t-\mu_t|^p-m_{p,t})$ for one or multiple powers $p\in\mathbb{R}^+$. 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.

@article{duda2025_2304.03069,
  title={ Adaptive Student's t-distribution with method of moments moving estimator for nonstationary time series },
  author={ Jarek Duda },
  journal={arXiv preprint arXiv:2304.03069},
  year={ 2025 }
}